This article has been selected for the prize of the journal  
''Rendiconti di Matematica e delle sue Applicazioni'' for the best PhD  
Thesis of the Academic Year 2008/2009
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\begin{document}

\begin{center}
{\large\textbf{Mean-field Limit and Semiclassical Approximation for  
Quantum Particle Systems }}
\end{center}
\vspace{0.1cm}
\begin{center}
F. Pezzotti\\{\footnotesize Departamento de Matem\'aticas, Universidad  
del Pa\'is Vasco, Spain - \emph{e-mail:} federica.pezzotti@ehu.es}
\end{center}








\tableofcontents













\section{Introduction}
\setcounter{equation}{0}
\def\theequation{1.\arabic{equation}}
In this thesis we discuss the connection between the mean-field limit  
and the semiclassical approximation for a system of $N$ identical  
quantum particles. More precisely, we look at a system of $N$  
identical particles (of mass $m=1$) interacting by means of the  
mean-field potential:
\begin{equation}\label{eq: meanINT}
U\left(X_N\right)=\frac{1}{2N}\sum_{l\neq j}^{N}\phi(x_l-x_j),\ \ \ \  
\phi:\R^3\rightarrow \R
\end{equation}
(where $X_N=\{x_1,\dots,x_N\},\ x_j\in\R^3,\ j=1,\dots, N $) in the  
limit $N\to\infty$. It is well known that the effective (limiting)  
dynamics of such a system is ruled by
the following nonlinear one-particle Schr$\ddot{\text{o}}$dinger equation:
\begin{equation}\label{eq: hartreeINTRO}
i\hbar\pa_t\psi_t=-\frac{\hbar^{2}}{2}\Delta\psi_t + \left(\phi \ast  
\vert\psi_t\vert^{2}\right)\psi_t,
\end{equation}
where
\begin{equation}\label{eq: hartreeINT}
\left(\phi \ast \vert\psi_t\vert^{2}\right)(x)=\int_{\R^3}\ud y\   
\phi(x-y)\vert \psi_t(y)\vert^2
\end{equation}
is the effective self-consistent interaction. Equation (\ref{eq:  
hartreeINTRO}) is known as the Hartree equation. The rigorous  
derivation of (\ref{eq: hartreeINTRO}) from the many-body evolution  
can be formulated in terms of convergence of $j$-particle Reduced  
Density Matrices (RDM).
In fact, by considering the $N$-particle wave function  
$\Psi_{N,t}=\Psi_{N,t}\left(X_N\right)$ solution of the  
Schr$\ddot{\text{o}}$dinger equation:
\begin{equation}\label{eq: mean}
i\hbar\pa_t\Psi_{N,t}=-\frac{\hbar^{2}}{2}\sum_{i=1}^{N}\Delta_{x_i}\Psi_{N,t}  
+ U\Psi_{N,t},
\end{equation}
with $U$ given by (\ref{eq: meanINT}) and completely factorized  
initial datum given by:
\begin{equation}\label{eq: meanINDAT}
\Psi_{N,0}(X_N)=\prod_{j=1}^{N}\psi_0(x_j),
\end{equation}
it can be proven that, for fixed $j$ (with $1\leq j\leq N$)
the $j$-particle reduced density matrix, defined as the trace class  
operator with kernel
\begin{equation}\label{eq: redDENSmatrN}
\rho_{N,t}^{(j)}\left(X_j,Y_j\right)=\int_{\R^{3(N-j)}} \ud X_{N-j}  
\overline{\Psi}_{N,t}\left(X_j, X_{N-j}\right)\Psi_{N,t}\left(Y_j,  
X_{N-j}\right),
\end{equation}
converges, in the limit $N\to\infty$, to the factorized state:
\begin{equation}\label{eq: redDENSmatrJlim}
\rho_t^{(j)}\left(X_j,Y_j;t\right)=\prod_{k=1}^{j}\overline{\psi}_t\left(x_k\right)\psi_t\left(y_k\right),
\end{equation}
where $\psi_t(x)$ solves the one-particle
Hartree equation (\ref{eq: hartreeINTRO}) with initial datum $\psi_0$.  
This feature is usually called ''propagation of chaos''.

The previous result was originally obtained for sufficiently smooth  
potentials (see \cite{HEPP}, \cite{GV}, \cite{SPOHN});
  then it has been
generalized to include Coulomb interactions (see \cite{BGM},  
\cite{EY}, \cite{BEGMY}). Furthermore, some results concerning the  
speed of convergence of the mean-field evolution to the Hartree  
dynamics (for all fixed times), have been proven more recently (see  
\cite{BENJIROD}, \cite{ErdBenji}).

The limit $N\to\infty$ for a classical system interacting by means of  
the same mean-field interaction (\ref{eq: meanINT}), can be considered  
as well (see
\cite{BH}, \cite{DOB},  \cite{Sp}, \cite{Neun} for the case of smooth  
potential, and \cite{JABIN} for more singular interactions).
In fact, considering as initial state of the system a completely  
factorized probability distribution  
$F_{N,0}=F_{N,0}\left(X_N,V_N\right)\ud X_N \ud V_N$ in the  
$N$-particle phase space $\R^{3N}\times \R^{3N}$, namely:
\begin{equation}\label{eq: classicINDAtN}
F_{N,0}\left(X_N,V_N\right)=
\prod_{j=1}^{N}f_0\left(x_j, v_j\right),\ \ \ \text{for some  
one-particle density $f_0$,}
\end{equation}
it is known that its evolution $F_{N}\left(X_N,V_N;t\right)$ at time  
$t>0$, is obtained by solving the Liouville equation:
\begin{equation}\label{eq: LiouvilleEQ}
\left(\pa_t +V_N\cdot \nabla_{X_N}\right)F_{N}(t)=\nabla_{X_N}U\cdot  
\nabla_{V_N}F_{N}(t),
\end{equation}
with $U$ given by (\ref{eq: meanINT}).
Then, the $j$-particle marginal at time $t>0$, defined as
\begin{equation}\label{eq: redCLASS}
F_{N}^{(j)}\left(X_j,V_j;t\right)=\int_{\R^{3(N-j)}\times \R^{3(N-j)}}  
\ud X_{N-j}\ud V_{N-j} F_N\left(X_j,  X_{N-j},V_j, V_{N-j};t\right),
\end{equation}
converges, as $N\to\infty$, to the product state:
\begin{equation}\label{eq: prodCLASS}
F^{(j)}\left(X_j,V_j;t\right)=\prod_{k=1}^{j}f\left(x_k, v_k;t\right),
\end{equation}
where $f(x,v ;t)$ is the solution of the Vlasov equation:
\begin{equation}\label{eq: VlasovINTRO}
\left(\pa_t+v\cdot \nabla_x\right)f(t)=\left(\nabla_{x}\phi\ast f(t)  
\right)\cdot\nabla_v f(t),
\end{equation}
(the convolution above is with respect to both the variables $x$ and  
$v$) with initial datum $f_0$.\\
Equation (\ref{eq: Vlasov}) is the classical analogue of the Hartree  
equation (\ref{eq: hartreeINTRO}).

Although the mean-field limit $N\to\infty$ is well understood for both  
classical and quantum systems,
there is a question which is
still open, namely, does that limit hold for quantum systems uniformly  
in $\hbar$, at least for systems having a reasonable classical analogue?
\\The proofs which are available up to now exhibit an error vanishing  
when $N\to\infty$ but diverging as $\hbar\to 0$, although in  
\cite{NS}, \cite{Graffi}, \cite{FGS}, \cite{FPK} some efforts in the  
direction of a better control of the error term have been done.

If one wants to deal with the classical and quantum case  
simultaneously, it is natural to work in the classical phase space by  
using the Wigner formalism.

The one-particle Wigner function associated with the wave function  
$\psi_t(x)$ is given by:
\begin{equation}\label{eq: WignerUNAINTRO}
f^{\hbar}\left(x, v;t\right)=(2\pi)^{-3}\int_{\R^{3}} \ud y\  
e^{iy\cdot v}\overline{\psi}_t\left(x+\frac{\hbar  
y}{2}\right)\psi_t\left(x-\frac{\hbar y}{2}\right),
\end{equation}
and, similarly, the $N$-particle Wigner function associated with the  
wave function $\Psi_{N,t}(X_N)$ is defined as:
\begin{equation}\label{eq: WignerN}
W_N^{\hbar}\left(X_N, V_N;t\right)=(2\pi)^{-3N}\int_{\R^{3N}} \ud Y_N  
\ e^{iY_N\cdot V_N}\overline{\Psi}_{N,t}\left(X_N+\frac{\hbar  
Y_N}{2}\right)\Psi_{N,t}\left(X_N-\frac{\hbar Y_N}{2}\right).
\end{equation}
Then, by using that $\psi_t(x)$ and $\Psi_{N,t}(X_N)$ solve equations  
(\ref{eq: hartreeINTRO}) and (\ref{eq: mean}) respectively, we find  
the equations:
\begin{equation}\label{eq: WigLIOUhartree1}
\left(\pa_t+v\cdot \nabla_x\right)f^{\hbar}(t)=T^{\hbar}f^{\hbar}(t)
\end{equation}
and
\begin{equation}\label{eq: WigLIOUhartreeN}
\left(\pa_t+V_N\cdot  
\nabla_{X_N}\right)W_N^{\hbar}(t)=T_N^{\hbar}W_N^{\hbar}(t),
\end{equation}
where $T^{\hbar}$ and $T_N^{\hbar}$ are suitable pseudodifferential operators.

The initial data for equations (\ref{eq: WigLIOUhartree1}) and  
(\ref{eq: WigLIOUhartreeN}) are
\begin{equation}\label{eq: WignerUNAindat}
f_0^{\hbar}\left(x, v\right)=(2\pi)^{-3}\int_{\R^{3}} \ud y\  
e^{iy\cdot v}\overline{\psi}_0\left(x+\frac{\hbar  
y}{2}\right)\psi_0\left(x-\frac{\hbar y}{2}\right),
\end{equation}
and
\begin{eqnarray}\label{eq: WignerNindat}
W_{N,0}^{\hbar}\left(X_N, V_N\right)&&=(2\pi)^{-3N}\int_{\R^{3N}} \ud  
Y_N \ e^{iY_N\cdot V_N}\overline{\Psi}_{N,0}\left(X_N+\frac{\hbar  
Y_N}{2}\right)\Psi_{N,0}\left(X_N-\frac{\hbar Y_N}{2}\right)\nonumber\\
&& = \prod_{j=1}^{N}f_0^{\hbar}\left(x_j, v_j\right),
\end{eqnarray}
respectively.

One can easily rephrase the result of \cite{SPOHN} by showing that
the $j$-particle Wigner function
\begin{equation}\label{eq: WignerJN}
W_{N,j}^{\hbar}\left(X_j,  
V_j;t\right)=\int_{\R^{3(N-j)}\times\R^{3(N-j)}} \ud X_{N-j} \ud  
V_{N-j}W_N^{\hbar}\left(X_j, X_{N-j},V_j,V_{N-j};t\right)
\end{equation}
converges, in a suitable sense, to
\begin{equation}\label{eq: WignerJlim}
f_j^{\hbar}\left(X_j, V_j;t\right)=\prod_{k=1}^{j}f^{\hbar}\left(x_k,  
v_k;t\right)\ \ \text{for any}\ t>0.
\end{equation}
However, the error in approximating the $N$-particle dynamics by the  
limiting one is diverging when $\hbar\to 0$ (for example, for  
sufficiently small times $t<t_0$ it is of the form  
$\frac{C_j}{N}e^{\frac{c}{\hbar}}$). The reason is that the operator
$T_N^{\hbar}$ appearing in
(\ref{eq: WigLIOUhartreeN}) is bounded on the space in which we can  
prove the convergence of (\ref{eq: WignerJN}), but its norm diverges  
as $\frac{c}{\hbar}$ when $\hbar\to 0$. On the other hand, the  
classical counterpart of this problem has been solved,
so that it seems natural to look for an asymptotic expansion for the  
$j$-particle ''marginals'' $W_{N,j}^{\hbar}$, namely:
\begin{equation}\label{eq: WignerJNexp}
W_{N,j}^{\hbar}\left(t\right)=W_{N,j}^{(0)}\left(t\right)+\hbar  
W_{N,j}^{(1)}\left(t\right)+\hbar^2 W_{N,j}^{(2)}\left(t\right)+\dots,
\end{equation}
and for an analogous expansion for the $j$-fold product of solutions  
of the equation (\ref{eq: WigLIOUhartree1}), namely:
\begin{equation}\label{eq: WignerJlimexp}
f_j^{\hbar}\left(t\right)=\left(f^\hbar\right)^{\otimes  
j}(t)=f_j^{(0)}\left(t\right)+\hbar f_j^{(1)}\left(t\right)+\hbar^2  
f_j^{(2)}\left(t\right)+\dots
\end{equation}
The zeroth order term in (\ref{eq: WignerJNexp}) is expected to  
correspond to what we previously denoted by $F_{N,t}^{(j)}$, namely,  
the $j$-particle marginals associated to the Liouville equation  
(\ref{eq: LiouvilleEQ}), while the function $f_j^{(0)}(t)$ appearing  
in (\ref{eq: WignerJlimexp}) is expected to be the $j$-fold product of  
solutions of the Vlasov equation (\ref{eq: VlasovINTRO})). Therefore,  
if at order zero in $\hbar$ we obtain the classical quantities, the  
classical mean-field theory ensures that the convergence of  
$W_{N,j}^{(0)}(t)$ to $f_j^{(0)}(t)$ is well established for all $t$  
and $j$. Then, it looks natural trying to show the convergence
\begin{equation}\label{eq: convergence}
W_{N,j}^{(k)}\left(t\right)\to f_j^{(k)}\left(t\right),\ \ \text{as  
$N\to\infty$, for any}\ k>0.
\end{equation}
This is the main goal of the present research.

A complete proof of the uniformity in $\hbar$ of the limit  
$N\to\infty$ would require a control of the remainder of the expansion  
(\ref{eq: WignerJNexp}), but we are not able to provide it. However,  
in proving (\ref{eq: convergence}) we characterize the quantum  
corrections to the classical mean-field limit and we prove that they  
are expressed in terms of classical quantities only.

The plan of the thesis is the following. \\

In Chapters 1 and 2 we discuss the mean-field model both in the  
classical framework and in the quantum context. First we introduce  
notation and technical tools that are needed to formulate the  
mean-field results to which we referred previously. Then, we give an  
outlook of the known results by discussing briefly the main approaches  
in facing the problem both for smooth and singular interactions. Thus,  
we focus on the case of sufficiently regular potential by showing
in detail the proof of the validity of ''propagation of chaos'' both  
in the classical and in the quantum case, accenting the main  
differences in the methods and, primarily, the inadequacy of the  
''BBGKY hierarchy method'' in facing the classical mean-field limit  
although in the quantum framework it plays a crucial role.  
Furthermore, we highlight the non uniformity with respect to $\hbar$  
of the error in the quantum mean-field approximation and we analyze in  
detail which is the estimate that, providing a bound which diverges as  
$\hbar\to 0$, is responsible for that.\\

In Chapter 3 we introduce the Wigner formalism. We accent first of all  
why it is appropriate in looking at the semiclassical behavior of  
quantum systems. Also, we point out the main difficulties of this  
formalism with respect to the wave function (Schr\"odinger) and the  
density matrix (Heisenberg) formulations introduced in Chapter 2.
Moreover, we rephrase the quantum mean-field result in the Wigner  
formalism and we note that the error in the mean-field approximation  
is still not uniform with respect to $\hbar$ and diverging when  
$\hbar\to 0$. This ''bad'' behavior is due to the failure of the same  
estimate we detected in Chapter 3, suitably rephrased in the Wigner  
framework. Finally, we discuss some known results concerning the  
connection between mean-field limit and semiclassical approximation.
\\

In Chapter 4 we prove our main result, namely, the convergence  
(\ref{eq: convergence}). More precisely, we do the semiclassical  
expansion both for the $N$-particle mean-field system (see (\ref{eq:  
WignerJNexp})) and for the Hartree dynamics (\ref{eq: WignerJlimexp})  
deriving explicitly what are the equations solved by the coefficients  
at each order in $\hbar$. Then, we introduce the initial datum
as a suitable mixtures of coherent states. This choice guarantees that  
the zeroth order coefficient of the $N$-particle expansion is a  
factorized probability distribution. Finally, we identify the higher  
order $N$-particle terms to be the expectation of certain derivatives  
of the empirical measure. Such an expectation is with respect to the  
probability distribution that we previously obtained from the  
$N$-particle zeroth order coefficient.
By virtue of that, we obtain the limit $N\to\infty$ by using the  
classical mean-field results presented in Chapter 1 and appropriate  
properties of the derivatives of the classical trajectories associated  
with the mean-field interaction.\\


In the last Chapter we present possible applications of our result  
(presented in Chapter 4) in considering suitable mixtures of WKB  
states (instead of coherent ones) and in dealing with other (related)  
problems.\\

I wish to thank my advisor, Prof. Mario Pulvirenti, for suggesting me  
a problem which I considered challenging from the very first moment,  
for offering me his precious experience and considering me not only as  
a student but indeed as a co-worker.


\section{Classical Mean-Field Limit}
\setcounter{equation}{0}
\def\theequation{2.\arabic{equation}}

In this chapter we analyze the mean-field limit for a many-body  
classical system. More precisely, we are looking at a system  
constituted by $N$ identical particles interacting by the potential
\begin{equation}\label{eq: class1}
U^{cl}(X_N)=\frac{1}{2N}\sum_{k\neq l}^{N}\phi(x_k-x_l),
\end{equation}
where we used the notation $X_N=(x_1,\dots, x_N)\in \R^{3N}$ for the  
positions of the $N$ particles
(here and henceforth we put the superscripts ''$cl$'' and ''$Q$'' to  
distinguish between classical and quantum quantities denoted by the  
same symbol).
We note that $U^{cl}$ is given by a sum over all interactions among  
pair of particles; the two-body interactions are governed by the  
potential $\phi$ which we assume to be spherically symmetric (as it is  
in all reasonable physical situations), namely $\phi(x)=\phi(\vert  
x\vert)\ \forall\ x\in \R^3$. We set the dimension to be equal to $3$  
but the results we are going to discuss hold in any dimension.  
Sometimes we will refer to the system under consideration as  
''mean-field system''.

We want to characterize the dynamics when the number of particles $N$  
is very big. In this sense we speak about ''macroscopic'' or  
''effective'' dynamics and from a mathematical point of view that  
purpose is realized by taking the limit $N\to\infty$.

As a second step, it is also important to describe the dynamics of the  
fluctuations of the $N$-particle evolution around the limiting one,  
but here we do not discuss in detail this topic which is analyzed, for  
example, in \cite{BH}. \\

\subsection{Setting of the problem: general features and known results}
\setcounter{equation}{0}
\def\theequation{2.1.\arabic{equation}}

A mean-field system is described by an $N$-body Hamiltonian of the form
\begin{equation}\label{eq: class2V}
H_N^{cl,V}(X_N,V_N)=\sum_{k=1}^{N}\(\frac{v_k^2}{2}+V^{cl}(x_k)\)+U^{cl}(X_N),
\end{equation}
where we used the notation $V_N=(v_1,\dots,v_N)\in \R^{3N}$ to  
indicate the velocities of the $N$ particles and, for the sake  
simplicity, the mass of the (identical) particles is chosen equal to  
one. The first part of the Hamiltonian is simply the sum of the  
one-body Hamiltonians associated with the motion of each particle (the  
function $V^{cl}$ describes
an external potential which acts in the same way on all $N$  
particles), while the remaining term involving $U^{cl}$ describes the  
interaction among the particles. For the sake of simplicity we assume  
that the force experienced by each particle is only that arising from  
the many-body interaction, namely, the one-particle potential $V^{cl}$  
is assumed to be equal to zero. We can do that without loss of  
generality because the results we are going to discuss can be  
generalized easily to the case $V^{cl}\neq 0$. \\Thus the Hamiltonian  
we consider is
\begin{equation}\label{eq: class2}
H_N^{cl}(X_N,V_N)=\sum_{k=1}^{N}\frac{v_k^2}{2}+U^{cl}(X_N),
\end{equation}
and we note that $H_N^{cl}$ is symmetric with respect to any  
permutation of the labeling.

The factor $1/N$ in the potential $U^{cl}$ (see (\ref{eq: class1}))  
forces the energy per particle to remain finite in the limit  
$N\to\infty$ and this is the crucial feature in order to obtain a  
well-defined but non-trivial limiting dynamics. Moreover we observe  
that $U^{cl}$ is such that the interaction among the particles is  
quite weak when $N$ is very big (the strength of the pair interaction  
is of the order $1/N$)
but it is long range (because the pair interaction potential $\phi$ is  
unscaled, thus its support remains of order one in the limit  
$N\to\infty$). Therefore when $N$ becomes large (for example, in the  
applications related to the gas dynamics we have $N\approx 10^{23}$)  
the mutual interaction turns to be weaker and weaker but the total  
effect of such an interaction is not negligible (the force experienced  
by a fixed particle because of the presence of all the others is  
proportional to $(N-1)/N\approx O(1)$) . We will see that these two  
features are responsible for the validity of ''propagation of chaos''  
and of the nonlinearity of the macroscopic equation we find in the  
limit (see Sections 1.3 and 1.4).\\

The dynamics of an $N$-particle system associated with the Hamiltonian  
(\ref{eq: class2}) is governed by the Newton equations
\begin{equation}\label{eq: class3}
\left\{
    \begin{aligned}
    & \dot{x}_i = v_i,\\
    & \dot{v}_i=  - \frac{1}{N}\sum_{k\neq i}^N\nabla_{x_i}  
\phi(x_i-x_k),\ \ i=1,\dots,N
    \end{aligned}
    \right.
  \end{equation}
Thus
we know that given an initial configuration $Z_N:=(X_N,V_N)\in  
\R^{3N}\times \R^{3N}$ in the $N$-particle phase-space, the  
time-evolved configuration $Z_N(t)$ up to some $t>0$ is obtained  
solving (\ref{eq: class3}) with initial datum $Z_N$. \\

As we have already noticed, in many interesting physical situations  
the number $N$ is very big thus it is not possible (and even not  
particularly relevant for the applications) to know which are the  
positions and the velocities of all particles at a certain time. In  
other words, it is quite difficult to determine a unique initial  
$N$-particle configuration for the time-evolution defined by (\ref{eq:  
class3}) and, even if one was able to provide that, it would be  
impossible to solve such a huge number of equations, even by using  
numerical methods.
Nevertheless, one can provide collective and more useful informations  
such as the probability to find $N_1$ particles ($N_1\leq N$) in a  
region $\Lambda_1\subset \R^{3N}$, the probability that $N_2$  
particles ($N_2\leq N$) have velocities belonging to some  
$\Lambda_2\subset \R^{3N}$ or the probability to find $N_3$ particles  
($N_3\leq N$) with positions belonging to some $\Lambda_3^x\subset  
\R^{3N}$ and velocities belonging to some $\Lambda_3^v\subset  
\R^{3N}$. In other words, one can give the $N$-particle probability  
distribution in the phase-space $\R^{3N}\times \R^{3N}$.
Therefore, denoting by $F_{N,0}(Z_N)\ud Z_N$ such a distribution, we  
have that $F_{N,0}$ is symmetric with respect to any permutation of  
the variables, $F_{N,0}\geq 0$ and:
\begin{equation}\label{eq: class3bis}
\int \ud Z_N F_{N,0}(Z_N)=1.
\end{equation}
Moreover, by computing the marginals of $F_{N,0}(Z_N)$ with respect to  
the velocities $V_N$ and to the positions $X_N$ one obtains  
respectively the spatial and the velocity probability density.\\

The time-evolved probability density $F_N(t):=F_{N}(Z_N;t)$ is  
obtained by solving the Liouville equation
\begin{equation}\label{eq: class4}
\(\pa_t +V_N\cdot\nabla_{X_N}\)F_{N}(t)=\nabla_{X_N}U\cdot\nabla_{V_N}F_N(t),
\end{equation}
with initial condition $F_{N,0}$, where $U$ is the potential defined  
in (\ref{eq: class1}). By denoting as $\Phi^t\(X_N, V_N\)$ the  
Hamiltonian flow associated with equations (\ref{eq: class3}), it is  
easy to verify that the solution of equation (\ref{eq: class4}) is  
obtained by propagating the initial datum $F_{N,0}$ through the  
characteristic curves of $\Phi^t\(X_N, V_N\)$, namely
\begin{equation}\label{eq: class5}
F_N(t)=F_{N,0}\left(\Phi^{-t}\left(X_N,V_N\right)\right).
\end{equation}
Thus we are guaranteed that starting from an $N$-particle probability  
density at time $t=0$, we have a probability density for each time  
$t>0$ and the evolution preserves also the symmetry with respect to  
permutations of the variables (because the Hamiltonian $H_N^{cl}$ is  
symmetric with respect to permutations).\\

In the classical framework observables of the $N$-particle system are  
represented by real functions defined on the phase-space  
$\R^{3N}\times \R^{3N}$. Then, if we know that the configuration of  
the system at a certain time $\tau$ is $Z_N(\tau)$, the value of the  
observable associated with a certain function $u_N$ at time $\tau$, is  
given by $u_N(Z_N(\tau))$. On the other side, if what we have is the  
$N$-particle probability distribution at a certain time $\tau_1$,  
namely $F_N(\tau_1)$, we are able to give probabilistic predictions  
about the value of the observables at time $\tau_1$. More precisely,  
the expectation of the observable associated with a certain function  
$u_N$ at time $\tau_1$, is given by
\begin{equation}\label{eq: class3tris}
\left\langle u_N\right\rangle_{F_{N}(\tau_1)}:=\int \ud Z_N  
u_N(Z_N)F_{N}(Z_N;\tau_1).
\end{equation}

By (\ref{eq: class3}) it is clear that to guarantee existence and  
uniqueness of the flow $\Phi^t\(X_N, V_N\)$ for each $t$ we need to  
assume $\phi\in \mathcal{C}_b^2\(\R^3\)$\footnote{Here and henceforth  
we denote by $\mathcal{C}_b^k\(\R^d\)$ the space of functions on  
$\R^d$ with continuous and uniformly bounded derivatives up to the  
order $k$}. Therefore the first rigorous results concerning the  
analysis of the limit $N\to\infty$ for the $N$-particles mean-field  
system have been proven under suitable smoothness assumption on the  
pair interaction potential $\phi$ (see for example \cite{BH} and  
\cite{Neun}).
Nevertheless, several systems of physical interest are described by  
more singular potential.
For example, a system of gravitating particles can be described by the  
potential (\ref{eq: class1}) where $\phi$ is the Coulomb interaction  
among the particles and,
in that case, the factor of $1/N$ in front of the potential energy can  
be justified
by the smallness of the gravitational
constant. \\Mean-field systems with singular interactions are clearly  
hard to face because one has to deal with a system of ODE (namely,  
(\ref{eq: class3})) with non regular fields. Quite recently some  
progress have been done in \cite{JABIN} where the mean-field limit is  
realized
by only assuming $\nabla_x \phi\approx 1/\vert x\vert^\alpha,\ \  
\alpha<1$ for the pair interaction $\phi$. On the other side, the  
assumptions on the initial datum are very strong and they are quite  
good for numerical purposes but not satisfying from a statistical  
physics point of view (for example, ''chaotic'' initial data are not  
admissible, namely it is not possible to consider initially factorized  
$N$-particle distribution). The problem involving the Coulomb  
potential is still open. \\

Here we will not discuss the ''singular case'' because for our  
purposes we need to deal with a smooth interaction potential and with  
a classical mean-field result involving ''chaotic'' initial data (see  
Chapter 3 and 4), thus from now on we will focus on the mean-field  
limit in the ''smooth case''.\\

In \cite{BH} and \cite{Neun} it is proved that the effective single  
particle dynamics of a mean-field system with smooth interaction  
potential ($\phi\in\mathcal{C}_b^2\(\R^3\)\  $) in the limit  
$N\to\infty$ is governed by the Vlasov equation:
\begin{equation}\label{eq: Vlasov}
\(\pa_t +v\cdot \nabla_x\)f(t)=\(\nabla_{x}\phi\ast f(t)\)\cdot\nabla_v f(t),
\end{equation}
where $f(t)=f(x,v;t)$ for each time $t\geq0$ is a one-particle  
probability density and here and in the rest of the chapter we denote  
by $\ast$ the convolution with respect to both position and velocity.  
By computing the marginals of $f(x,v;t)$ with respect to the velocity  
$v$ and the position $x$ one finds respectively the spatial and the  
velocity probability density.

It is remarkable that the results proven in \cite{BH} describe both  
the continuum limit of the point particle dynamics associated with the  
mean-field interaction, as we specified previously, and the so called  
''propagation of chaos'' for the many-body mean-field system.  
Moreover, in \cite{BH} it has been proven that the fluctuations of a  
certain class of observables (called ''intensive observables'')  
converge to a gaussian stochastic process. We will not discuss this  
last feature, on the contrary we will show in detail the emergence of  
the Vlasov dynamics as the limit of the $N$-particle evolution and the  
proof of propagation of chaos.

\subsection{The Vlasov equation}
\setcounter{equation}{0}
\def\theequation{2.2.\arabic{equation}}

Let us consider a one-particle density $f_0\in C^1(\R^6)$ and let us  
look at the solution $f(t)$ of the Vlasov equation (\ref{eq: Vlasov})  
with initial datum $f_0$.
Denoting by $\Phi^t_V(x,v)$ the flow associated with the system:
\begin{equation}\label{eq: NewtonVlasov}
\left\{
    \begin{aligned}
    & \dot{x} = v,\\
    & \dot{v}=  - \nabla_{x} \phi\ast f(t),
    \end{aligned}
    \right.
  \end{equation}
one can easily verify that $f(t)$ is obtained by propagating  
$f_0(x,v)$ through the characteristic curves of the flow  
$\Phi^t_V(x,v)$, namely
\begin{equation}\label{eq: class6}
f(t)=f(x,v;t)=f_0\left(\Phi^{-t}_V\left(x,v\right)\right).
\end{equation}
Therefore in proving existence and uniqueness of the solution of  
(\ref{eq: Vlasov}) one has to deal with a system of ODE with a  
self-consistent field (see (\ref{eq: NewtonVlasov})) and the  
smoothness of the potential $\phi$ is not sufficient to make a  
standard fixed point argument to be successful. One needs a more  
involved analysis and it has been done
by R.L. Dobrushin in \cite{DOB}. It is remarkable that the Vlasov  
equation (\ref{eq: Vlasov}) makes sense even for a generic probability  
measure $\nu$ because $\nabla\phi \ \ast\  \nu$ is sufficiently smooth
(thanks to the regularity of $\phi$) then the proof presented in  
\cite{DOB} ensures existence and uniqueness of the solution in this  
framework. In particular, if the initial datum is an absolutely  
continuous measure with respect to the Lebesgue measure in  
$\R^3\times\R^3$ with a smooth density $f_0$
(which is the case we discussed previously), the solution $f(t)$ is a  
strong solution whose regularity depends on that of $f_0$ and $\phi$ (  
$f_0\in C^1\(\R^6\)$ and $\phi\in C^2_b\(\R^3\)$, at least).
Furthermore, introducing the Wasserstein distance $\mathcal{W}$,
in \cite{DOB} it has been proven the following stability result for  
solutions of the Vlasov equation:
\begin{eqnarray}\label{eq: DOBineq}
&&\mathcal{W}(\nu^t_1,\nu^t_2)\leq e^{Ct}\mathcal{W}(\nu^0_1,\nu^0_2)
\end{eqnarray}
where $\nu^0_1$ and $\nu^0_2$ are two probability measures and  
$\nu^t_1$ and $\nu^t_2$ are the weak solutions of the Vlasov equation  
with initial data given by $\nu^0_1$ and $\nu^0_2$ respectively. The  
metric induced by $\mathcal{W}$ on the space of probability measures  
on $\R^3\times\R^3$ is equivalent to the weak topology of probability  
measures, namely we have to look at measures tested versus functions  
in $\mathcal{C}_b^{0}(\R^3\times\R^3)$ (the space of continuous and  
uniformly bounded functions). Thus by (\ref{eq: DOBineq}) it follows  
that
\begin{eqnarray}\label{eq: class7}
&&\int u(x,v)\nu^0_n(\ud x\ud v)\to\int u(x,v)\nu^0(\ud x\ud v)\ \  
\text{as}\ n\to\infty,\ \forall\ u\in \mathcal{C}_b^{0}(\R^3\times\R^3)
\end{eqnarray}
implies
\begin{eqnarray}\label{eq: class8}
&&\int u(x,v)\nu^t_n(\ud x\ud v)\to\int u(x,v)\nu^t(\ud x\ud v)\ \  
\text{as}\ n\to\infty,\ \forall\ u\in \mathcal{C}_b^{0}(\R^3\times\R^3),
\end{eqnarray}
where $\{\nu^0_n\}_{n\geq 0}$ is a sequence of probability measures  
converging to some $\nu^0$ when the parameter $n$ goes to infinity and  
$\nu^t_n$ and $\nu^t$ are the weak solutions of the Vlasov equation  
with initial data given by $\nu^0_n$ and $\nu^0$ respectively. In the  
sequel we will denote the weak convergence of probability measures by  
the symbol $\stackrel{M}{\rightarrow}$.

\subsection{The Vlasov dynamics as the continuum limit of the  
$N$-particle Mean-Field dynamics}
\setcounter{equation}{0}
\def\theequation{2.3.\arabic{equation}}

Let us introduce the empirical measure associated with an $N$-particle  
configuration $Z'_N$
\begin{eqnarray}\label{eq: class9}
&&\mu_N(z\vert Z'_N)=\frac{1}{N}\sum_{i=1}^{N}\delta\left(z-z'_i\right),
\end{eqnarray}
where $z:=(x,v)$ is the generic point in the one-particle phase-space  
$\R^3\times\R^3$ and $Z'_N=(z'_1,\dots, z'_N)\in \R^{3N}\times\R^{3N}$.
By definition $\mu_N$ is a measure on the one-particle phase-space  
but, as it is clear by (\ref{eq: class9}), it depends on all the  
configuration $Z'_N$. Then let us consider an initial $N$-particle  
configuration $Z_N$ for equations (\ref{eq: class3}) distributed  
according to a factorized (smooth) $N$-particle measure $F_{N,0} \ud  
Z_N$, namely
\begin{eqnarray}\label{eq: class10}
&&F_{N,0}(Z_N)=\prod_{i=1}^N f_0(z_i)=f_0^{\otimes N},\ \ \ \ \ \  
f_0\in C^1(\R^6).
\end{eqnarray}
We denote by $\mu_N^0$ the empirical measure associated with $Z_N$ and  
by considering the empirical measure $\mu_N(t)=\mu_N(z\vert Z_N(t))$  
associated with the time-evolved configuration $Z_N(t)$ (solution of  
equations (\ref{eq: class3}) with initial datum $Z_N$), it is easy to  
verify that $\mu_N(t)$ is the unique (weak) solution of the Vlasov  
equation (\ref{eq: Vlasov}) with initial datum $\mu_N^0$. In fact, by  
integrating versus $\mu_N(t)$ versus a smooth test function  
$u=u(x,v)$, we find:
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAK}
&&\(u,\mu_N(t)\)=\int\ud z\ \mu_N(z\vert  
Z_N(t))=\frac{1}{N}\sum_{i=1}^N u(z_i(t))=\frac{1}{N}\sum_{i=1}^N  
u(x_i(t),v_i(t)).
\end{eqnarray}
Then, we obtain
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKI}
\frac{d}{dt}\(u,\mu_N(t)\)&&=\frac{1}{N}\sum_{i=1}^N  
\frac{d}{dt}u(x_i(t),v_i(t))=\nonumber\\
&&=\frac{1}{N}\sum_{i=1}^N \[\nabla_x u(x_i(t),v_i(t))\dot{x}_i(t)+  
\nabla_v u(x_i(t),v_i(t))\dot{v}_i(t)\],\nonumber\\
&&
\end{eqnarray}
that, by virtue of (\ref{eq: class3}), implies
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKII}
\frac{d}{dt}\(u,\mu_N(t)\)=\frac{1}{N}\sum_{i=1}^N \nabla_x  
u(x_i(t),v_i(t))v_i(t)- \frac{1}{N}\sum_{i=1}^N \nabla_v  
u(x_i(t),v_i(t))\(\frac{1}{N}\sum_{k\neq i}^{N}\nabla_{x_i}\phi(x_i  
-x_k)\).\nonumber\\
&&
\end{eqnarray}
Following (\ref{eq: muSOLVESvlasovWEAK}), the equation (\ref{eq:  
muSOLVESvlasovWEAKII}) becomes
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKIV}
\frac{d}{dt}\(u,\mu_N(t)\)=\(v\cdot\nabla_x u,\mu_N(t)\)-  
\(\(\nabla\phi\ast \mu_N(t)\) \cdot\nabla_v u,\mu_N(t)\),
\end{eqnarray}
where
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKV}
&&\(\nabla\phi\ast \mu_N(t)\)(x)=\int \ud y \ud  
w\nabla_x\phi(x-y)\(\frac{1}{N}\sum_{k=1}^N  
\delta(y-x_k(t))\delta(w-v_k(t))\)=\nonumber\\
&&=\frac{1}{N}\sum_{k=1}^N \int \ud y \nabla_x\phi(x-y)  
\delta(y-x_k(t))=\frac{1}{N}\sum_{k=1}^N\nabla_{x}\phi(x-x_k(t)).
\end{eqnarray}
Therefore, $\mu_N(t)$ verifies (\ref{eq: muSOLVESvlasovWEAKIV}) for  
any function $u$ sufficiently smooth and
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKVI}
&&\(u,\mu_N(t)\)\vert_{t=0}=\(u,\mu_N^0\)=\frac{1}{N}\sum_{i=1}^N u(x_i, v_i).
\end{eqnarray}
In other words, $\mu_N(t)$ satisfies the following weak equation
\begin{eqnarray}\label{eq: muSOLVESvlasovWEAKVII}
\pa_t \mu_N(t) + v\cdot\nabla_x\mu_N(t)= \(\nabla_{x}\phi\ast  
\mu_N(t)\) \cdot\nabla_v \mu_N(t),
\end{eqnarray}
with initial datum $\mu_N^0$, and we note that (\ref{eq:  
muSOLVESvlasovWEAKVII}) is precisely the Vlasov equation (\ref{eq:  
Vlasov}).\\

By the Strong Law of Large Numbers (SLLN) we know that
\begin{eqnarray}\label{eq: class11}
&&\mu^0_N\stackrel{M}{\rightarrow}f_0\ \ \text{as}\ N\to\infty,\  
\text{a.e with respect to the product measure }\ f_0^{\otimes \infty},
\end{eqnarray}
therefore, by (\ref{eq: class11}), by knowing that $\mu_N(t)$ solves  
the Vlasov (weak) equation (\ref{eq: muSOLVESvlasovWEAKVII})
and by (\ref{eq: DOBineq}), it follows that
  \begin{eqnarray}\label{eq: class12}
&&\mu_N(t)\stackrel{M}{\rightarrow}f(t)\ \ \text{as}\ N\to\infty,\  
\text{a.e with respect to the product measure }\ f_0^{\otimes \infty},
\end{eqnarray}
where $f(t)$ is the (strong) solution of the Vlasov equation (\ref{eq:  
Vlasov}) with initial datum $f_0$. \\

 From now on, we will say that a configuration $Z_N$ is ''typical''  
with respect to the measure $f_0$ if the empirical measure $\mu_N^0$  
associated with $Z_N$ verifies
\begin{eqnarray}\label{eq: class11NUOVA}
&&\mu^0_N\stackrel{M}{\rightarrow}f_0\ \ \text{as}\ N\to\infty.
\end{eqnarray}

\subsection{Hierarchies and Propagation of Chaos}
\setcounter{equation}{0}
\def\theequation{2.4.\arabic{equation}}

In the previous paragraph we proved that the Vlasov equation arises  
from the continuum limit of a system of $N$ particles interacting by  
the mean-field potential (\ref{eq: class1}). This is precisely what  
convergence (\ref{eq: class12}) tells us and it can be seen as a  
one-particle effect, namely (\ref{eq: class12}) provides the equation  
governing the single-particle dynamics in the limit.\\

Now we want to show how (\ref{eq: class12}) works in order to  
characterize the effective dynamics of a subsystem
made by a fixed number $j$ of particles.
This is a natural approach in looking at the macroscopic behavior of  
many-body systems because we want to look at the limit $N\to\infty$  
and we need to deal with quantities depending on a number of variables  
which remains finite in the limit. \\
In this perspective, for any $j=1,\dots,N$ we introduce the  
''$j$-particle marginal density'' (or simply ''$j$-particle marginal''  
) associated with an $N$-particle density $F_N(X_N,V_N)$ as
\begin{eqnarray}\label{eq: class13}
&&F_N^{(j)}(X_j,V_j)=\int_{\R^{3(N-j)}\times\R^{3(N-j)}}\ud X_{N-j}\ud  
V_{N-j}F_N(X_N,V_N),
\end{eqnarray}
where we used the notation $X_{j}=(x_{1},\dots, x_j),  
V_{j}=(v_{1},\dots, v_j) \in \R^{3j}$ and $X_{N-j}=(x_{j+1},\dots,  
x_N),$ $V_{N-j}=(v_{j+1},\dots, v_N)\in \R^{3(N-j)}$. Indeed, the  
marginal $F_N^{(j)}$ is obtained by integrating $F_N$ with respect to  
the ''last'' $N-j$ variables thus it is a $j$-particle probability  
density (we remind that all quantities under consideration are  
symmetric with respect to permutations of the variables then, without  
loss of generality, in order to refer to any subsystem made by $N-j$  
particles we can consider the last $N-j$). Clearly if $j=N$ we have  
$F_N^{(N)}=F_N$.

For fixed $j < N$, the $j$-particle
marginal does not contain the full information about the $N$-particle  
configuration described by $F_N$. Knowledge
of the $j$-particle marginal $F_N^{(j)}$, however, is sufficient to  
compute the expected value of every $j$-particle
observable in the configuration described by the probability  
distribution $F_N\ud Z_N$. In fact, if $u_j$ denotes an arbitrary
continuous and uniformly bounded function on $\R^{3j}$, and if  
$u_j\otimes 1_{N-j}$ denotes the function on $\R^{3N}$ which is  
associated with the $N$-particle observable corresponding to $u_j$ for  
the first $j$ particles and to $1_{N-j}$ for the last $(N-j)$ particles,
we have
\begin{eqnarray}\label{eq: class24}
&&\left\langle u_j\otimes 1_{N-j}\right\rangle_{F_N}=\int \ud Z_N  
u_j(Z_j)F_N(Z_N)=\int \ud Z_j u_j(Z_j)F_N^{(j)}(Z_j)=\left\langle  
u_j\right\rangle_{F_N^{(j)}},\nonumber\\
&&
\end{eqnarray}
where we denoted by $\left\langle u_j\otimes  
1_{N-j}\right\rangle_{F_N}$ the expected value of the $N$-particle  
observable corresponding to $ u_j\otimes 1_{N-j}$ with respect to the  
distribution $F_N\ud Z_N$ and with $\left\langle  
u_j\right\rangle_{F_N^{(j)}}$ the expected value of the $j$-particle  
observable corresponding to $u_j$ with respect to $F_N^{(j)}\ud Z_j$.
Thus, $F_N^{(j)}$ is sufficient to compute the expectation of  
arbitrary observables which depend non-trivially
on at most $j$ particles (because of the permutation symmetry, it is  
not important on which particles
it acts, just that it acts at most on $j$ particles).\\

We are interested in characterizing the time-evolution of the  
marginals $F_{N}^{(j)}(t):=F_{N}^{(j)}(Z_j;t)$ associated with the  
solution $F_N(t)$ of the Liouville equation (\ref{eq: class4}).
By integrating the Liouville equation versus the variables  
$Z_{N-j}=(X_{N-j},V_{N-j})$ we find
the following family of equations (one for each $j=1,\dots,N$)
\begin{eqnarray}\label{eq: class14}
&&\(\pa_t+V_j\cdot  
\nabla_{X_j}\)F_{N}^{(j)}(t)=T_{N,j}^{cl}F_{N}^{(j)}(t)+\frac{N-j}{N}C_{j,j+1}^{cl}F_{N}^{(j+1)}(t),
\end{eqnarray}
where $T_{N,j}^{cl}$ is precisely the $j$-particle Liouville operator,  
namely $T_{N,j}^{cl}=\nabla_{X_j}U^{cl}(X_j)\cdot \nabla_{V_j}$, while  
the operator $C_{j,j+1}^{cl}$ maps $j+1$-particle densities in  
$j$-particle ones (if $j=N$ we find $C_{N,N+1}^{cl}\equiv 0$).
The family of equations (\ref{eq: class14}) is known as BBGKY  
hierarchy (in honor of the authors who independently derived it: Born,  
Bogoliubov, Green, Kirkwood, Yvon)
and it is called ''hierarchy'' because we can see that the equation  
for the $j$-particle marginal is linked to the subsequent one by the  
term $C_{j,j+1}^{cl}F_{N}^{(j+1)}(t)$. The physical meaning is clear:  
the variation in time of $F_{N}^{(j+1)}(t)$ is due to the free motion  
of the $j$ particles, which is encoded in the free-transport term  
$V_j\cdot \nabla_{X_j}F_{N}^{(j)}(t)$, to the interaction among  
themselves, which is modeled by the term $T_{N,j}^{cl}F_{N}^{(j)}(t)$,  
and to the interaction among the $j$-particle subsystem and the  
remaining $N-j$ particles, which is encoded in the term  
$\frac{N-j}{N}C_{j,j+1}^{cl}F_{N}^{(j+1)}(t)$ (the factor $1/N$ is  
precisely the factor appearing in the potential $U^{cl}$ (see  
(\ref{eq: class1})) while the interaction with the last $N-j$  
particles can be modeled by $N-j$ times the interaction with the  
$j+1$-th because of the symmetry with respect to permutations of the  
labeling (which follows from the fact that we are dealing with $N$  
identical particles).

Writing explicitly the action of the operators $T_{N,j}^{cl}$ and  
$C_{j,j+1}^{cl}$, we find:
\begin{eqnarray}\label{eq: class15}
&&\left(T_{N,j}^{cl}F_{N}^{(j)}\right)(X_j, V_j)=\frac{1}{N}\sum_{  
k\neq l}^{ j}\nabla_{x_k}\phi(x_k-x_l)\cdot  
\nabla_{v_k}F_{N}^{(j)}(X_j,V_{j}),
\end{eqnarray}
and
\begin{eqnarray}\label{eq: class16}
&&\left(C_{j,j+1}^{cl}F_{N}^{(j+1)}\right)(X_j, V_j)=\nonumber\\
&&=\sum_{k=1}^{j}\int_{\R^3\times\R^3}\ud x_{j+1}\ud  
v_{j+1}\nabla_{x_k}\phi(x_k-x_{j+1})\cdot   
\nabla_{v_k}F_{N}^{(j+1)}(X_j,x_{j+1},V_j,v_{j+1}).
\end{eqnarray}
By these expressions we can argue that the operator $T_{N,j}^{cl}$  
gives a vanishing contribution in the limit because it is of size  
$j^2/N$, while the operator $C_{j,j+1}^{cl}$ is of order one in the  
limit and the factor $(N-j)/N$ appearing in (\ref{eq: class14}) is  
also of order one. Therefore denoting by $F^{(j)}(t)$ the expected  
limit of $F_N^{(j)}(t)$  when $N\to\infty$, the formal limit of the  
BBGKY hierarchy (\ref{eq: class14}) is
\begin{eqnarray}\label{eq: class17}
&&\(\pa_t+V_j\cdot \nabla_{X_j}\)F^{(j)}(t)=C_{j,j+1}^{cl}F^{(j+1)}(t),
\end{eqnarray}
which in the case $j=1$ is equal to:
\begin{eqnarray}\label{eq: class18}
&&\(\pa_t+v_1\cdot \nabla_{x_1}\)F^{(1)}(t)=\int\ud x_{2}\ud  
v_{2}\nabla_{x_1}\phi(x_1-x_{2})\cdot   
\nabla_{v_1}F^{(2)}(x_1,x_{2},v_1,v_{2};t).
\end{eqnarray}
We observe that the Vlasov equation (\ref{eq: Vlasov}) can be rewritten as
\begin{eqnarray}\label{eq: class19}
&&\(\pa_t+v\cdot \nabla_{x}\)f^t=\int\ud x_{2}\ud  
v_{2}\nabla_{x}\phi(x-x_{2})\cdot  \nabla_{v}f^t(x,v)f^t(x_{2},v_{2}).
\end{eqnarray}
Replacing $(x,v)$ by $(x_1,v_1)$, $f^t$ by $F^{(1)}(t)$ and the  
product $f^t\ f^t$ by $F^{(2)}$ we realize that (\ref{eq: class19}) is  
precisely the same of (\ref{eq: class18}). Thus the equation of the  
hierarchy (\ref{eq: class17}) corresponding to $j=1$ is properly the  
Vlasov equation, provided that the the two-particle distribution  
$F^{(2)}(t)$ is factorized, and for this reason (\ref{eq: class17})  
(which is an infinite hierarchy because $j$ can be equal to any  
positive number
) is usually called ''Vlasov hierarchy''. More precisely, by  
considering (\ref{eq: class17})
and by assuming the marginals $\{F^{(j)}(t)\}_{j\geq 1}$ to be  
factorized, namely
\begin{eqnarray}\label{eq: class20}
&&F^{(j)}(t)=f(t)^{\otimes j}\ \ \forall\ j,
\end{eqnarray}
it is easy to verify that $f(t)$ has to solve the Vlasov equation.  
Conversely, if we consider a time dependent one-particle density  
$f(t)$ solving the Vlasov equation and we take the $j$-particle  
densities $F^{(j)}(t)=f(t)^{\otimes j}$, for $j=1,2\dots$, we find  
that the sequence $\{F^{(j)}(t)\}_{j\geq 1}$ solves the hierarchy  
(\ref{eq: class17}). \\

An interesting problem is that of the uniqueness of the solution of  
the Vlasov hierarchy which plays an important role in facing the mean  
field limit when a generic (namely, non factorized) initial datum is  
considered for the many-body dynamics (such a case is also studied in  
\cite{BH}). This topic  has been discussed in \cite{NS}, under strong  
smoothness assumptions on the interaction potential, and in \cite{Sp}  
by assuming $\phi\in \mathcal{C}_b^2(\R^3)$.
Here we will not enter into details because we are going to show that  
in the present context there is no need to prove the uniqueness of the  
solution of the Vlasov hierarchy in order to establish the validity of  
propagation of chaos. (In the next chapter we will see that the  
situation in the quantum case can be very different).\\

First of all let us explain what we mean by ''propagation of chaos''.

As we have already specified, we consider as initial datum for the  
Liouville equation (\ref{eq: class4}) the factorized $N$-particle  
probability density (\ref{eq: class10}). This choice means that we are  
assuming that the particles are identically and independently  
distributed at time $t=0$, or equivalently, the particles are  
initially uncorrelated. This is quite reasonable from the physical  
point of view and this is what is usually called ''hypotheses of  
molecular chaos''.
Because of the interaction among the particles, the factorization  
(\ref{eq: class10}) is not preserved by the
time evolution because some correlations are introduced by the  
dynamics; in other words, the evolved $N$-particle density $F_N(t)$ is  
not given by the product of one-particle
densities, if $t\neq 0$. However, due to the mean-field character of  
the interaction each
particle interacts very weakly (we remind that the strength of the  
interaction is of the order $1/N$) with all other
$(N - 1)$ particles.
For this reason, we may expect that, in
the limit of large $N$, the total interaction force experienced by a  
typical particle in the system
can be effectively replaced by an averaged, mean-field, force, and  
therefore that factorization is
approximately, and in an appropriate sense, preserved by the time  
evolution. In other words, we
may expect that, in a sense to be made precise,
\begin{eqnarray}\label{eq: class21}
&&F_N(t)\approx f(t)^{\otimes N} \ \ \text{as}\ N\to\infty
\end{eqnarray}
for an evolved one-particle density $f(t)=f(x,v;t)$. This asymptotic  
factorization is precisely what is called ''propagation of chaos''
Assuming (\ref{eq: class21}), it is simple to derive a self-consistent  
equation
for the time-evolution of the one-particle density $f(t)$. In fact,  
(\ref{eq: class21}) states that, for every fixed time $t$,
the $N$ particles are independently distributed in space according to  
the density $\rho(x;t)=\int \ud v f(x,v;t)$. If this is true, the total
force experienced, for example, by the first particle can be approximated by
\begin{eqnarray}\label{eq: class22}
&&\frac{1}{N}\sum_{k\geq 2}
\nabla_{x_1}
\phi(x_1 - x_k)\approx\frac{1}{N}\sum_{k\geq 2}\int \ud y\nabla_{x_1}  
\phi(x_1 - y)\rho(y;t)=\nonumber\\
&&=\frac{N-1}{N}\int \ud y\ud w\nabla_{x_1} \phi(x_1 -  
y)f(y,w;t)=\frac{N-1}{N}\(\nabla_{x_1}\phi\ast  
f(t)\)\approx\(\nabla_{x_1}\phi\ast f(t)\),\nonumber\\
&&
\end{eqnarray}
as $N\to\infty$. It follows that, if (\ref{eq: class21}) holds true,  
the one-particle density $f(t)$ must satisfy the self-consistent
equation
\begin{eqnarray}\label{eq: class23}
&&\(\pa_t +v\cdot \nabla_x\)f(t)=\(\nabla_{x}\phi\ast f(t)\)\cdot  
\nabla_v f(t)
\end{eqnarray}
with initial data $f(t)\vert_{t=0}=f_0$ given by (\ref{eq: class10}).  
Equation (\ref{eq: class23}) is precisely the Vlasov equation
and we have just presented an heuristic argument to explain
how it is related to the propagation of chaos. We observe that the  
Vlasov equation is a nonlinear Liouville equation on $\R^3\times  
\R^3$. Therefore starting from the linear Liouville equation
(\ref{eq: class4}) on $\R^{3N}\times \R^{3N}$, we obtain, for the  
evolution of factorized densities, a nonlinear Liouville equation on  
$\R^3\times \R^3$; the nonlinearity in the Vlasov equation is a  
consequence of the many-body effects
in the linear dynamics. \\

The validity of propagation of chaos (namely, the precise statement  
concerning the asymptotic factorization (\ref{eq: class21})) is  
expressed in terms of convergence of the $j$-particle marginal  
densities associated with the solution of the Liouville equation  
(\ref{eq: class4}) to the $j$-fold product of solutions of the Vlasov  
equation when $N\to\infty$. We are going to show that it is a  
straightforward consequence of the convergence (\ref{eq: class12})  
(e.g. \cite{BH}). \\

Let us consider the $j$-particle marginal $F_N^{(j)}(t)$ associated  
with the solution $F_N(t)$ of the Liouville equation with factorized  
initial datum $F_{N,0}$ given by (\ref{eq: class10}).

We want to look at the behavior of $F_N^{(j)}(t)$ when $N\to\infty$.  
Denoting by $\mathbb{E}_N$ the expectation with respect to the initial  
$N$-particle distribution $F_{N,0}(Z_N)$, after straightforward  
computations, we obtain:
\begin{eqnarray}\label{eq: EprodEMP}
\mathbb{E}_N\left[\mu_N\(z'_1\vert Z_N(t)\)\dots\mu_N\(z'_j\vert  
Z_N(t)\)\right]&&=\frac{N(N-1)\dots(N-j+1)}{N^j}\  
F_N^{(j)}(Z'_j;t)+\nonumber\\
&&+\ O\left(\frac{1}{N}\right),\nonumber\\
&&
\end{eqnarray}
where
$F_N^{(j)}(Z'_j;t)=F_N^{(j)}(\Phi^{-t}(Z'_j)=F_N^{(j)}(Z'_j(-t))$ (see  
(\ref{eq: class5})).
Consider now a typical sequence $Z_N$ with respect to $f_0$, namely  
such that (\ref{eq: class11NUOVA}) holds. By the Strong Law of Large  
Numbers (\ref{eq: class11}) we know that (\ref{eq: class11NUOVA})  
holds a.e. with respect to the product measure $f_0^{\otimes \infty}$.  
Then,
by (\ref{eq: class12}) and (\ref{eq: EprodEMP}) we have:
\begin{eqnarray}\label{eq: prodEMP}
&&\lim_{N\to\infty}\mathbb{E}_N\left[\mu_N\(z'_1\vert Z_N(t)\)\dots  
\mu_N\(z'_j\vert Z_N(t)\)\right]=\lim_{N\to\infty}  
F_N^{(j)}(Z'_j;t)=\prod_{k=1}^{j}f(z'_k;t),\nonumber\\
&&
\end{eqnarray}
in the weak topology of probability measures, where $f(z'_k;t)=f(t)$  
solves the Vlasov equation with initial datum $f_0$. Thus propagation  
of chaos holds. In the end, we found that starting from an initial  
uncorrelated state (\ref{eq: class10}) for the $N$-particle system,  
for times $t>0$ we loose the factorization, but it is recovered in the  
limit because the correlations created by the dynamics are smaller and  
smaller when $N\to\infty$. On the other side, the effect of the  
many-body interaction is ''translated'' into the self-consistent force  
appearing in the Vlasov equation.\\

The convergence (\ref{eq: prodEMP}) of $F_N^{(j)}(t)$ to  
$f(t)^{\otimes j}$ implies that:
\begin{eqnarray}\label{eq: convOBSj}
&&\left\langle u_j\otimes 1_{N-j}\right\rangle_{ F_N(t)}=\left\langle  
u_j\right\rangle_{ F_N^{(j)}(t)}\rightarrow \left\langle  
u_j\right\rangle_{f(t)^{\otimes j}}\ \ \text{as}\ N\to\infty,\nonumber\\
&&
\end{eqnarray}
for each $u_j\in \mathcal{C}^{0}_{b}\(\R^{3j}\times\R^{3j}\)$. In  
other words, we are able to compute the ''macroscopic'' expected value  
of $j$-particle observables.\\

A remarkable fact is that the validity of propagation of chaos has  
been proven without using the hierarchies and this is really a big  
advantage because to deal with the hierarchies (\ref{eq: class14}),  
(\ref{eq: class17}) seems to be quite difficult. A priori, one could  
think to prove the convergence (\ref{eq: convOBSj}) of the marginals  
$F_N^{(j)}(t)$ to the products $f(t)^{\otimes j}$, by using that the  
first ones solve the BBGKY hierarchy (\ref{eq: class14}) and the  
second ones solve the Vlasov hierarchy (\ref{eq: class17}). Thus, if  
one would be able to prove the convergence of solutions of the  
$N$-dependent hierarchy to the Vlasov one, by knowing that the  
limiting hierarchy has factorized solutions arising from the Vlasov  
equation (as we previously discussed), the final step for proving the  
propagation of chaos would be to show the uniqueness of the solution  
of the Vlasov hierarchy over the class in which one is able to prove  
convergence. As regard to the ''convergence problem'' the difficulty  
is that the BBGKY hierarchy involves operators which are unbounded, at  
least in reasonable spaces, thus it does not seem possible to apply  
any compactness argument to ensure the convergence of the solution. On  
the other side, concerning the ''uniqueness problem'' for the limiting  
hierarchy (\ref{eq: class17}), the crucial point is the connection  
between the space in which one could show convergence and those in  
which it would be possible to prove uniqueness. Therefore, the problem  
of realizing the classical mean-field limit by dealing with the  
hierarchies is quite hard. On the other side, we have just seen that  
it can be faced more naturally by using two crucial tools: the Law of  
Large Numbers (\ref{eq: class11}) and the continuity of solutions of  
the Vlasov equations with respect to the weak convergence of measures  
(\ref{eq: DOBineq}).\\

In the next chapter we will see that in the quantum case to deal with  
hierarchies is not so difficult and a possible approach to realize the  
limit (indeed the one that has been used more in the last years) is  
properly the one we have just described (convergence + uniqueness),  
particularly to deal with singular pair interaction potentials.
\\

\section{Quantum Mean-Field Limit}
\setcounter{equation}{0}
\def\theequation{3.\arabic{equation}}

This chapter is devoted to the analysis of the macroscopic properties  
of the dynamics of a quantum system constituted by $N$ identical  
particles interacting by a mean-field potential in the limit  
$N\to\infty$. As in the previous chapter, we set the dimension of the  
system equal to $3$ but the main results we are going to discuss hold  
in any dimension.
\\
The mean-field interaction potential is represented by the (right hand  
side) multiplication operator
\begin{equation}\label{eq: quant1}
U^{Q}(X_N)=\frac{1}{2N}\sum_{k\neq l}^N\phi(x_k-x_l),
\end{equation}
where, as in the previous chapter, we denote by  
$X_N=\(x_1,\dots,x_N\)\in \R^{3N}$ the positions of the $N$ particles  
and we assume $\phi$ to be spherically symmetric. We want to  
characterize the effective dynamics of such a system for large $N$.

The problem of investigating the error in the approximation of the  
many-body evolution with the limiting macroscopic dynamics, which we  
do not discuss here, has been studied in \cite{BENJIROD} and  
\cite{ErdBenji}.

\subsection{Setting of the problem: general features and known results}
\setcounter{equation}{0}
\def\theequation{3.1.\arabic{equation}}

The state of an $N$-particle quantum mechanical system in $\R^3$
can be described by a complex valued wave function $\Psi_N\in L^2\(\R^{3N}\)$.
Physically the absolute value squared of $\Psi_N(x_1,\dots,x_N)$ is  
interpreted as the
probability density for finding particle one at $x_1$, particle two at  
$x_2$, and so on. Moreover the absolute value squared of the Fourier  
transform $\hat{\Psi}_N(v_1,\dots,v_N)$ is interpreted as the
probability density for having particle one with velocity $v_1$,  
particle two with velocity $v_2$, and so on (for the sake simplicity  
we always consider identical particles with mass $m=1$ thus velocities  
are always equal to momenta). Because of this
probabilistic interpretation, we will always consider wave functions  
$\Psi_N$ with $L^2$-norm equal to one.\\

In nature there exist two different types of particles; bosons and  
fermions. Bosonic systems are
described by wave functions which are symmetric with respect to  
permutations, in the sense that
\begin{equation}\label{eq: quant3}
\Psi_N(x_{\pi(1)},\dots,x_{\pi(N)})=\Psi_N(x_1,\dots,x_N),
\end{equation}
for every permutation $\pi$ acting on $1,\dots,N$. Fermionic systems,  
on the other hand, are described by antisymmetric
wave functions satisfying
\begin{equation}\label{eq: quant4}
\Psi_N(x_{\pi(1)},\dots,x_{\pi(N)})=(-1)^{\sigma(\pi)}\Psi_N(x_1,\dots,x_N),
\end{equation}
for every permutation $\pi$ acting on $1,\dots,N$
where $\sigma(\pi)=0$ if $\pi$ is even (in the sense that it can be written
as the composition of an even number of transpositions) and  
$\sigma(\pi)=1$ if it is odd. In the sequel we will denote by  
$L_s^2\(\R^{3N}\)$ the space of bosonic wave functions (namely the  
subspace of $L^2\(\R^{3N}\)$ consisting of all functions satisfying  
(\ref{eq: quant3})).

Equations (\ref{eq: quant3}) and (\ref{eq: quant4}) are responsible  
for substantial differences between an $N$-particle bosonic system and  
a fermionic one. Actually these features determine a different way to  
look at the limit $N\to\infty$ in the mean-field context, the use of  
different techniques leading to (a bit) different effective dynamics  
(see paragraph ''Joint limit $N\to \infty$ and $\var\to 0$'' in  
Section 3.4). Furthermore,
the different nature of bosons and fermions is crucial in the  
perspective of looking at the connection between mean-field limit and  
semiclassical approximation (
as we will observe in Section 3.4) and, at least from this point of  
view, bosonic systems seem to be more difficult to treat.

Anyway, here and henceforth we consider undistinguishable quantum  
particles by neglecting the statistics. In particular, in some cases  
the states we consider are indeed admissible for bosons.\\

We know that the observables of an $N$-particle system are represented  
by self adjoint operators $A$ on $L^2\(\R^{3N}\)$, then the expectation
\begin{equation}\label{eq: EXP}
\left\langle A\right\rangle_{\Psi_N}=<\Psi_N,A\Psi_N>=\int  
\bar{\Psi}_N(X_N)\(A\Psi_N\)(X_N)\ud X_N
\end{equation}
gives the value of the observable represented by $A$ in the state  
described by $\Psi_N$. \\

The Hamiltonian of an $N$-particle system interacting by (\ref{eq:  
quant1}), assuming the mass of the particles to be equal to one, is  
the standard quantization of (\ref{eq: class2V}), namely
\begin{equation}\label{eq: quant2V}
H_N^{Q,V}=-\sum_{k=1}^{N}\(\frac{\var^2 \Delta_k}{2}+V^Q(x_k)\)+U^{Q}(X_N),
\end{equation}
where we denoted by $\Delta_k$ the Laplace operator acting on the  
variable $x_k$, $k=1,2,\dots,N$ and here and henceforth we denote the  
Planck constant by $\var$. The potentials $V^Q$ and $\phi$ (appearing  
in (\ref{eq: quant1}) are such that the Hamiltonian $H_N^{Q,V}$ is  
guaranteed to be a
self-adjoint operator acting on the Hilbert space $L^2\(\R^{3N}\)$ and  
it is invariant with respect to any permutation of the labeling  
(namely, the Hamiltonian is symmetric in the exchange of particle  
names).
The first part of $H_N^{Q,V}$ is a sum of one-body operators  
(operators acting on one particle
only); the sum of the Laplacians is the kinetic part of the  
Hamiltonian. The function $V^Q$ describes
an external potential which acts in the same way on all $N$ particles.
The second part of the Hamiltonian describes the interaction among the  
particles.

As in the classical case, we can assume without loss of generality  
that the potential experienced by each particle is only that arising  
from the many-body interaction, namely, the one-particle potential  
$V^Q$ is assumed to be equal to zero. Thus the Hamiltonian we consider  
is
\begin{equation}\label{eq: quant2}
H_N^{Q}=-\sum_{k=1}^{N}\frac{\var^2\Delta_k}{2}+U^{Q}(X_N).
\end{equation}
The Hamiltonian (\ref{eq: quant2}) is the observable associated with  
the energy of the
$N$-particle system interacting by the mean-field potential (\ref{eq:  
quant1}), thus the expectation
\begin{eqnarray}\label{eq: quant7}
&&\left\langle H_N^Q\right\rangle_{\Psi_N}=<\Psi_N,H_N^Q\Psi_N>=\int  
\ud X_N \bar{\Psi}_N(X_N)\(H_N^Q\Psi_N\)(X_N)
\end{eqnarray}
gives the energy of the system in the state described by the wave  
function $\Psi_N$.\\

The considerations we did in the previous chapter as regard to the  
scaling of the potential hold also in the quantum context. Therefore  
we are guaranteed that the energy per particle is of order one for  
large $N$, as it is crucial in looking for a non-trivial and  
well-defined limiting dynamics, and we realize that the basic features  
of the model are that the mutual interaction among the particles is  
weak (again of size $1/N$) and of long range type (unscaled support of  
$\phi$). Again, as a consequence of such two effects we will have
propagation of chaos and nonlinearity of the equation governing the  
limiting one-particle dynamics respectively (see Section 2.2).\\

The time evolution of a wave function $\Psi_{N}\in L^2\(\R^{3N}\)$  
associated with the $N$-particle system whose Hamiltonian is (\ref{eq:  
quant2}) is governed by the linear Schr\"odinger
equation
\begin{equation}\label{eq: quant5}
i\var\pa_t\Psi_{N,t}=H_N^Q\Psi_{N,t},
\end{equation}
and, since $H_N^Q$ is a self-adjoint operator, the time-evolution  
associated with the equation (\ref{eq: quant5}) preserves the  
$L^2$-norm of the wave function.

The solution to
(\ref{eq: quant5}), with initial condition $\Psi_{N,t}\vert_{t=0} =  
\Psi_{N,0}\in L^2\(\R^{3N}\)$, can be written by means of the unitary  
group generated
by $H_N^Q$ as
\begin{equation}\label{eq: quant6}
\Psi_{N,t}=e^{-i\frac{t}{\var}H_N^Q}\Psi_{N,0}\ \ \text{for all}\ t\in \R.
\end{equation}
The global well-posedness of (\ref{eq: quant5}) is not an issue here.  
The study of (\ref{eq: quant5}) is focused, therefore, on
other questions concerning the qualitative and quantitative behavior  
of the solution $\Psi_{N,t}$. Despite
the linearity of the equation, these questions are usually quite hard  
to answer, because in physically
interesting situation the number of particles $N$ is very large; for  
example, in applications related to the study of boson stars we have  
$N \approx 10^{30}$. For such huge values of $N$,
it is of course impossible to compute the solution (\ref{eq: quant5})  
explicitly; numerical methods are completely
useless as well (unless the interaction among the particles is switched off).

Fortunately, also from the point of view of physics, it is not so  
important to know the precise solution
to (\ref{eq: quant5}); it is much more important, for physicists  
performing experiments, to have information
about the macroscopic properties of the system, which describe the  
typical behavior of the particles,
and result from averaging over a large number of particles.  
Restricting the attention to macroscopic
quantities simplifies the study of the solution $\Psi_{N,t}$, but it  
still does not make it accessible to mathematical
analysis. To further simplify matters, we are going to let the number  
of particles $N$ tend
to infinity. The macroscopic properties of the system, computed in the  
limiting regime $N\to\infty$, are
then expected to be a good approximation for the macroscopic  
properties observed in experiments,
where the number of particles $N$ is very large, but finite (explicit
bounds on the difference between the limiting behavior as $N\to\infty$  
and the behavior for large but
finite $N$ are obtained in \cite{BENJIROD} and \cite{ErdBenji}).

\subsubsection{The density matrix formalism}
To consider the limit of large $N$, we are going to make use of the  
Reduced (or Marginal) Density
Matrices (RDM) associated with an $N$-particle wave function  
$\Psi_N\in L^2\(\R^{3N}\)$. First of all, we define the
density matrix $\hat{\rho}_N=\vert \Psi_N><\Psi_N\vert$ associated  
with $\Psi_N$ as the orthogonal projection onto $\Psi_N$; we use
here and henceforth the notation $\vert \psi><\psi\vert$ to indicate  
the orthogonal projection onto $\psi$ (Dirac bracket
notation).
Therefore $\hat{\rho}_N$ is a non-negative integral operator acting  
from $L^2\(\R^{3N}\)$ to $L^2\(\R^{3N}\)$ with kernel given by
\begin{equation}\label{eq: BenjiPRELdensMATRIX1}
\rho_N(X_N;Y_N)=\bar{\Psi}_N(X_N)\Psi_N(Y_N),
\end{equation}
where $Y_N=(y_1,\dots,y_N)\in \R^{3N}$. Note that, by virtue of the $L^2$-
normalization of $\Psi_N$, we have
\begin{equation}\label{eq: BenjiPRELdensMATRIX1bis}
\texttt{Tr}\hat{\rho}_N=\int \ud X_N \rho_N(X_N;X_N)=\int \ud X_N  
\bar{\Psi}_N(X_N)\Psi_N(X_N)=\left\|\Psi_N\right\|_{L^2\(\R^{3N}\)}^2=1.
\end{equation}
Thus $\hat{\rho}_N\in\mathscr{L}^1\(L^2\(\R^{3N}\)\)$, where  
$\mathscr{L}^1\(L^2\(\R^{3N}\)\)$ is the Banach space (with respect to  
the norm  
$\left\|\cdot\right\|_{\mathscr{L}^1\(L^2\(\R^{3N}\)\)}=\texttt{Tr}\left\vert  
\cdot \right\vert$) of the trace class operators acting on  
$L^2\(\R^{3N}\)$. Moreover,
the positivity of $\hat{\rho}_N$ implies $\left\|\hat{\rho}_N  
\right\|_{\mathscr{L}^1\(L^2\(\R^{3N}\)\)} =\texttt{Tr}\hat{\rho}_N  
=1$.\\

It turns out that the state of a quantum mechanical system can be  
equivalently represented in the wave function (Schr\"odinger) picture  
and in the density matrix (Heisenberg) formalism and the expectation  
$\left\langle A\right\rangle_{\Psi_N}=<\Psi_N,A\Psi_N>$ of an  
observable $A$ in the state described by $\Psi_N$, expressed through  
the density matrix $\hat{\rho}_N$,
can be written as \texttt{Tr}$ A\hat{\rho}_N$. For example, the energy  
of the mean-field system in the state described by $\hat{\rho}_N$ is
\begin{eqnarray}\label{eq: quant7conRHO}
&&\left\langle  
H_N^Q\right\rangle_{\Psi_N}=<\Psi_N,H_N^Q\Psi_N>=\texttt{Tr}  
H_N^Q\hat{\rho}_N,\nonumber\\
&&
\end{eqnarray}
$H_N^Q$ defined in (\ref{eq: quant2}).\\

The time evolution of a density matrix
describing the state of the $N$-particle mean-field system
is governed by the linear equation
\begin{equation}\label{eq: quant8}
i\var\pa_t\hat{\rho}_{N,t}=\[H_N^Q,\hat{\rho}_{N,t}\],
\end{equation}
where
$\[H_N^Q,\hat{\rho}_{N,t}\]$ denotes the commutator between $H_N^Q$  
and $\hat{\rho}_{N,t}$, namely  
$\[H_N^Q,\hat{\rho}_{N,t}\]=H_N^Q\hat{\rho}_{N,t}-\hat{\rho}_{N,t}H_N^Q$.  
Equation (\ref{eq: quant8}) is usually called Heisenberg equation and,  
by knowing that $\hat{\rho}_{N,t}=\vert\Psi_{N,t}><\Psi_{N,t}\vert $,  
it can be derived easily by the Schr\"odinger equation (\ref{eq:  
quant5}) solved by $\Psi_{N,t}$. The self-adjointness of the  
Hamiltonian $H_N^Q$, responsible for conservation of the $L^2$-norm of  
the wave function, implies that positivity and trace of the density  
matrix are also preserved in time. \\

We remind that we are looking at systems constituted by  
undistinguishable particles. Then we consider density matrices  
$\hat{\rho}_N$ such that their kernel $\rho_N(x_1, . . . , x_N; y_1 .  
. . , y_N)$ is symmetric in the exchange of particle names, namely
\begin{equation}\label{eq: quant3NEW}
\rho_N(x_{\pi(1)},\dots,x_{\pi(N)};y_{\pi(1)},\dots,y_{\pi(N)})=\rho_N(x_1,\dots,x_N;y_1,\dots,y_N),
\end{equation}
for every permutation $\pi$ acting on $1,\dots,N$. By the definition  
of the time-evolution (\ref{eq: quant8}) it is easy to verify that  
this property is preserved in time. \\

The solution to
(\ref{eq: quant8}), with initial condition  
$\hat{\rho}_{N,t}\vert_{t=0} =\hat{\rho}_{N,0}$,
can be written by means of the unitary group generated
by $H_N^Q$ as
\begin{equation}\label{eq: quant9}
\hat{\rho}_{N,t}=
e^{-i\frac{t}{\var}H_N^Q}\hat{\rho}_{N,0}e^{i\frac{t}{\var}H_N^Q}\ \  
\text{for all}\ t\in \R.
\end{equation}

The main advantage in describing the state and the dynamics of an  
$N$-particle system by using the density matrix formalism is that it  
gives the possibility to investigate the properties of subsystems made  
by a fixed number of variables.
The way to do that is to introduce the Reduced Density Matrices (RDM).  
For $j = 1,\dots,N$, we define the $j$-particle marginal
density $\hat{\rho}_N^{(j)}$ associated with $\hat{\rho}_N$ as the  
partial trace of $\hat{\rho}_N$ over the degrees of freedom of the last
$(N - j)$ particles:
\begin{equation}\label{eq: BenjiPRELdensMATRIX2}
\hat{\rho}_N^{(j)}=\texttt{Tr}_{j+1}\hat{\rho}_N
\end{equation}
where $\texttt{Tr}_{j+1}$ denotes the partial trace over the particles  
$j + 1, j + 2, \dots ,N$. In other words, $\hat{\rho}_N^{(j)}$ is  
defined as the non-negative trace class operator on $L^2\(\R^{3j}\)$  
with kernel given by
\begin{equation}\label{eq: BenjiPRELdensMATRIX3}
\rho_N^{(j)}(X_j;Y_j)=\int \ud X_{N-j}\rho_N(X_j,X_{N-j};Y_j, X_{N-j}).
\end{equation}
The last equation can be considered as the definition of partial  
trace. As in the previous chapter, we used the notation
$X_j = (x_1,\dots, x_j), Y_j= (y_1,\dots,y_j)\in \R^{3j}$ and $X_{N-j}  
= (x_{j+1},\dots, x_N) \in \R^{3(N-j)}$. By definition,  
$\texttt{Tr}\hat{\rho}_N^{(j)} = 1$ for all $N$ and for all $j = 1,  
\dots,N$ (clearly, if $j=N$ we find $\hat{\rho}_N^{(N)}=\hat{\rho}_N$)  
thus $\hat{\rho}_N^{(j)}\in \mathscr{L}^1\(L^2\(\R^{3j}\)\)$ for all  
$N$ and for all $j$.

\begin{rem}\label{REMARKsuNORMALIZZAZIONE}
\textnormal{Note that, in the physics literature, one normally
uses a different normalization for the reduced density matrices. If  
the statistics are taken into account, the reduced density matrices  
are defined as expectation of bosonic and fermionic fields in the  
framework of the ''second quantization formalism''.  }
\end{rem}












For fixed $j < N$, the $j$-particle
density matrix does not contain the full information about the state  
described by $\hat{\rho}_N$. Knowledge
of the $j$-particle marginal $\hat{\rho}_N^{(j)}$, however, is  
sufficient to compute the expectation of every $j$-particle
observable in the state described by the density matrix  
$\hat{\rho}_N$. In fact, if $A^{(j)}$ denotes an arbitrary
bounded operator on $L^2\(\R^{3j}\)$, and if $A^{(j)} \otimes  
1^{(N-j)}$ denotes the operator on $L^2(\R^{3N})$ which acts as
$A^{(j)}$ on the first $j$ particles, and as the identity on the last  
$(N -j)$ particles, we have
\begin{equation}\label{eq: BenjiPRELdensMATRIX4}
\texttt{Tr}\(A^{(j)} \otimes  
1^{(N-j)}\)\hat{\rho}_N=\texttt{Tr}A^{(j)}\hat{\rho}_N^{(j)}.
\end{equation}
Thus, $\hat{\rho}_N^{(j)}$ is sufficient to compute the expectation of  
arbitrary observables which depend non-trivially
on at most $j$ particles (because of the permutation symmetry, it is  
not important on which particles
it acts, just that it acts at most on $j$ particles).\\

Marginal densities play an important role in the analysis of the  
$N\to\infty$ limit because, in contrast
to the wave function $\Psi_N$ and to the density matrix  
$\hat{\rho}_N$, the $j$-particle marginal $\hat{\rho}_N^{(j)}$ can  
have, for
every fixed $j\in\mathbb{N}$, a well-defined limit as $N\to\infty$  
(because, if we fix $j\in \mathbb{N}$, $\{\hat{\rho}_N^{(j)}\}_N$  
defines a sequence
of operators all acting on the same space $L^2(\R^{3j})$). In other  
words, $\hat{\rho}_N^{(j)}$ is a function of a fixed number of  
variables (which remains finite in the limit $N\to\infty$), while  
$\Psi_N$ and $\hat{\rho}_N$ are functions of $N$ variables thus in the  
limit we would have to deal with functions of an infinite number of  
variables and clearly it prevents the possibility to find a  
well-defined limit for them.

\subsubsection{Mixed states}
In the previous analysis we have always considered systems whose state  
is described by a density matrix  
$\hat{\rho}\in\mathscr{L}^1\(L^2\(\R^{3d}\)\)$ which is the orthogonal  
projection onto a wave function $\Psi\in L^2\(\R^{3d}\) $ with  
$\texttt{Tr}\hat{\rho}=\left\|\Psi\right\|_{L^2\(\R^{3d}\)}=1$ (we had  
$d=N$). Such kind of states are called ''pure'' states. Indeed, we say  
that a system is in a pure state whenever we know that it is described  
by a uniquely determined wave function with probability equal to one.  
As a consequence, the density matrix describing a ''pure'' state is a  
rank-one projection on $L^2\(\R^{3d}\)$, namely  
$\left\vert\Psi><\Psi\right\vert$. Nevertheless, in some cases it is  
not possible to know precisely (namely, with probability equal to one)  
which is the wave function describing the state of a system but one  
only has probabilistic predictions about that.
For example, one can have a certain number (possibly infinite) $k$ of  
known wave functions $\Psi^{1},\dots,\Psi^{k}\in L^2\(\R^{3d}\)$, with  
$\left\|\Psi^{1}\right\|_{L^2\(\R^{3d}\)}=\dots=  
\left\|\Psi^{k}\right\|_{L^2\(\R^{3d}\)}=1$, and a sequence of  
non-negative numbers $\lambda_1,\dots, \lambda_k$ such that it is  
known that the state can be described by $\Psi^{1}$ with probability  
equal to $\lambda_1$, by $\Psi^{2}$ with probability equal to  
$\lambda_2$ and so on..., where $\lambda_s\leq 1$ for $s=1,\dots,k$  
and $\sum_s \lambda_s=1$. These kind of states are called ''mixed''  
states (or equivalently, one can say that the state ''associated  
with'' the sequence $\Psi^{1},\dots,\Psi^{k}$ is a ''mixture'' of the  
pure states $\Psi^{1},\dots,\Psi^{k}$). \\
It turns out that one of the advantages of the density matrix  
formalism is that it encodes both the case of pure states and the case  
of mixtures. In fact, denoting by $\hat{\rho}_{s}$ the orthogonal  
projection onto $\Psi^s$, for $s=1,\dots,k$, the density matrix  
$\hat{\rho}_{mix}$ associated with the system under consideration is  
given by
\begin{equation}\label{eq: mixture}
\hat{\rho}_{mix}=\sum_{s=1}^{k}\lambda_s  
\hat{\rho}_{s}=\sum_{s=1}^{k}\lambda_s  
\left\vert\Psi^{s}><\Psi^{s}\right\vert.
\end{equation}
Then, since $\left\|\Psi^{s}\right\|_{L^2\(\R^{3d}\)}=1$ for any $s$, we have
\begin{equation}\label{eq: CORREZIONE1}
\texttt{Tr}\hat{\rho}_{mix}=\sum_{s=1}^{k}\lambda_s  
\texttt{Tr}\hat{\rho}_{s}=\sum_{s=1}^{k}\lambda_s  
\left\|\Psi^{s}\right\|_{L^2\(\R^{3d}\)}= \sum_{s=1}^{k}\lambda_s=1,
\end{equation}
namely, $\hat{\rho}_{mix}\in \mathscr{L}^1\(L^2\(\R^{3d}\)\)$ and  
$\left\|\hat{\rho}_{mix}\right\|_{\mathscr{L}^1\(L^2\(\R^{3d}\)\)}=1$.  
\\
Clearly, a mixed state reduces to a pure state if  
$\lambda_{\bar{s}}=1$ for some $\bar{s}$ and $\lambda_{s}=0$ for  
$s\neq \bar{s}$.

By virtue of (\ref{eq: CORREZIONE1}), it turns out that the analysis  
done previously by starting from a pure state can be generalized  
straightforward to the case of mixtures.

Furthermore, there are also states that are made by a ''continuum''  
mixture of pure states. Indeed, let us take the parameter $s$ in  
(\ref{eq: mixture}) as a continuum variable varying in a certain set  
$\Lambda\subset \R^n$, for some $n$ (for example, in the initial state  
considered in Section 4.5 we have $s=(x_0,v_0)\in \Lambda=\R^6$), and  
let us consider a function $g=g(s)$ such that $g\ud s$ is a  
probability distribution on $\Lambda$. If we have a family of  
$L^2$-normalized wave functions $\left\{\Psi^s\right\}_{s\in \Lambda}$  
on $\R^d$ (for example, in Section 4.5 we considered the family of  
coherent states ''centered'' in $(x_0, v_0)$), we can costruct a mixed  
state which is the ''continuum'' mixture of the pure states  
$\left\{\Psi^s\right\}_{s\in \Lambda}$ through the probability  
distribution $g\ud s$. It turns out that the kernel $\rho_{mix}$ of  
the density matrix $\hat{\rho}_{mix}$ describing the mixed state under  
consideration is
\begin{equation}\label{eq: mixtureDENScontinue}
\rho_{mix}(X;Y)=\int_\Lambda \ud s\  g(s)\rho^s(X;Y)=\int_\Lambda \ud  
s\  g(s)\overline{\Psi}^{s}(X) \Psi^{s}(Y),
\end{equation}
where $X\in \R^d,\ Y\in \R^d$. Clearly, all considerations we did for  
''discrete'' mixtures hold also in that case.

\subsubsection{The limit $N\to\infty$}
We will discuss several known results about the study of the limiting  
dynamics when $N\to\infty$ for a mean-field
system and we will see that what has been established, by using  
different techniques and various formalisms, is that the effective  
single-particle dynamics is governed by a cubic nonlinear  
Schr\"odinger equation
\begin{equation}\label{eq: Hartree}
i\var\pa_t \psi_t=-\frac{\var^2}{2}\Delta \psi_t+\(\phi\ast \vert  
\psi_t\vert^2\)\psi_t
\end{equation}
which is known as Hartree equation. Clearly, in that case the symbol  
''$\ast$'' denotes the convolution with respect to the spatial  
variable, namely
\begin{equation}\label{eq: HartreePOT}
\(\phi\ast \vert \psi_t\vert^2\)(x)=\int \ud y\ \phi(x-y)\vert  
\psi_t(y)\vert^2.
\end{equation}

The first rigorous results establishing a relation between the many  
body Schr\"odinger evolution
and the nonlinear Hartree dynamics were obtained by K. Hepp in  
\cite{HEPP}(for smooth interaction potentials)
and then generalized by J. Ginibre and G. Velo to singular potentials  
in \cite{GV}. These works were inspired
by techniques used in quantum field theory. We will not discuss this  
method because we want to focus on other techniques which are more  
related to the topic we are going to face in the next chapters (the  
connection between mean-field limit and semiclassical approximation).

The first proof of the emergence of the Hartree dynamics by using the  
RDM formalism (''RDM-convergence'') was obtained by H. Spohn in  
\cite{SPOHN}, for bounded potentials (see Theorem \ref{teoSPOHN} in  
Section 2.3).
The method introduced by Spohn was then extended to singular  
potentials: in \cite{EY}, L. Erd{\H{o}}s and H. T. Yau faced the RDM-  
convergence for a Coulomb potential $\phi (x) = �1/\vert x\vert$;  
partial results for this kind of interaction were
also obtained by C. Bardos, F. Golse and N. Mauser in \cite{BGM} (note  
that recently a new proof in the
case of a Coulomb interaction has been proposed by J. Fr\"ohlich, A.  
Knowles, and S. Schwarz in \cite{AnntiNEW}).


A different approach to the proof of the rigorous derivation of the  
Hartree equation from a mean-field bosonic system has been proposed by  
Fr\"ohlich, Schwarz and
Graffi in \cite{FGS}. By using the Wigner formalism (see Chapter 3)
they can consider the mean-field limit uniformly in the Planck
constant $\var$ (up to an exponential error depending on time); this  
allows them to combine the semiclassical
limit and the mean field limit by assuming restrictive assumptions on  
the pair interaction potential (we will come back on this result in  
Chapter 3). It is also interesting to remark that the mean-field limit  
can be
interpreted as a Egorov-type theorem; this was observed
in \cite{FPK} for sufficiently smooth potentials and in  
\cite{AnntiNEW} for the Coulomb interaction.


\subsection{Quantum BBGKY hierarchy and its formal limit as $N\to\infty$}
\setcounter{equation}{0}
\def\theequation{3.2.\arabic{equation}}


We have already remarked that, for any $j=1,\dots,N$, the marginal  
densities $\hat{\rho}_{N,t}^{(j)}$ associated with the solution  
$\hat{\rho}_{N,t}$ of the equation (\ref{eq: quant8}), are crucial  
tools in studying the mean-field limit because they can have, for
every fixed $j$, a well-defined limit as $N\to\infty$. Thus, we are  
interested in their time-evolution as $N\to\infty$.

By taking the partial trace over the degrees of freedom of the last  
$N-j$ particles in the Heisenberg equation (\ref{eq: quant8}) we find  
the following family of equations (one for each $j=1,\dots,N$)
\begin{eqnarray}\label{eq: quant10}
&&i\var \pa_t  
\hat{\rho}_{N,t}^{(j)}=\sum_{k=1}^{j}\[-\frac{\var^2}{2}\Delta_k,\hat{\rho}_{N,t}^{(j)}\] +  
T_{N,j}^{Q}\hat{\rho}_{N,t}^{(j)}+\frac{N-j}{N}C_{j,j+1}^Q\hat{\rho}_{N,t}^{(j)},
\end{eqnarray}
where the operator $T_{N,j}^{Q}$ acts on $\mathscr{L}^1\(L^2_s\(\R^{3j}\)\)$
while the operator $C_{j,j+1}^Q$ maps $j+1$-particle densities in  
$j$-particle ones (if $j=N$ we find $C_{N,N+1}^Q\equiv 0$).
The family of equations (\ref{eq: quant10}) is called BBGKY hierarchy  
in analogy to the classical case and, again, it is called  
''hierarchy'' because we can see that the equation for the  
$j$-particle marginal density is linked to the subsequent one by the  
term $C_{j,j+1}^Q \hat{\rho}_{N,t}^{(j)}$. The physical meaning is the  
same we discussed in the classical case: the variation in time of  
$\hat{\rho}_{N,t}^{(j)}$ is due to the free motion of the $j$  
particles, to their interaction among themselves and to the  
interaction among the $j$-particle subsystem and the remaining $N-j$  
particles. The first effect is modeled by the l.h.s and by the first  
term in the r.h.s of (\ref{eq: quant10}), the second one is encoded in  
$T_{N,j}^{Q}\hat{\rho}_{N,t}^{(j)}$), while the interaction between  
the $j$-particle subsystem and the remaining $N-j$ particles is  
modeled by $\frac{N-j}{N}C_{j,j+1}^Q\hat{\rho}_{N,t}^{(j)}$. The  
factor $1/N$ in front of $C_{j,j+1}^Q\hat{\rho}_{N,t}^{(j)}$ arises  
from the scaling of the potential $U^{Q}$ (see (\ref{eq: quant1})  
while the factor $N-j$ is due to the symmetry with respect to  
permutations of the labeling (we remind that we are dealing with $N$  
identical particles): indeed, the interaction of the $j$ particles  
under consideration with the last $N-j$ can be modeled by $N-j$ times  
the interaction with the $j+1$-th particle.

Writing explicitly the action of the operators $T_{N,j}^{Q}$ and  
$C_{j,j+1}^{Q}$, we find:
\begin{eqnarray}\label{eq: quant11}
&&T_{N,j}^{Q}\hat{\rho}_{N,t}^{(j)}=\frac{1}{2N}\sum_{k\neq  
l}^{j}\[\phi(x_k-x_l),\hat{\rho}_{N,t}^{(j)}\],
\end{eqnarray}
and
\begin{eqnarray}\label{eq: quant12}
&&C_{j,j+1}^Q\hat{\rho}_{N,t}^{(j)}=\sum_{k=1}^{j}\texttt{Tr}_{j+1}\left\{\[\phi(x_k-x_{j+1})
,\hat{\rho}_{N,t}^{(j+1)}\]\right\}.
\end{eqnarray}
By (\ref{eq: quant11})
we can argue that the operator $T_{N,j}^{Q}$ gives a vanishing  
contribution in the limit because it is of size $j^2/N$, while the  
operator $C_{j,j+1}^Q$ is of order one in the limit and the factor  
$(N-j)/N$ appearing in (\ref{eq: quant10}) is also of order one.  
Therefore denoting by $\hat{\rho}_{t}^{(j)}$ the expected limit of  
$\hat{\rho}_{N,t}^{(j)}$  when $N\to\infty$, the formal limit of the  
BBGKY hierarchy (\ref{eq: quant10}) is
\begin{eqnarray}\label{eq: quant13}
&&i\var\pa_t\hat{\rho}_{t}^{(j)}=\sum_{k=1}^{j}\[-\frac{\var^2}{2}  
\Delta_k,\hat{\rho}_{t}^{(j)}\]+ C_{j,j+1}^Q\hat{\rho}_{t}^{(j+1)},
\end{eqnarray}
which in the case $j=1$ is equal to:
\begin{eqnarray}\label{eq: quant14}
&&i\var\pa_t\hat{\rho}_{t}^{(1)}=\[-\frac{\var^2}{2}  
\Delta_1,\hat{\rho}_{t}^{(1)}\]+\texttt{Tr}_{2}\left\{\[\phi(x_1-x_{2})
,\hat{\rho}_{t}^{(2)}\]\right\}.
\end{eqnarray}
We observe that the Hartree equation (\ref{eq: Hartree}) in the  
density matrix formalism (''Heisenberg form'') is
\begin{eqnarray}\label{eq: quant15}
&&i\var\pa_t\hat{\rho}_{t}=\[-\frac{\var^2}{2}  
\Delta,\hat{\rho}_{t}\]+\texttt{Tr}_{2}\left\{\[\phi(x-x_{2})
,\hat{\rho}_{t}\otimes\hat{\rho}_{t}\]\right\}.
\end{eqnarray}
Replacing $x$ by $x_1$, $\hat{\rho}_{t}$ by $\hat{\rho}_{t}^{(1)}$ and  
the product $\hat{\rho}_{t}\otimes\hat{\rho}_{t}$ by  
$\hat{\rho}_{t}^{(2)}$ we realize that (\ref{eq: quant15}) is  
precisely the same of (\ref{eq: quant14}). Thus the equation of the  
hierarchy (\ref{eq: quant13}) corresponding to $j=1$ is properly the  
Hartree equation, provided that the the two-particle density  
$\hat{\rho}_{t}^{(2)}$ is factorized, and for this reason (\ref{eq:  
quant13})
is usually called ''Hartree hierarchy''. More precisely, by  
considering (\ref{eq: quant13})
and by assuming the reduced density matrices $\hat{\rho}_{t}^{(j)}$,  
$j=1,2,\dots$, to be factorized, namely
\begin{eqnarray}\label{eq: quant16}
&&\hat{\rho}_{t}^{(j)}=\hat{\rho}_{t}^{\otimes j}\ \ \forall\ j,
\end{eqnarray}
it is easy to verify that $\hat{\rho}_{t}$ has to solve the Hartree  
equation. Conversely, if we consider a time dependent one-particle  
density $\hat{\rho}_{t}$ solving the Hartree equation (\ref{eq:  
quant15}) and we take the $j$-particles densities  
$\hat{\rho}_{t}^{(j)}=\hat{\rho}_{t}^{\otimes j}$, $j=1,2\dots$, we  
find that the sequence $\{\hat{\rho}_{t}^{(j)}\}_{j\geq 1}$ solves the  
hierarchy (\ref{eq: quant13}). \\

An interesting problem is that of the uniqueness of the solution of  
the Hartree hierarchy. The situation in the quantum case is quite  
different from that of the classical one. In fact, as we will see in  
the next section, the Hartree hierarchy is much more controllable than  
the Vlasov one because the operators involved are bounded with respect  
to the norms appropriate to study the convergence of the sequence of  
reduced density matrices to the solution of the limiting hierarchy.  
Thus, it is possible to follow the approach we described briefly at  
the end of the previous chapter (convergence + uniqueness) in order to  
prove ''propagation of chaos'' in the quantum context, namely,  
asymptotic factorization of the dynamics (in the sense specified in  
the forthcoming paragraph). Nonetheless, the proof of uniqueness of  
the solution of the quantum limiting hierarchy is very far to be a  
trivial stuff. Indeed, in the case of bounded interaction the problem  
is quite easy to face and, in particular, by following the strategy of  
\cite{SPOHN} (originally introduced by O. Lanford for the derivation  
(for short times) of the Boltzmann equation from the hard-sphere  
dynamics (see \cite{LANFORD})) it is possible to prove ''at the same  
time'' convergence and uniqueness (see Theorem \ref{teoSPOHN}). On the  
contrary, in case of more singular interactions (as the Coulomb one),  
the proof of propagation of chaos consists really of two steps:  
proving the convergence of solutions of the BBGKY hierarchy to the  
Hartree hierarchy and showing the uniqueness of the solution of such a  
hierarchy (which implies factorization of the limiting $j$-particle  
density matrices because, as we have already remarked, the Hartree  
hierarchy admits factorized solutions as (\ref{eq: quant16})). In the  
Coulomb case, the ''uniqueness' problem is quite hard to deal with  
because of the singularity of the interaction (see \cite{EY}).

Anyway, we will come back later on the rigorous proof of propagation  
of chaos, analyzing in detail the case of bounded potentials. Moreover  
we will discuss briefly the Coulomb case, accenting which are the main  
new tools with respect to the bounded case, why there is need of them  
and in which way they make the proof harder requiring a more refined  
analysis of the limiting hierarchy. \\

Now let us clarify what we mean by ''propagation of chaos'' in the  
quantum framework.

Let us consider as initial datum for the Schr\"odinger equation  
(\ref{eq: quant5}) a factorized $N$-particle wave function
\begin{equation}\label{eq: quant17bis}
\Psi_{N,0}=\psi_0^{\otimes N},\ \ \  \ \ \ \text{for some}\ \  
\psi_0\in L^2\(\R^3\).
\end{equation}
This assumption, rephrased in the density matrix formalism, leads to  
consider the following factorized $N$-particle density matrix
\begin{equation}\label{eq: quant17}
\hat{\rho}_{N,0}=\vert  
\Psi_{N,0}><\Psi_{N,0}\vert=\hat{\rho}_0^{\otimes N},\ \ \ \text{with}  
  \ \hat{\rho}_0=\vert \psi_0><\psi_0\vert,
\end{equation}
as initial datum for the Heisenberg equation (\ref{eq: quant8}).
As in the classical context, (\ref{eq: quant17bis}) (or equivalently  
(\ref{eq: quant17})) is called ''hypotheses of molecular chaos''  
because we are assuming that the particles are initially uncorrelated.  
Furthermore, they are all in the same (one-particle) state at time  
$t=0$ and clearly $\Psi_{N,0}\in L^2_s\(\R^{3N}\)$. Thus, (\ref{eq:  
quant17bis}) is an admissible state for bosons (while for fermions it  
is prevented by the Pauli exclusion principle).
The physical motivation for studying the evolution of factorized wave  
functions
is that states close to the ground state of $H_N^Q$ (the eigenvector  
associated with the lowest eigenvalue),
which are the most accessible and thus the most interesting states,  
can be approximately described
by wave functions like (\ref{eq: quant17bis}) (some of the results  
which we are going to discuss in the following sections do not require
strict factorization as in (\ref{eq: quant17bis}); instead asymptotic  
factorization of the initial wave function in the sense of  
$\mathscr{L}^1$-convergence of the RDM to the $j$-fold product of  
one-particle densities
would be sufficient (see Theorem \ref{teoSPOHN} and the discussion below)).\\

Because of the interaction among the particles, the factorization  
(\ref{eq: quant17bis}) (or equivalently (\ref{eq: quant17})) is not  
preserved by the
time evolution; in other words, the evolved $N$-particle wave function  
$\Psi_{N,t}$ is not given by the product of one-particle
wave functions, if $t\neq 0$. All considerations done in the classical  
case concerning the mean-field (weak) character of the interaction  
hold, then we may expect that, in
the limit of large $N$, the total interaction potential experienced by  
a typical particle in the system
can be effectively replaced by an averaged, mean-field, potential, and  
therefore that factorization is
approximately, and in an appropriate sense, preserved by the time  
evolution. In other words, we
may expect that, in a sense to be made precise,
\begin{eqnarray}\label{eq: quant18}
&&\Psi_{N,t}\approx \psi_t^{\otimes N} \ \ \text{as}\ N\to\infty
\end{eqnarray}
or
\begin{eqnarray}\label{eq: quant18tris}
&&\hat{\rho}_{N,t}\approx \hat{\rho}_t^{\otimes N} \ \ \text{as}\  
N\to\infty, \ \ \text{with} \ \ \hat{\rho}_{N,t}=\vert  
\Psi_{N,t}><\Psi_{N,t}\vert,\ \ \ \hat{\rho}_t=\vert \psi_t><\psi_t\vert
\end{eqnarray}
for an evolved one-particle wave function $\psi_t$. This asymptotic  
factorization
is precisely what is called ''propagation of chaos''. Assuming  
(\ref{eq: quant18}), it is simple to derive a self-consistent equation
for the time-evolution of the wave function $\psi_t$. In fact,  
(\ref{eq: quant18}) states that, for every fixed time $t$,
the $N$ bosons are independently distributed in space according to the  
density $\vert \psi_t(x)\vert^2$. If this is true, the total
potential experienced, for example, by the first particle can be  
approximated by
\begin{eqnarray}\label{eq: quant18bis}
&&\frac{1}{N}\sum_{k\geq 2}
\phi(x_1 - x_k)\approx\frac{1}{N}\sum_{k\geq 2}\int \ud y \phi(x_1 -  
y)\vert\psi_t(y)\vert^2=\frac{N-1}{N}\( \phi\ast  
\vert\psi_t\vert^2\)\approx\(\phi\ast \vert\psi_t\vert^2 \),\nonumber\\
&&
\end{eqnarray}
as $N\to\infty$. It follows that, if (\ref{eq: quant18}) holds true,  
the one-particle wave function $\psi_t$ must satisfy the self-consistent
equation
\begin{eqnarray}\label{eq: quant19}
&& i\var \pa_t \psi_t = -\frac{\var^2}{2}\Delta \psi_t +\(\phi\ast  
\vert\psi_t\vert^2 \)\psi_t
\end{eqnarray}
with initial datum $\psi_0$ given by (\ref{eq: quant17bis}). Equation  
(\ref{eq: quant19}) is precisely the Hartree equation
and we have just presented an heuristic argument to explain
how it is related to the propagation of chaos. We observe that the  
Hartree equation is a nonlinear Schr\"odinger equation on $\R^3\times  
\R^3$. Therefore starting from the linear Schr\"odinger equation
(\ref{eq: quant5}) on $\R^{3N}\times \R^{3N}$, we obtain, for the  
evolution of factorized densities, a nonlinear Schr\"odinger equation  
on $\R^3\times \R^3$; the nonlinearity in the Hartree equation is a  
consequence of the many-body effects
in the linear dynamics.\\

The validity of propagation of chaos (namely, the precise statement  
concerning the asymptotic factorization (\ref{eq: quant18}) or  
(\ref{eq: quant18tris})) is expressed in terms convergence in  
$\mathscr{L}^1\(L^2(\R^{3j})\)$ of the $j$-particle marginal densities  
associated with the solution of the Heisenberg equation (\ref{eq:  
quant8}) to the $j$-fold product of solutions of the Hartree equation  
when $N\to\infty$, namely
\begin{equation}\label{eq: spohn2PRIMA}
\left\| \hat{\rho}_{N,t}^{(j)}-\hat{\rho}_t^{\otimes  
j}\right\|_{\mathscr{L}^1\(L^2\(\R^{3j}\)\)}
\rightarrow 0,\ \ \text{as}\ N\to\infty,
\end{equation}
$\hat{\rho}_t\in\mathscr{L}^1\(L^2\(\R^{3}\)\)$ solving the Hartree  
equation (in the ''Heisenberg form'') (\ref{eq: quant15})
with initial datum $\hat{\rho}_0$  given by (\ref{eq: quant18tris}).  
Clearly, $\hat{\rho}_t=\vert \psi_t><\psi_t\vert$, $\psi_t$ solving  
the Hartree equation (\ref{eq: quant19}) with initial datum $\psi_0$  
given by (\ref{eq: quant18}). \\

We have already remarked that, for fixed $j < N$, the $j$-particle RDM  
$\hat{\rho}_{N,t}^{(j)}$
does not contain the full information about the $N$-particle system  
described by $\hat{\rho}_{N,t}$. Nonetheless,
$\hat{\rho}_{N,t}^{(j)}$ is sufficient to compute the expectation of  
arbitrary observables of the form $A_j\otimes 1_{N-j}$ which depend  
non-trivially
on at most $j$ particles (because of the permutation symmetry, it is  
not important on which particles
it acts, just that it acts at most on $j$ particles).

Therefore the convergence (\ref{eq: spohn2PRIMA}) implies that:
\begin{eqnarray}\label{eq: convOBSjQUANTUM}
&&\left\langle A_j\otimes 1_{N-j}\right\rangle_{  
\Psi_{N,t}}=\texttt{Tr}\(A_j\otimes  
1_{N-j}\)\hat{\rho}_{N,t}=\texttt{Tr} A_j\hat{\rho}_{N,t}^{(j)}\  
\rightarrow \ \texttt{Tr} A_j\hat{\rho}_{t}^{\otimes j}= \left\langle  
A_j\right\rangle_{ \psi_{t}^{\otimes j}}\ \text{as}\  
N\to\infty,\nonumber\\
&&
\end{eqnarray}
for each bounded operator $A_j$ acting on $L^2\(\R^{3j}\)$. In other  
words, (\ref{eq: spohn2PRIMA}) allows to know the ''macroscopic''  
expected value of $j$-particle observables for an $N$-particle
system interacting by a men-field potential.






\subsection{Mean-Field limit for bounded potentials}
\setcounter{equation}{0}
\def\theequation{3.3.\arabic{equation}}



We consider, in this section, the dynamics generated by the mean field  
Hamiltonian (\ref{eq: quant2}) under the
assumption that the interaction potential is a bounded operator. We  
will assume, in other words,
that $\phi\in L^\infty\(\R^3\)$ (recall that the operator norm of the  
multiplication operator $\phi (x_k - x_l)$ is given
by the $L^\infty$-norm of the function $\phi$ ).

In the sequel we will use the notation $\phi_{kl}:=\phi (x_k - x_l)$.

\begin{teo} \label{teoSPOHN}\textnormal{{ \bf [Spohn 1980]}}\ \
Let the pair interaction potential $\phi$ be in $L^\infty\(\R^3\)$ and  
the initial state of the system be described by a factorized  
$N$-particle wave function $\Psi_{N,0}\in L^2_s\(\R^{3N}\)$, namely
\begin{equation}\label{eq: teoSPOHNwave}
\Psi_{N,0}=\psi_0^{\otimes N},\ \ \  \ \ \ \text{for some}\ \  
\psi_0\in L^2\(\R^3\):\ \ \left\| \psi_{0}\right\|_{L^2\(\R^{3}\)}=1.
\end{equation}
This implies that the initial $N$-particle density matrix  
$\hat{\rho}_{N,0}\in \mathscr{L}^1\(L^2\(\R^{3N}\)\)$ is given by
\begin{equation}\label{eq: spohn1}
\hat{\rho}_{N,0}=\vert  
\Psi_{N,0}><\Psi_{N,0}\vert=\hat{\rho}_0^{\otimes N},\ \ \  
\hat{\rho}_0=\vert\psi_0><\psi_0\vert.
\end{equation}

Then, for any fixed $j$,
\begin{equation}\label{eq: spohn2}
\left\| \hat{\rho}_{N,t}^{(j)}-\hat{\rho}_t^{\otimes  
j}\right\|_{\mathscr{L}^1\(L^2\(\R^{3j}\)\)}
\longrightarrow 0,\ \ as\ N\to\infty,
\end{equation}
where $\hat{\rho}_{N,t}^{(j)}$ solves the BBGKY hierarchy (\ref{eq:  
quant10}) with initial datum $\hat{\rho}_{0}^{\otimes j}$
and $\hat{\rho}_t\in\mathscr{L}^1\(L^2\(\R^{3}\)\)$ is the solution of  
the Hartree equation (in the ''Heisenberg form'')
\begin{eqnarray}\label{eq: spohn3}
i\var \pa_t  
\hat{\rho}_t=\[-\frac{\var^2}{2}\Delta,\hat{\rho}_t\]+\textnormal{\texttt{Tr}}_2\{\[\phi(x-x_2)
,\hat{\rho}_t\otimes\hat{\rho}_t\]\},
\end{eqnarray}
with initial datum $\hat{\rho}_0$. In terms of wave functions, we find  
that $\hat{\rho}_t=\vert\psi_t><\psi_t\vert$, $\psi_t$ solving the  
Hartree equation (\ref{eq: Hartree}) with initial datum $\psi_0$.
\end{teo}
\begin{dem}
Let $\hat{\rho}^{(N)}
\in \mathscr{L}^1\(L^2\(\R^{3N}\)\) $ be a trace class operator with  
kernel $\rho^{(N)}$ invariant under permutations of the labeling.
For fixed $j$, let $\hat{\rho}_{j}^{(N)}\in\mathscr{L}^1\(L^2\(\R^{3j}\)\)$ be
\begin{equation}\label{eq: spohn4}
\hat{\rho}_{j}^{(N)}=\texttt{Tr}_{j+1}\hat{\rho}^{(N)}.
\end{equation}
Then, by considering the time-evolution  
$\hat{\rho}^{(N)}(t)=e^{-\frac{i}{\var}H_N^Q  
t}\hat{\rho}^{(N)}e^{\frac{i}{\var}H_N^Q t}$, $H_N^Q$ defined in  
(\ref{eq: quant2}), it is also invariant under permutations of the  
labeling and
the $j$-particle trace class operator  
$\hat{\rho}_{j}^{(N)}(t)=\texttt{Tr}_{j+1}\hat{\rho}^{(N)}(t)$  
satisfies the differential equation
\begin{eqnarray}\label{eq: spohn5}
i\var \pa_t  
\hat{\rho}_j^{(N)}(t)&&=\[\sum_{k=1}^{j}\(-\frac{\var^2}{2}\Delta_k\)+\frac{1}{2N}\sum_{k\neq l}^j\phi_{k  
l},\hat{\rho}_j^{(N)}(t)\]+\nonumber\\
&&+\(\frac{N-j}{N}\)\sum_{k=1}^{j}\texttt{Tr}_{j+1}\left\{\[\phi_{k j+1}
,\hat{\rho}_{j+1}^{(N)}(t)\]\right\}.
\end{eqnarray}
This is what we previously called BBGKY hierarchy (see (\ref{eq:  
quant10})) as it can be seen by using the ''compact'' notation
\begin{eqnarray}\label{eq: spohn6}
\pa_t  
\hat{\rho}_j^{(N)}(t)=-\frac{i}{\var}\[\sum_{k=1}^{j}\(-\frac{\var^2}{2}\Delta_k\),\hat{\rho}_j^{(N)}(t)\]-\frac{i}{\var}T_{N,j}^Q\hat{\rho}_j^{(N)}(t)-\frac{i}{\var}\(\frac{N-j}{N}\)C_{j,j+1}^Q\hat{\rho}_{j+1}^{(N)}(t),\nonumber\\
&&
\end{eqnarray}
$T_{N,j}^Q$ and
$C_{j,j+1}^Q$ as in (\ref{eq: quant11}) and (\ref{eq: quant12}) respectively.

Let $S_j^{(N)}(t)$ is the flow associated with the equation:
\begin{eqnarray}\label{eq: spohn6bis}
\pa_t  
\hat{\rho}_j^{(N)}(t)=-\frac{i}{\var}\[H_{N,j}^Q,\hat{\rho}_j^{(N)}(t)\],
\end{eqnarray}
with
\begin{eqnarray}\label{eq: HAMaJ}
H_{N,j}^Q:=\sum_{k=1}^{j}\(-\frac{\var^2}{2}\Delta_k\)+T_{N,j}^Q.
\end{eqnarray}
Thus, $S_{j}^{(N)}(t)\hat{\rho}_{j}=e^{-\frac{i}{\var}H_{N,j}^Q  
t}\hat{\rho}_{j}e^{\frac{i}{\var}H_{N,j}^Q t}$, for any  
$\hat{\rho}_{j}\in\mathscr{L}^1\(L^2\(\R^{3j}\)\) $. By the Duhamel  
formula, the solution of (\ref{eq: spohn6}) can be written as
\begin{eqnarray}\label{eq: spohn7prima}
\hat{\rho}_j^{(N)}(t)=S_{j}^{(N)}(t)\hat{\rho}_j^{(N)}+
\(\frac{N-j}{N}\)\(-\frac{i}{\var}\)\int_{0}^{t}\ud t_1\   
S_j^{(N)}(t-t_1)C_{j,j+1}^Q \hat{\rho}_{j+1}^{(N)}(t_1).
\end{eqnarray}
Iterating the integral equation (\ref{eq: spohn7prima}),
we obtain the
series
\begin{eqnarray}\label{eq: spohn7}
\hat{\rho}_j^{(N)}(t)&&=S_{j}^{(N)}(t)\hat{\rho}_j^{(N)}+\nonumber\\
&&+\sum_{n=1}^{N-j}\int_{0\leq t_n\leq \dots \leq t_1\leq t}\ud  
t_n\dots\ud t_1\  
S_j^{(N)}(t-t_1)\(\frac{N-j}{N}\)\(-\frac{i}{\var}\)C_{j,j+1}^Q\dots\nonumber\\
&&\qquad \quad\dots  
\(\frac{N-j-n+1}{N}\)\(-\frac{i}{\var}\)C_{j+n-1,j+n}^Q\  
S_{j+n}^{(N)}(t_n) \hat{\rho}_{j+n}^{(N)}.
\end{eqnarray}

Let $\left\|\cdot \right\|_j$ denote the trace norm in  
$\mathscr{L}^1\(L^2\(\R^{3j}\)\)$. Since $S_j^{(N)}(t)$ preserves the  
$\left\|\cdot \right\|_j$ norm (because $H_{N,j}^Q$ is a self-adjoint  
operator on $L^2\(\R^{3j}\)$), by the expression (\ref{eq: quant12})  
for $C_{j,j+1}^Q$, it is easy to verify that the $n$-th term of the  
series (\ref{eq: spohn7}) is bounded by
\begin{eqnarray}\label{eq: spohn8}
\frac{t^n}{n!}\ j(j+1)\dots  
(j+n-1)\(\frac{2\left\|\phi\right\|_{L^\infty}}{\var}\)^n
\left\|\hat{\rho}_{j+n}^{(N)}\right\|_{j+n}.
\end{eqnarray}
If one assumes
\begin{eqnarray}\label{eq: spohn9}
\mathbf{P1)}\ \left\|\hat{\rho}_{j}^{(N)}\right\|_{j}\leq a^j\ \ \  
\text{for any}\ j,
\end{eqnarray}
then the series (\ref{eq: spohn7}) converges in trace norm for $\vert  
t\vert\leq t_0$ with  
$t_0<\frac{\var}{4\left\|\phi\right\|_{L^\infty}a}$.

For any
$\hat{\rho}_{j}\in\mathscr{L}^1\(L^2\(\R^{3j}\)\) $, let  
$S_{j}(t)\hat{\rho}_{j}=e^{-\frac{i}{\var}H_{j}t}\hat{\rho}_{j}e^{\frac{i}{\var}H_j  
t}$,
where $H_{j}=\sum_{k=1}^{j}\(-\frac{\var^2}{2}\Delta_k\)$ is the  
$j$-particle free Hamiltonian. We note that
\begin{eqnarray}\label{eq: spohn8opTbound}
\left\|T_{N,j}^Q\hat{\rho}_{j}^{(N)}\right\|\leq  
\frac{j(j-1)}{2N}\|[\phi, \hat{\rho}_{j}^{(N)}]\|_{j}\leq  
\frac{j(j-1)}{N}\left\|\phi\right\|_{L^\infty}
\left\|\hat{\rho}_{j}^{(N)}\right\|_{j},\nonumber\\
&&
\end{eqnarray}
then by Property $\mathbf{P1)}$ we find
\begin{eqnarray}\label{eq: spohn8opTboundII}
\left\|T_{N,j}^Q
\right\|\leq \frac{j(j-1)}{N}\left\|\phi\right\|_{L^\infty}a^j
\ \to 0\ \ \ \text{as}\ \ N\to\infty,
\end{eqnarray}
where $\left\|\cdot\right\|$ is the operator norm on  
$\mathscr{L}^1\(L^2\(\R^{3j}\)\)$.

We note that, for any $\hat{\rho}_{j}\in\mathscr{L}^1\(L^2\(\R^{3j}\)\)$,
\begin{eqnarray}\label{eq: nuova}
\left\|S_j^{(N)}(t)\hat{\rho}_{j}-S_{j}(t)\hat{\rho}_{j}\right\|_j&&\leq\frac{1}{\var}\int_0^t\ud \tau \left\|S_{j}(t-\tau)T_{N,j}^Q\hat{\rho}_{j}(\tau)\right\|_j\leq  
\nonumber\\
&&\leq\frac{1}{\var}\int_0^t\ud \tau  
\left\|T_{N,j}^Q\hat{\rho}_{j}(\tau)\right\|_j,
\end{eqnarray}
where we use that $S_j(t)$ preserves the trace norm. Then, from  
(\ref{eq: spohn8opTboundII}), it follows that
\begin{eqnarray}\label{eq: spohn10ante}
\lim_{N\to\infty}\left\|S_j^{(N)}(t)-S_{j}(t)\right\|=0.
\end{eqnarray}

If one assumes
\begin{eqnarray}\label{eq: spohn10}
\mathbf{P2)}\  
\lim_{N\to\infty}\left\|\hat{\rho}_{j}^{(N)}-\hat{\rho}_{j}\right\|_{j}=0,
\end{eqnarray}
for some $\hat{\rho}_{j}\in \mathscr{L}^1\(L^2\(\R^{3j}\)\)$,
then by (\ref{eq: spohn7}) and (\ref{eq: spohn10ante}), it follows  
that $\hat{\rho}_{j}^{(N)}(t)$ converges as $N\to\infty$ in trace norm  
to
\begin{eqnarray}\label{eq: spohn11}
\hat{\rho}_{j}(t)&&=\sum_{n=0}^{+\infty}\int_{0\leq t_n\leq \dots \leq  
t_1\leq t}\ud t_n\dots\ud t_1\  
S_{j}(t-t_1)\(-\frac{i}{\var}\)C_{j,j+1}^Q\dots\nonumber\\
&&\qquad \qquad\dots \(-\frac{i}{\var}\)C_{j+n-1,j+n}^Q\ S_{j+n}(t_n)  
\hat{\rho}_{j+n},
\end{eqnarray}
for $\vert t\vert\leq t_0$. We note that the $n$-th term of the above  
series in bounded in trace norm by (\ref{eq: spohn8}), then for short  
times $\vert t\vert\leq t_0$ we are ensures that (\ref{eq: spohn11})  
converges in trace norm.\\
Let $\hat{\rho}^{(N)}$ be a density matrix. Then  
$\left\|\hat{\rho}_{j}^{(N)}(t)\right\|_{j}=\left\|\hat{\rho}_{j}^{(N)}\right\|_{j}$ by preservation of positivity and trace. Therefore, if for the initial state the bound $\mathbf{P1)}$ is satisfied, it remains valid for all times, and the argument just given can be iterated to prove convergence of $\hat{\rho}_{j}^{(N)}(t)$ to $\hat{\rho}_{j}(t)$ as $N\to\infty$ for all times. Furthermore, $\hat{\rho}_{j}(t)$ is uniquely determined for all times because by iteration we prove that (\ref{eq: spohn11}) converges in trace norm for all times.  
\\

One checks that for the particular initial state $\hat{\rho}_{N,0}$ in  
(\ref{eq: spohn1}) the conditions $\mathbf{P1)}$ and $\mathbf{P2)}$  
are satisfied with $a=1$ and $\hat{\rho}_j=\hat{\rho}_0^{\otimes j}$.  
Therefore, we can claim that the solution $\rho_{N,t}^{(j)}$ of the  
BBGKY hierarchy (\ref{eq: spohn6}) with initial datum  
$\hat{\rho}_0^{\otimes j}$ converges in trace norm to the unique  
$j$-particle density matrix $\hat{\rho}_{j}(t)$ identified by the  
series (\ref{eq: spohn11}) with $  
\hat{\rho}_{j+n}=\hat{\rho}_0^{\otimes j+n}$ \ $\forall\ n$.  
Differentiating (\ref{eq: spohn11}) with respect to $t$, one obtains  
the limiting hierarchy of equations
\begin{eqnarray}\label{eq: spohn12}
i\var \pa_t  
\hat{\rho}_j(t)&&=\[\sum_{k=1}^{j}\(-\frac{\var^2}{2}\Delta_k\),\hat{\rho}_j(t)\]+
C_{j,j+1}^Q\ \hat{\rho}_{j+1}(t),
\end{eqnarray}
whose unique trace class solution is $\hat{\rho}_{j}(t)$
with initial datum $\hat{\rho}_0^{\otimes j}$.
Moreover, (\ref{eq: spohn12}) preserves the factorization property for  
all $t$ according to the Hartree equation (in the Heisenberg form)  
(\ref{eq: spohn3}). This ensures the validity of propagation of chaos,  
namely $\hat{\rho}_j(t)=\hat{\rho}_t^{\otimes j}$, $\hat{\rho}_t$  
solving the nonlinear Heisenberg equation (\ref{eq: spohn3}) with  
initial datum $\hat{\rho}_0=\vert \psi_0><\psi_0\vert$. Then,  
$\hat{\rho}_t=\vert \psi_t><\psi_t\vert$, $\psi_t$ solving the Hartree  
equation (\ref{eq: Hartree}) with initial datum $\psi_0$.
\end{dem}

\begin{rem} \label{Remark S.1}:

\textnormal{By looking at the the proof above it is clear that in  
order to let Theorem \ref{teoSPOHN} to hold there is no need of
strict factorization of the initial datum as in (\ref{eq: spohn1}).
Instead asymptotic factorization in the sense of
\begin{eqnarray}\label{eq: REMS.1}
\lim_{N\to\infty}\left\|\hat{\rho}_{N,0}^{(j)}-\hat{\rho}_{0}^{\otimes  
j}\right\|_{j}=0
\end{eqnarray}
would be sufficient. We remind that (\ref{eq: REMS.1}) is a reasonable  
''physical'' condition because states close to the ground state of  
$H_N^Q$,
which are the most accessible and thus the most interesting states,  
can be approximately described
by factorized wave functions, and then, by factorized density matrices}.
\end{rem}


\begin{rem} \label{Remark S.2}:

\textnormal{In proving the
convergence of the series (\ref{eq: spohn7}) to (\ref{eq: spohn11})  
the crucial tools have been the boundedness of the operator  
$T_{N,j}^Q:\mathscr{L}^1\(L^2\(\R^{3j}\)\)\to\mathscr{L}^1\(L^2\(\R^{3j}\)\)$  
(see (\ref{eq: spohn8opTbound})) and property $\mathbf{P1)}$ for the  
RDM. In particular, the bound (\ref{eq: spohn8opTboundII}) on the  
operator norm of $T_{N,j}^Q$ provides the rate of convergence to the  
Hartree dynamics by means of (\ref{eq: nuova}).}
\textnormal{Then, by observing that the estimate obtained in (\ref{eq:  
nuova}) is not uniform with respect to $\var$ and it fails when  
$\var\to 0$, it follows that the error in approximating the  
$N$-particle dynamics with the limiting one is diverging when $\var\to  
0$ (for short times it is of the form $\frac{C_j}{N}e^{Ct/\var}$)}.
\end{rem}

In the next sections we will discuss some other results concerning the  
mean-field limit starting from factorized initial datum as in  
(\ref{eq: teoSPOHNwave}), both for bounded interactions and for the  
Coulomb potential and we will see that considerations done in Remark  
\ref{Remark S.2} still hold.
This means that
all results concerning the mean-field limit exhibit an error in  
approximating the $N$-particle dynamics by the limiting one which is  
not uniform with respect to $\var$ and diverging when $\var\to 0$.  
This is a quite surprising feature because it seems that, roughly  
speaking, the accuracy of the approximation depends on ''how much''  
the system can be considered quantum or not and, except for fermionic  
systems, there are no reasonable motivations for that. In fact, we  
will see in Chapter 3 that in the fermionic case it is quite natural  
to look at a joint limit: $N\to\infty$ and $\var\to 0$ as in \cite{NS}  
and \cite{fermioni}. On the contrary, in looking at systems of  
undistiguishable particles or even bosonic systems
the fact that the mean-field limit and the semiclassical approximation  
seems to be so strictly connected is an open problem (except for  
specific scalings of the potential as in \cite{Graffi}). Furthermore,  
in the classical case (see Chapter 1) everything works, so it is quite  
natural to ask if, at least for quantum systems having a reasonable  
classical analogue, it is possible to realize the limit $N\to\infty$  
uniformly with respect to $\var$. This is the main motivation of our  
research and in the next section we will focus on that topic,  
discussing some known results and presenting what we did in this  
perspective.





\subsubsection{An alternative approach}
 From the proof of Theorem \ref{teoSPOHN} presented above, we notice  
that the expansion of the BBGKY hierarchy
in (\ref{eq: spohn7}) is much more involved than the corresponding  
expansion (\ref{eq: spohn11}) of the infinite hierarchy
(\ref{eq: spohn12}). It turns out that it is possible to avoid the  
expansion of the BBGKY hierarchy making use
of a simple compactness argument; this will be especially important  
when dealing with singular potentials.
In the following we explain the main steps of this alternative proof  
to Theorem \ref{teoSPOHN}. Then,
in the next section, we will illustrate how to extend it to potentials  
with a Coulomb singularity.
The idea, which was first presented in \cite{BGM}, \cite{EY}, \cite{BEGMY},
consists in characterizing the limit of the RDM
$\hat{\rho}_{N,t}^{(j)}$
as the unique solution to the infinite hierarchy of equations  
(\ref{eq: spohn12}); combined with the compactness,
this information provides a proof of Theorem \ref{teoSPOHN}. \\More  
precisely, the proof is divided into three
main steps:

$i)$\  First of all, one shows the compactness of the sequence  
$\{\hat{\rho}_{N,t}^{(j)}\}_{j= 1}^N$ with respect to an
appropriate weak topology.

$ii)$\ Then, one proves that an arbitrary limit point  
$\{\hat{\rho}_{\infty,t}^{(j)}\}_{j\geq 1}$ of the sequence  
$\{\hat{\rho}_{N,t}^{(j)}\}_{j= 1}^N$
  is a solution to the infinite hierarchy (\ref{eq: spohn12}) (one  
proves, in other words, the convergence
to the infinite hierarchy).

$iii)$\ Finally, one shows the uniqueness of the solution to the  
infinite hierarchy
(\ref{eq: spohn12}).\\

We have already observed that the factorized family  
$\{\hat{\rho}_{t}^{\otimes j}\}_{j\geq 1}$
is a solution of the infinite hierarchy with factorized initial datum  
$\hat{\rho}_{0}^{\otimes j}$. In particular, if $\hat{\rho}_{0}=\vert  
\psi_0><\psi_0\vert$, as in the present case, we find that  
$\hat{\rho}_{t}=\vert \psi_t><\psi_t\vert$, $\psi_t$ solving the  
Hartree equation (\ref{eq: Hartree}). Then, by proving that the  
solution of the infinite hierarchy is unique, we are guaranteed that  
it is factorized according to the solution of the Hartree equation.

Therefore, by $ii)$, it follows immediately that  
$\hat{\rho}_{N,t}^{(j)}\to \hat{\rho}_{t}^{\otimes j}=\(\vert  
\psi_t><\psi_t\vert\)^{\otimes j}$ as $N\to\infty$ (at first only in  
the weak topology with respect to which we have compactness; since the  
limit is an
orthogonal rank one projection, it is however simple to check that  
weak convergence implies strong
convergence, in the sense (\ref{eq: spohn2})). Next, we discuss these  
three main steps (compactness, convergence,
and uniqueness) in some more details in order to show that, even  
following this approach, the estimates that ensure the convergence are  
not uniform with respect to $\var$ and they fail if $\var\to 0$. \\

\underline{\textbf{Compactness}}:  By knowing that, for any $j$ and  
$N$, $\left\|\hat{\rho}_{N,t}^{(j)}\right\|_{\mathscr{L}^1}=1$ for  
fixed $t$, thanks to standard abstract and compactness results of  
functional analysis we prove that the sequence  
$\Gamma_{N,t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$ is compact with  
respect to a suitable topology. More precisely, for an arbitrary fixed  
$T>0$, we denote by
$\mathcal{C}\([0, T],\mathscr{L}^1\(L^2(\R^{3j})\)\)$ the space of  
functions of $t \in [0, T]$ with values in  
$\mathscr{L}^1\(L^2(\R^{3j})\)$ which are continuous in time with  
respect to a suitable metric $\eta_j$ on  
$\mathscr{L}^1\(L^2(\R^{3j})\)$ (it can be constructed explicitly in  
such a way that the topology generated by $\eta_j$ is equivalent to  
the weak*-topology of $\mathscr{L}^1\(L^2(\R^{3j})\)$). By $\eta_j$ we  
can easily define a metric $\hat{\eta}_j$ on $\mathcal{C}\([0,  
T],\mathscr{L}^1\(L^2(\R^{3j})\)\)$ and we consider the topology  
$\tau_{prod}$ on
$\bigoplus_{j\geq 1}\mathcal{C}\([0,  
T],\mathscr{L}^1\(L^2(\R^{3j})\)\)$ given by the product of the
topologies generated by the metrics $\hat{\eta}_j$ on  
$\mathcal{C}\([0, T],\mathscr{L}^1\(L^2(\R^{3j})\)\)$. The topology  
$\tau_{prod}$ is precisely the topology with respect to which we prove  
compactness of the sequence $\{\Gamma_{N,t}\}_{N\in \N}$ and this is  
equivalent to the following
\begin{prop}\label{PROPcompactness}
Fix an arbitrary time $T > 0$.
For every sequence $\{M_m\}_{m\in\N}$ there exists a subsequence  
$\{N_m\}_{m\in\N }\subset\{M_m\}_{m\in\N}$
and a limit point  
$\Gamma_{\infty,t}=\{\hat{\rho}_{\infty,t}^{(j)}\}_{j\geq 1}$ for  
$\Gamma_{N_m,t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^{N_m}$ such that
\begin{eqnarray}\label{eq: BenjiALT0}
\hat{\rho}_{\infty,t}^{(j)}\geq 0,\ \ \  
\textnormal{\texttt{Tr}}\hat{\rho}_{\infty,t}^{(j)}\leq 1,\ \ \forall\  
j\geq 1,
\end{eqnarray}
$\hat{\rho}_{\infty,t}^{(j)}$ (for any $j$) is symmetric with respect  
to permutations of the labeling.
\end{prop}

Let $\mathcal{K}_j \equiv \mathcal{K}\(L^2(\R^{3j})\)$ be the space of  
compact operators on $L^2(\R^{3j})$, equipped with the
operator norm. The claim of Proposition \ref{PROPcompactness} is  
equivalent to the affirm that, passing to a subsequence,\\

\emph{
For every fixed $j\geq 1$ and for every fixed compact operator  
$J^{(j)}\in \mathcal{K}_j$, }
\begin{eqnarray}\label{eq: BenjiALT1}
\textnormal{\texttt{Tr}}\  
J^{(j)}\(\hat{\rho}_{N,t}^{(j)}-\hat{\rho}_{\infty,t}^{(j)}\)  
\rightarrow 0\ \ \ as\ \ N\to\infty
\end{eqnarray}

\emph{uniformly in $t$ for $t \in [0, T]$.}
\vspace{0.5cm}

\underline{\textbf{Convergence}}: The second main step consists in  
characterizing the limit points of the (compact) sequence  
$\Gamma_{N,t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$ as solutions to the  
infinite hierarchy of equations (\ref{eq: spohn12}) with initial datum  
$\hat{\rho}_0^{\otimes j}$, $\hat{\rho}_0=\vert \psi_0><\psi_0\vert$.\\

\begin{prop}\label{convALT}
Suppose that $\phi\in L^\infty(\R^3)$ such that $ \phi(x)\to 0$ as  
$|x|\to \infty$.
Assume moreover
that $\Gamma_{\infty,t}=\{\hat{\rho}_{\infty,t}^{(j)}\}_{j\geq 1}\in  
\bigoplus_{j\geq 1}\mathcal{C}\([0,  
T],\mathscr{L}^1\(L^2(\R^{3j})\)\)$ is a limit point of the sequence  
$\Gamma_{N,t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$  in the sense  
(\ref{eq: BenjiALT1}). Then \begin{eqnarray}\label{eq: BenjiALT2}
\hat{\rho}_{\infty,t}^{(j)}=S_{j}(t)\hat{\rho}_{\infty,0}^{(j)}+
\(-\frac{i}{\var}\)\int_{0}^{t}\ud t_1\  S_j(t-t_1)C_{j,j+1}^Q\   
\hat{\rho}_{\infty,t_1}^{(j+1)}.
\end{eqnarray}
for all $j\geq 1$, with  
$\hat{\rho}_{\infty,0}^{(j)}=\hat{\rho}_{N,0}^{(j)}=\hat{\rho}_{0}^{\otimes  
j}$. Here $S_{j}(t)$ is the flow associated with the $j$-particle free  
dynamics and $C_{j,j+1}^Q$ is defined as in (\ref{eq: quant12}).  
Therefore equation (\ref{eq: BenjiALT2}) evaluated for all $j\geq 1$  
gives rise precisely to a solution of the Hartree hierarchy (\ref{eq:  
spohn12}) with factorized initial datum.
\end{prop}
Note that in Proposition \ref{convALT} we assume the potential to  
vanish at infinity. This condition, which
was not required in Theorem \ref{teoSPOHN}, is not essential but it  
simplifies the proof and it is also satisfied by the Coulomb  
interaction for which the derivation of the Hartree equation has been  
proven (e.g. \cite{EY}) by following the present strategy (together  
with crucial technical tools that are necessary to deal with the  
singularity of the potential ).\\

\begin{dem}
Passing to a subsequence we can assume that  
$\Gamma_{N,t}\to\Gamma_{\infty,t}$ as $N\to\infty$, in the sense  
(\ref{eq: BenjiALT1}); this implies immediately that  
$\hat{\rho}_{\infty,0}^{(j)}=\hat{\rho}_{N,0}^{(j)}=\hat{\rho}_{0}^{\otimes  
j}$. To prove (\ref{eq: BenjiALT2}), on the other
hand, it is enough to show that for every fixed $j\geq 1$, and for  
every fixed $J^{(j)}$ from a dense subset
of $\mathcal{K}_{j}$,
\begin{eqnarray}\label{eq: BenjiALT3}
\texttt{Tr}J^{(j)}\hat{\rho}_{\infty,t}^{(j)}=\texttt{Tr}J^{(j)}S_{j}(t)\hat{\rho}_{\infty,0}^{(j)}+
\(-\frac{i}{\var}\)\int_{0}^{t}\ud t_1\   
\texttt{Tr}J^{(j)}S_j(t-t_1)C_{j,j+1}^Q\   
\hat{\rho}_{\infty,t_1}^{(j+1)}.
\end{eqnarray}
To demonstrate (\ref{eq: BenjiALT3}), we start from the BBGKY  
hierarchy (\ref{eq: spohn6}) which leads to
\begin{eqnarray}\label{eq: BenjiALT4}
\texttt{Tr}J^{(j)}\hat{\rho}_{N,t}^{(j)}&&=\texttt{Tr}J^{(j)}S_{j}(t)\hat{\rho}_{N,0}^{(j)}+
-\frac{i}{\var}\int_{0}^{t}\ud t_1\   
\texttt{Tr}J^{(j)}S_j(t-t_1)T_{N,j}^Q  
\hat{\rho}_{N,t_1}^{(j)}+\nonumber\\
&&\(-\frac{i}{\var}\)\frac{(N-j)}{N}\int_{0}^{t}\ud t_1\   
\texttt{Tr}J^{(j)}S_j(t-t_1)C_{j,j+1}^Q\  \hat{\rho}_{N,t_1}^{(j+1)}.
\end{eqnarray}
Since, by assumption, the l.h.s. and the first term on the r.h.s. of  
the last equation converge, as
$N\to\infty$, to the l.h.s. and, respectively, to the first term on  
the r.h.s. of (\ref{eq: BenjiALT3}) (for every compact
operator $J^{(j)}$), (\ref{eq: BenjiALT2}) follows if we can prove that
\begin{eqnarray}\label{eq: BenjiALT5}
-\frac{i}{\var}\int_{0}^{t}\ud t_1\   
\texttt{Tr}J^{(j)}S_j(t-t_1)T_{N,j}^Q \hat{\rho}_{N,t_1}^{(j)}\to 0
\end{eqnarray}
and that
\begin{eqnarray}\label{eq: BenjiALT6}
&&\(-\frac{i}{\var}\)\frac{(N-j)}{N}\int_{0}^{t}\ud t_1\   
\texttt{Tr}J^{(j)}S_j(t-t_1)C_{j,j+1}^Q\   
\hat{\rho}_{N,t_1}^{(j+1)}\to \(-\frac{i}{\var}\)\int_{0}^{t}\ud t_1\   
\texttt{Tr}J^{(j)}S_j(t-t_1)C_{j,j+1}^Q\   
\hat{\rho}_{\infty,t_1}^{(j+1)}\nonumber\\
&&
\end{eqnarray}
as $N\to\infty$. Eq. (\ref{eq: BenjiALT5}) follows because, by the  
expression (\ref{eq: quant11}) of $T_{N,j}^Q$, we have
\begin{eqnarray}\label{eq: BenjiALT7}
&&\left\vert \frac{i}{\var}  \texttt{Tr}J^{(j)}S_j(t-t_1)T_{N,j}^Q  
\hat{\rho}_{N,t_1}^{(j)}\right\vert\leq \frac{1}{\var 2N}\sum_{k\neq  
l}^{ j}\left\vert \texttt{Tr}J^{(j)}S_j(t-t_1)\[\phi(x_k -  
x_l),\hat{\rho}_{N,t_1}^{(j)}\]\right\vert\leq\nonumber\\
&&\qquad\leq
\frac{j^2}{\var  
N}\left\|J^{(j)}\right\|\left\|\phi\right\|\texttt{Tr}\left\vert\hat{\rho}_{N,t_1}^{(j)}\right\vert=
\frac{j^2}{\var N}\left\|J^{(j)}\right\|\left\|\phi\right\|\to 0
\end{eqnarray}
because the product $\left\|J^{(j)}\right\|\left\|\phi\right\|$ is  
finite and uniformly bounded with respect to $N$  
($\left\|J^{(j)}\right\|$ and $\left\|\phi\right\|$ being the operator  
norms of $J^{(j)}$ and of the multiplication operator $\phi$). To  
prove (\ref{eq: BenjiALT6}) one can use a similar argument, combined  
with the
observation that, by the expression (\ref{eq: quant12}) of $C_{j,j+1}^Q $,
\begin{eqnarray}\label{eq: BenjiALT8}
&& \frac{i}{\var}   
\texttt{Tr}J^{(j)}S_j(t-t_1)C_{j,j+1}^Q\(\hat{\rho}_{N,t_1}^{(j+1)}-\hat{\rho}_{\infty,t_1}^{(j+1)}\)=\nonumber\\
&& = \frac{i}{\var}\sum_{1\leq k\leq j}  
\texttt{Tr}\[\(J^{(j)}S_j(t-t_1)\),\phi(x_k -  
x_{j+1})\]\(\hat{\rho}_{N,t_1}^{(j+1)}-\hat{\rho}_{\infty,t_1}^{(j+1)}\)\to  
0,\nonumber\\
&&
\end{eqnarray}
as $N\to\infty$. This does not follow directly from the assumption  
that $\Gamma_{N,t}\to \Gamma_{\infty,t}$ in the sense (\ref{eq:  
BenjiALT1}) because the operator $\[\(J^{(j)}S_j(t-t_1)\),\phi(x_k -  
x_{j+1})\]$ is not compact on $L^2(\R^{3(j+1}))$. Instead it is  
necessary to apply an approximation argument which is made simpler by  
the assumption that $\phi (x)\to 0$ as $|x|\to \infty$ (that is the  
reason for which we did it).
The details of this approximation
argument can be found, for example, in
\cite{GP}.
\end{dem}

\underline{\textbf{Uniqueness}}: to conclude the proof of Theorem  
\ref{teoSPOHN}, we still have to prove the uniqueness of the
solution to the infinite (Hartree) hierarchy (\ref{eq: BenjiALT2}).

\begin{prop}
Fix $\Gamma_{\infty, 0}=\{\hat{\rho}_{\infty,0}^{(j)}\}_{j\geq 1}\in  
\bigoplus_{j\geq 1}\mathscr{L}^1 \(L^2(\R^3)\)$. Then there exists at  
most one solution $\Gamma_{\infty,  
t}=\{\hat{\rho}_{\infty,t}^{(j)}\}_{j\geq 1}\in \bigoplus_{j\geq  
1}\mathcal{C}\([0,T],\mathscr{L}^1 \(L^2(\R^3)\)\)$
  to the infinite (Hartree) hierarchy (\ref{eq: BenjiALT2}) such that  
$\hat{\rho}_{\infty,t}^{(j)}\vert_{t=0}=\hat{\rho}_{\infty,0}^{(j)}$
  and
   
$\textnormal{\texttt{Tr}}\left\vert\hat{\rho}_{\infty,t}^{(j)}\right\vert\leq  
1$
  for all $j\geq 1$ and all $t \in [0, T]$.
  \end{prop}
  \begin{dem}
The proof is exactly the same we did in proving Theorem  
\ref{teoSPOHN}. Indeed, we write the solution of the Hartree hierarchy  
by iterating the Duhamel formula (\ref{eq: BenjiALT2}) and we observe  
that the series we obtain is uniformly bounded in trace norm by a  
geometric series converging for short times $t<t_0$ (see (\ref{eq:  
spohn11})). This proves uniqueness locally in time. Then, by noting  
that $t_0$ does not depend on the initial condition (except for the  
trace norm of $\hat{\rho}_{\infty,0}$ which is clearly preserved in  
time) but only on the $L^\infty$-norm of the interaction potential  
$\phi$, we can iterate the same argument, obtaining uniqueness for all  
times.
\end{dem}

\vspace{0.3cm}
We realize that the estimate ensuring the convergence is (\ref{eq:  
BenjiALT7}) and, as in the proof of Theorem \ref{teoSPOHN} presented  
in the previous paragraph, we note that it is due to the boundness of  
the operator  
$T_{N,j}^Q:\mathscr{L}^1\(L^2\(\R^{3j}\)\)\to\mathscr{L}^1\(L^2\(\R^{3j}\)\)$.  
Indeed, its norm is bounded by $C/N$, thus vanishing when $N\to  
\infty$. Nevertheless, by looking at (\ref{eq: BenjiALT7}) we see that  
there is a factor $1/\var$ in front of $T_{N,j}^Q$, thus we obtain a  
bound of order $1/\var$ which is diverging if $\var\to 0$. Therefore,  
even by using this approach, it is clear that considerations done  
previously as regard to the uniformity of the mean-field approximation  
with respect to $\var$, still hold.

We conclude this paragraph by observing that in estimating the  
operator norm of $T_{N,j}^Q$ on $\mathscr{L}^1\(L^2\(\R^{3j}\)\)$ we  
have to deal with  
$\frac{1}{\var}\texttt{Tr}\[\phi,\hat{\rho}_{N,t}^{(j)}\]$ and we find  
a uniform bound with respect to $N$, but diverging in $\var$, by using  
that in any Hilbert space $\mathcal{H}$:
\begin{eqnarray}\label{eq: CRUCIAL}
\texttt{Tr}\[A,B\]\leq 2\left\|A\right\|\texttt{Tr}\left\vert  
B\right\vert, \ \ \forall\ \ A\in \mathscr{L}^\infty\(\mathcal{H}\),  
B\in\mathscr{L}^1\(\mathcal{H}\),
\end{eqnarray}
where $\mathscr{L}^\infty\(\mathcal{H}\)$ is the space of bounded  
operators on $\mathcal{H}$ ($\left\|\cdot\right\|$ being the operator  
norm in $\mathscr{L}^\infty\(\mathcal{H}\)$) and  
$\mathscr{L}^1\(\mathcal{H}\)$ is the space of trace class operators  
on $\mathcal{H}$ equipped with the norm $\texttt{Tr}\left\vert  
\cdot\right\vert$.
Essentially the crucial question in looking for an estimate of the  
error in the mean-field approximation (for bounded or even smooth  
potentials) which is uniform (or at least ''better diverging'') with  
respect to $\var$ is: can we improve (\ref{eq: CRUCIAL}) by taking  
into account further properties of the operators $A$ and $B$ we have  
to deal with, possibly considering suitable ''semiclassical'' initial  
data?

\subsubsection{Mean-field limit for the Coulomb potential}

The result presented in \cite{EY} concerning the mean-field limit in  
the case of Coulomb interaction, $\phi=1/\vert x\vert$, is formulated  
as in Theorem \ref{teoSPOHN}, except for the fact that the initial one  
particle wave function $\psi_0$ is assumed to be in $H_1(\R^3)$ (the  
Sobolev space $W^{1,2}(\R^3)$ of functions il $L^2(\R^3)$ whose  
derivatives are also in $L^2\(\R^3\)$) and that the theorem holds for  
dimensions $d\geq 2$. Even the general strategy of the proof is the  
same we outlined in the previous paragraph.  First one proves the  
compactness of the sequence of marginal
$\Gamma_{N, t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$
  with respect to
an appropriate weak topology (the product topology $\tau_{prod}$  
previously introduced), then one shows
that an arbitrary limit point $\Gamma_{\infty,  
t}=\{\hat{\rho}_{\infty,t}^{(j)}\}_{j\geq 1}$
of the sequence $\Gamma_{N, t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$ is  
a solution to the infinite hierarchy
of equations
\begin{eqnarray}\label{eq: BenjiALT2bis}
\hat{\rho}_{\infty,t}^{(j)}=S_{j}(t)\hat{\rho}_{0}^{j}+
\(-\frac{i}{\var}\)\int_{0}^{t}\ud t_1\  S_j(t-t_1)C_{j,j+1}^Q\   
\hat{\rho}_{\infty,t_1}^{(j+1)}.
\end{eqnarray}
where $S_j$ is the free evolution defined in Section 2.3, and the  
collision map $C_{j,j+1}^Q$ is now given by
\begin{eqnarray}\label{eq: quant12BIS}
&&C_{j,j+1}^Q\hat{\rho}_{N,t}^{(j)}=\lambda\sum_{k=1}^{j}\texttt{Tr}_{j+1}\left\{\[\frac{1}{\vert x_k-x_{j+1}\vert  
}
,\hat{\rho}_{N,t}^{(j+1)}\]\right\},
\end{eqnarray}
where $\lambda$ is a coupling constant that can be positive (as in the  
most interesting physical case: the attractive Coulomb interaction) or  
not (repulsive case).
Finally, one proves the uniqueness of the solution to (\ref{eq:  
BenjiALT2bis}). Although the proof of the compactness
and of the convergence also require several changes with respect to  
what we discussed in the previous paragraph,
the main difficulty one has to face when the bounded potential is  
replaced by the Coulomb interaction
is the proof of the uniqueness of the solution to the infinite  
hierarchy. The key idea introduced by
Erd\H{o}s and Yau in \cite{EY} was to restrict the class of densities  
for which uniqueness must be proven. In
Theorem \ref{teoSPOHN}, uniqueness is proven in the class of densities  
with $\texttt{Tr} |\hat{\rho}_t^{(j)}| \leq 1$ for all $j\geq 1$, and
all $t \in[0, T]$ (but the same argument works under the weaker  
assumption $Tr |\hat{\rho}_t^{(j)}
|\leq  C^j$, for some
constant $C < +\infty$). In \cite{EY}, in the case of a Coulomb  
potential the uniqueness of (\ref{eq: BenjiALT2bis}) has been proven  
in the (smaller) class of densities $\Gamma_{  
t}=\{\hat{\rho}_{t}^{(j)}\}_{j\geq 1}$ satisfying the a-priori bound
\begin{eqnarray}\label{eq: nuovaNORMA}
\texttt{Tr}\left\vert\(1-\Delta_1\)^{1/2}\dots \(1-\Delta_j\)^{1/2}  
\hat{\rho}_{t}^{(j)}\(1-\Delta_j\)^{1/2}\dots  
\(1-\Delta_1\)^{1/2}\right\vert\leq C^j
\end{eqnarray}
for all $j\geq 1$ and for all $t \in [0, T]$.

There is, of course, a price to pay in order to restrict the proof of  
the uniqueness to this class of
densities. In fact, to apply this uniqueness result to prove the  
convergence of the RDM to the $j$-fold product of solutions of the  
Hartree equation, one has to show that
an arbitrary limit point $\Gamma_{\infty,  
t}=\{\hat{\rho}_{\infty,t}^{(j)}\}_{j\geq 1}$ of the sequence of  
densities $\Gamma_{ N,t}=\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$  associated
with $\hat{\rho}_{N,t}$ satisfies the a-priori bound (\ref{eq:  
nuovaNORMA}). Due to the Coulomb singularity, this is actually not so
simple and requires an additional approximation argument and suitable  
energy estimates.\\

Anyway, even if in the Coulomb case strong technical tools are needed  
in proving the mean-field result (much more with respect to the  
bounded interaction case), it is not difficult to realize that the  
estimates ensuring the convergence are not uniform with respect to  
$\var$ and they fail if $\var\to 0$. To see this, for concreteness in  
the case $d=3$, we observe that the Coulomb potential is controlled in  
three dimensions by the Laplacian (by virtue of an operator inequality  
of Hardy type). This is the reason for considering the class of  
density matrices such that (\ref{eq: nuovaNORMA}) holds.
Roughly speaking, by (\ref{eq: nuovaNORMA}) it follows that,  
considering the operator $
\frac{1}{\vert x_i - x_k\vert} \hat{\rho}_{N,t}^{(j)}$ (appearing both  
in the BBGKY and in the Hartree hierarchy) multiplied in a suitable  
way by some operators $(C-\Delta_k)^{1/2}$ ($C>1$) and taking the  
trace, one obtains estimates which are uniform with respect to $N$.  
Such estimates are crucial in proving
the uniqueness of the solution of the Hartree hierarchy and even the  
convergence of the BBGKY hierarchy to the limiting one.
In particular, they provide the rate of convergence to the Hartree  
dynamics in terms of the number of particles $N$ and indipendently of  
$\var$. Nevertheless,
by looking at the explicit computations, we find a factor $1/\var$ in  
front of the interaction potential, thus we have again diverging  
estimates when $\var\to 0$.\\\\

 From now on, we will focus on the case of smooth pair interaction  
potential, primarily, because in this case both the quantum and the  
classical mean-field limit have been rigorously established, therefore  
it is quite reasonable to look at that situation in investigating the  
connection between mean-field limit and semiclassical approximation  
(which we are going to discuss in Chapter 3 and 4). On the other side,  
we will see that for our purposes we need to deal with a smooth  
potential (see Chapter 4).





\section{Mean-field limit VS Semiclassical Approximation}

In this chapter we discuss the problem of ''connecting'' mean-field  
limit and semiclassical approximation which, as we saw previously,  
emerges quite naturally from the analysis of the quantum mean-field  
limit results. If one wants to deal with the classical and quantum  
case simultaneously, it is natural to
work in the classical phase space by using the so called ''Wigner formalism''.

\subsection{The Wigner formulation}
\setcounter{equation}{0}
\def\theequation{4.1.\arabic{equation}}


By the Heisenberg uncertainty principle, it follows that it is not  
possible to determine simultaneously the position
and the momentum of a quantum particle, thus the concept of classical  
phase space density
does not generalize directly to quantum mechanics. Nevertheless one  
can define a substitute for
it, namely the Wigner transform. For any wave function $\psi\in  
L^2\(\R^{d}\)$ we define the Wigner transform
of $\psi$ as
\begin{equation}\label{eq: WignerUNA}
f^\var_\psi\left(x, v\right)=(2\pi)^{-d}\int_{\R^{d}} \ud y\  
e^{iy\cdot v}\overline{\psi}\left(x+\frac{\var  
y}{2}\right)\psi\left(x-\frac{\var y}{2}\right),
\end{equation}
and we still interpret it as ''quantum phase space density'' (see  
\cite{Wigner} ).
It is easy to check that $f^\var_\psi$ is always real but in general  
is not positive (thus it cannot be
the density of a positive measure - in coincidence with the Heisenberg  
principle). However, its
marginals reconstruct the position and momentum space densities, as  
the following formulas
can be easily checked:
\begin{equation}\label{eq: WignerUNAerd}
\int f^\var_\psi(x, v)dv = |\psi(x)|^2,\ \ \ \ \ \int f^\var_\psi(x,  
v)dx = |\hat{\psi}(v)|^2
\end{equation}
$\hat{\psi}(v)$ being the Fourier transform of $\psi$, namely, by  
integrating versus the velocity variable we obtain the quantum spatial  
probability density and by integrating with respect to the position  
variable we find the velocity (or momentum) probability density.
In particular
\begin{equation}\label{eq: WignerUNAerd1}
\int f^\var_\psi(x, v)dvdx = 1
\end{equation}
for normalized wave functions.
More generally, if $J(x, v)$ is a classical phase space observable,  
the scalar product
\begin{equation}\label{eq: WignerUNAerd2}
\left\langle J, f^\var_\psi\right\rangle=\int J(x,v)f^\var_\psi(x, v)dvdx
\end{equation}
can be interpreted as the expected value of $J$ in state described by  
$\psi$. Recall that ''honest'' quantum
mechanical
observables are self-adjoint operators $O$ on $L^2(\R^d)$ and their  
expected value is
given by
\begin{equation}\label{eq: WignerUNAerd3}
\left\langle O\right\rangle_\psi= \int \bar{\psi}(x)\(O\psi\)(x)\ud x
\end{equation}
For a large class of observables there is a natural relation between  
observables $O$ and their
phase space representations (called symbols) that are functions on the  
phase space like $J(x, v)$.
For example, if $J$ depends only on $x$ or only on $v$, then the  
corresponding operator is just the
standard quantization, i.e.
\begin{equation}\label{eq: WignerUNAerd4}
\int J(x)f^\var_\psi(x, v)dxdv = \left\langle \psi,J\psi\right\rangle
\end{equation}
where $J$ is a multiplication operator on the right hand side,
\begin{equation}\label{eq: WignerUNAerd5}
\int J(v)f^\var_\psi (x, v)dxdv = \left\langle \psi,J(-i\var  
\nabla)\psi\right\rangle
\end{equation}
and similar relations hold for the Weyl quantization of any symbol $J(x, v)$.
We also remark that the map $\psi\to f^\var_\psi$ is invertible, i.e.  
one can fully reconstruct the
wave function from its Wigner transform. On the other hand, not every  
real function of two
variables $(x, v)$ is the Wigner transform of some wave function.\\

The correspondence between wave functions and their Wigner transform  
can be easily rephrased for density matrices. Indeed, if  
$\hat{\rho}=\vert\psi><\psi\vert$ for some $\psi\in L^2\(\R^d\)$, then  
formula (\ref{eq: WignerUNA}) can be rewritten as
\begin{equation}\label{eq: WignerUNAconRHO}
f^\var_\rho\left(x, v\right)=(2\pi)^{-d}\int_{\R^{3}} \ud y\  
e^{iy\cdot v}\rho\left(x+\frac{\var y}{2}, x-\frac{\var y}{2}\right),
\end{equation}
where $\rho(x,y)=\bar{\psi}(x)\psi(y)$ is the integral kernel of $\hat{\rho}$.
Furthermore, formula (\ref{eq: WignerUNAconRHO}) holds for any density  
matrix $\hat{\rho}\in \mathscr{L}^1\(L^2\(\R^d\)\)$, even for those  
which are associated with mixed states and (\ref{eq: WignerUNAerd1})  
holds because of positivity and trace norm normalization of the  
density matrix. Vice versa, starting from a quantum system whose state  
is described by a Wigner function $f^\var\left(x, v\right)$, it is  
possible to compute the corresponding density matrix (actually, its  
integral kernel) by the Weyl quantization rule
\begin{equation}\label{eq: Weyl}
\rho_{f^\var}\left(x, y\right)=(2\pi)^{-d}\int_{\R^{d}} \ud v\  
e^{i\frac{v}{\var}\cdot (x-y)}f^\var\left(\frac{x+ y}{2}, v\right).
\end{equation}
Therefore the Wigner transform and the Weyl quantization rule provide  
an invertible map $\hat{\rho}\leftrightarrow f^\var_\rho$ between  
density matrices and Wigner functions and it is simple to check that
\begin{equation}\label{eq: propDENSisometry}
\left\|\rho\right\|_{L^2\(\R^d\times\R^d\)}=  
\left\|f^\var_{\rho}\right\|_{L^2\(\R^d\times\R^d\)}.
\end{equation}
This is particularly meaningful because for any density matrix  
$\hat{\rho}$ we have
\begin{equation}\label{eq: propDENS1}
\hat{\rho}\geq 0,\ \hat{\rho}\in \mathscr{L}^1\(L^2\(\R^d\)\),\  
\text{with}\ \  
\left\|\hat{\rho}\right\|_{\mathscr{L}^1\(L^2\(\R^d\)\)}=1\Rightarrow\left\|\hat{\rho}\right\|_{\mathscr{L}^2\(L^2\(\R^d\)\)}\leq  
1,
\end{equation}
where $\mathscr{L}^2\(L^2\(\R^d\)\)$ is the Hilbert space of  
Hilbert-Schmidt operators on $L^2\(\R^d\)$ and for any operator  
$\Gamma\in\mathscr{L}^2\(L^2\(\R^d\)\)$ with kernel  
$\gamma=\gamma(x,y)$ we find
\begin{equation}\label{eq: propDENS2}
\left\|\Gamma\right\|_{\mathscr{L}^2\(L^2\(\R^d\)\)}=\left\|\gamma\right\|_{L^2\(\R^d\times\R^d\)}.
\end{equation}
Therefore by (\ref{eq: propDENS1}), (\ref{eq: propDENS2}) and by  
(\ref{eq: propDENSisometry}) it follows that
\begin{eqnarray}\label{eq: propDENS3}
&&\left\|\rho\right\|_{L^2\(\R^d\times\R^d\)}\leq 1\ \ \forall\ \  
\text{density matrix}\ \ \hat{\rho}\ \Rightarrow\ \  
\left\|f^\var\right\|_{L^2\(\R^d\times\R^d\)}\leq 1
\ \ \forall\ \ \text{Wigner function}\ \ f\nonumber\\
&&
\end{eqnarray}
\newpage
\begin{rem} \label{remW.1}

\textnormal{By (\ref{eq: Weyl}) it follows that one can fully  
reconstruct a density matrix from its Wigner transform but, in  
general, by knowing the Wigner function associated with the state of a  
quantum system it is not possible to reconstruct such a state in the  
wave function picture. More precisely, if we know that the system is  
in a \emph{pure state} and we know that it is described by a certain  
Wigner function $f^\var$, we can reconstruct the density matrix  
$\hat{\rho}$ which will be given by $\hat{\rho}=\vert \psi><\psi\vert$  
for some $L^2$-function $\psi$. On the contrary, if the system is in a  
\emph{mixed state}, by knowing the Wigner function we can only  
reconstruct the density matrix but there is no way to know which are  
the wave functions ''composing'' it}.
\end{rem}


\begin{rem}\label{remW.2}

\textnormal{The correspondence between density matrices and Wigner  
functions is quite useful but one has to be careful in using that. In  
fact, by considering a density matrix $\hat{\rho}$ one can compute its  
Wigner transform $f^\var_{\rho}$ and it will be for sure a real  
function on the classical phase space with the properties specified  
above. On the contrary, a real function on the classical phase space  
does not correspond necessarily to an admissible quantum state,  
namely, it is not necessarily the Wigner transform of a density  
matrix}.\\
\end{rem}

Let us consider a density matrix $\hat{\rho}^0\in  
\mathscr{L}^1\(L^2\(\R^d\)\)$ representing the initial state of a  
system whose Hamiltonian $H$ is
\begin{eqnarray}\label{eq: propDENS4ante}
&& H=-\frac{\var^2}{2}\Delta_x + U(x)
\end{eqnarray}
and the potential $U$ is such that $H$ is a self-adjoint operator on  
$L^2\(\R^d\)$. We know that the time evolution for the density matrix  
$\hat{\rho}^0$ is determined by
\begin{eqnarray}\label{eq: propDENS4}
&&i\var \pa_t \hat{\rho}^t=[H,\hat{\rho}^t],
\end{eqnarray}
and it is easy to check that it preserves the Hilbert-Schmidt norm of  
$\hat{\rho}^0$, namely the $L^2$-norm of the kernel $\rho^0$. Thus, by  
looking at the initial Wigner function $f^\var_{\rho^0}(x,v)$ ($x,v\in  
\R^d$) and at the time-evolved $f_t^\var(x,v)=f^\var_{\rho^t}(x,v)$,  
the $L^2$-norm has to be also preserved in time (by (\ref{eq:  
propDENSisometry}). We can verify this property by looking at the  
equation solved by $f^\var_t$. By applying the Wigner transform  
defined in (\ref{eq: WignerUNA}) to (\ref{eq: propDENS4}), we find the  
equation
\begin{eqnarray}\label{eq: propDENS5}
&&\(\pa_t +v\cdot\nabla_x \)f^\var_t=T^\var f^\var_t,
\end{eqnarray}
where
\begin{eqnarray}\label{eq: propDENS6}
&&\(T^\var f^\var_t\)(x,v)=i\int_{-1/2}^{1/2}\ud \lambda\int \ud k  
\hat{U}(k) e^{i\ k\cdot x}\(k\cdot \nabla_v\)f^\var_t(x,v+\var\lambda  
k),
\end{eqnarray}
and we denoted by $\hat{U}$ the Fourier transforms of $U$, namely:
\begin{equation}\label{eq: fourierU}
\hat{U}(k)=\int_{\R^d}\ud x\ e^{-i\ k\cdot x} U(x).
\end{equation}
By noting that both $v\cdot\nabla_x$ and $T^\var$ are skewsymmetric  
operators and reminding that $f_t^\var(x,v)\in \R$ for any $t$, we find
\begin{eqnarray}\label{eq: propDENS8}
&&\frac{1}{2}\frac{d}{d  
t}\left\|f^\var_t\right\|_{L^2\(\R^d\times\R^d\)}^2=\(f_t^\var,\pa_t  
f_t^\var\)=\(f_t^\var,-v\cdot\nabla_x f_t^\var\)+\(f_t^\var,T^\var  
f_t^\var\)=0,\nonumber\\
&&
\end{eqnarray}
namely the $L^2$-norm is conserved. It can be also proved that  
$H_s$-estimates hold for (\ref{eq: propDENS5})  
($H_s(\R^{3N}\times\R^{3N})$ being the Sobolev space  
$W^{s,2}(\R^{3N}\times\R^{3N})$) by assuming the potential $\phi$ to  
be sufficiently smooth (see for example \cite{PULVIRENTI}) in the  
sense that the $H_s$-norm of the time evolved Wigner function is  
controlled by the $H_s$-norm of the initial datum, up to a constant  
depending on time (but finite for any time interval) and of a suitable  
norm of the potential.\\

Equation (\ref{eq: propDENS5}) looks like a classical kinetic equation  
but the crucial facts are that $f^\var_t$ is not a probability density  
in the phase space $\R^d\times\R^d$ and we have to deal with a  
pseudodifferential operator instead of a differential one as it is  
usual in kinetic theory. It is immediate to check that
\begin{eqnarray}\label{eq: propDENS7}
&&\int \ud x\int \ud v\ f^\var_t(x,v)=\int \ud x \rho^\var_t(x)=1 \ \  
\forall\ t>0,
\end{eqnarray}
with
\begin{eqnarray}\label{eq: propDENS7bis}
\rho^\var_t(x)=\int \ud v\ f^\var_t(x,v), \ \ \rho^\var_t\geq 0\ \  
\forall\ t,\ \ \ \rho^\var_t(x)\ud x :=\text{spatial probability  
distribution, }
\end{eqnarray}
and (\ref{eq: propDENS7}) follows from conservation of ''mass'' and  
from the fact that, because of the trace norm normalization of  
$\hat{\rho}^0$, we have $\int \ud x\int \ud v\  
f^\var_{\rho^0}(x,v)=\texttt{Tr}\hat{\rho}^0=1$.\\\\

\subsection{The Mean-Field system in the Wigner formalism}
\setcounter{equation}{0}
\def\theequation{4.2.\arabic{equation}}


The Wigner formalism introduced in the previous section is an  
alternative way of describing the state and the dynamics of a quantum  
system and it is precisely equivalent
to the density matrix (or Heisenberg) description, and, for pure  
states, to the wave function (or Schr\"odinger) picture. As we have  
observed, the advantage in using the Wigner formalism in looking at  
semiclassical approximation of quantum systems is that Wigner  
functions ''live'' on the classical phase space and for suitable  
''semiclassical'' quantum states the Wigner functions can have a well  
defined limit when $\var\to 0$ (see for example \cite{Paul}).\\

Thus, in the perspective of looking at the semiclassical limit, we  
rephrase the quantum mean-field model discussed in Chapter 2 by using  
the Wigner formulation.

By applying the Wigner transform (\ref{eq: WignerUNAconRHO}) to the  
Heisenberg equation (\ref{eq: quant8}) we find
\begin{equation}\label{eq: WigLIOUhartreeNsec3}
\left(\pa_t+V_N\cdot \nabla_{X_N}\right)W_N^{\var}(t)=T_N^{\var}W_N^{\var}(t),
\end{equation}
where $W_N^{\var}(t):=W_N^{\var}(X_N,V_N;t)$ is the Wigner function  
describing the state of the system (namely, the Wigner transform of  
the density matrix $\hat{\rho}_{N,t}$),
$$
X_N=(x_1,\dots,x_N)\in \R^{3N}, \ \ V_N=(v_1,\dots,v_N)\in \R^{3N},
$$
and the pair $Z_N:=(X_N, V_N)$ denotes the generic point in the  
classical $N$-particle phase space. Moreover,
\begin{eqnarray}\label{eq: opT_N}
&&\(T_N^{\var}W_N^{\var}\)(Z_N)=\frac{i}{(2\pi)^{3N}}\int_{-1/2}^{1/2}\ud  
\lambda\int \ud K_N \hat{U}^Q(K_N) e^{iK_N\cdot V_N}\left(K_N\cdot  
\nabla_{V_N}\right)W_N^{\var}(X_N,V_N+\lambda \var K_N),\nonumber\\
&&
\end{eqnarray}
where $K_N=(k_1,\dots,k_N)\in \R^{3N}$, $U^Q$ is the (mean-field)  
interaction potential (\ref{eq: quant1}), and $\hat{U}^Q$ is the  
Fourier transform of $U^Q$, namely:
\begin{equation}\label{eq: fourierUQ}
  \hat{U}^Q(k)=\int_{\R^{3N}}\ud X_N\ e^{-i\ K_N\cdot X_N} \ U^Q(X_N).
\end{equation}
We note that (\ref{eq: WigLIOUhartreeNsec3}) is the analogue of the  
classical Liouville equation (\ref{eq: class4}) and, roughly speaking,  
by setting ''$\var=0$'' in (\ref{eq: opT_N}) we obtain precisely the  
Liouville operator appearing in (\ref{eq: class4}). From now on, we  
will refer to (\ref{eq: WigLIOUhartreeNsec3}) as ''$N$-particle  
Wigner-Liouville equation''.\\

We remind that we are dealing with undistinguishable particles, then  
we consider $N$-particle Wigner functions $W_N$ which are invariant in  
the exchange of particle names, namely
\begin{equation}\label{eq: quant3NEWwig}
W_N(x_{\pi(1)},\dots,x_{\pi(N)},  
v_{\pi(1)},\dots,v_{\pi(N)})=W_N(x_1,\dots,x_N, v_1,\dots,v_N),
\end{equation}
for every permutation $\pi$ acting on $1,\dots,N$. It is easy to  
verify that this property is preserved by the evolution (\ref{eq:  
WigLIOUhartreeNsec3}).

\subsubsection{The Wigner BBGKY hierarchy}
For any fixed $j$ we introduce the $j$-particle ''marginals'':
\begin{equation}\label{eq: marginaliJ}
W_{N,j}^{\var}(t):=W_{N,j}^{\var}(X_j,  
V_j;t)=\int_{\R^{3(N-j)}\times\R^{3(N-j)}} \ud X_{N-j}\ud  
V_{N-j}W_{N}^{\var}(X_j, X_{N-j}, V_j, V_{N-j};t).
\end{equation}
It is easy to check that $\{W_{N,j}^{\var}(t)\}_{j=1}^{N}$ are  
precisely the Wigner transforms of the RDM  
$\{\hat{\rho}_{N,t}^{(j)}\}_{j=1}^N$.
Furthermore, by integrating the Wigner-Liouville equation (\ref{eq:  
WigLIOUhartreeNsec3}) with respect to the last $N-j$ variables we find  
the following sequence of equations:
\begin{eqnarray}\label{eq: hierarchyN}
&&\left(\pa_t +V_j\cdot  
\nabla_{X_j}\right)W_{N,j}^{\var}(t)=T_{N,j}^{\var}W_{N,j}^{\var}(t)+\left(\frac{N-j}{N}\right)C_{j,j+1}^{\var}W_{N,j+1}^{\var}(t),\ \ j=1,2,\dots,  
N,\nonumber\\
&&\\
&&\text{with}\ \ \ W_{N,N}^{\var}(t)=W_N^{\var}(t)\ \ \text{and}\ \  
C_{N,N+1}^{\var}\equiv 0,\nonumber
\end{eqnarray}
which is precisely the BBGKY hierarchy (\ref{eq: quant10}) rephrased  
in the Wigner formalism and it can be seen as the quantum analogue of  
the classical BBGKY hierarchy (\ref{eq: class14}).\\
The operator $T_j^{\var}$ (for a fixed $j$), describing the  
interaction of the first $j$ particles,
is given by
\begin{eqnarray}\label{eq: opT_j}
&&\left(T_{N,j}^{\var}W_{N,j}^{\var}\right)(X_j, V_j)=\nonumber\\
&&=\frac{i(2\pi)^{-3N}}{N}\sum_{l\neq r}^{j}\int_{-1/2}^{1/2}\ud  
\lambda \int_{\R^3}\ud k\  \hat{\phi}(k)e^{ik\cdot (x_l-x_r)}(k\cdot  
\nabla_{v_l})W_{N,j}^{\var}(X_j,V_{l-1},v_l+\lambda \var  
k,V_{j-l}),\nonumber\\
&&
\end{eqnarray}
while the collision operator $C_{j,j+1}^{\var}$ is
\begin{eqnarray}\label{eq: opC_j+1}
&&\left(C_{j,j+1}^{\var}W_{N,j+1}^{\var}\right)(X_j, V_j)=\nonumber\\
&&=i(2\pi)^{-3N}\sum_{l=1}^{j}\int_{-1/2}^{1/2}\ud \lambda  
\int_{\R^3}\ud k \ \hat{\phi}(k)\int_{\R^3\times\R^3}\ud x_{j+1}\ud  
v_{j+1}\ e^{ik\cdot( x_l-x_{j+1})}\nonumber\\
&&\qquad\qquad\qquad\qquad(k\cdot  
\nabla_{v_l})W_{N,j+1}^{\var}(X_j,x_{j+1},V_{l-1},v_l+\lambda \var  
k,V_{j-l},v_{j+1}),\nonumber\\
&&
\end{eqnarray}
and in (\ref{eq: opT_j}) and (\ref{eq: opC_j+1}) we denoted by  
$\hat{\phi}$ the Fourier transform of the pair interaction potential  
$\phi$, namely:
\begin{equation}\label{eq: fourierPERphi}
\hat{\phi}(k)=\int_{\R^3}\ud x\ e^{-i\ k\cdot x} \phi(x).
\end{equation}

By using (iteratively) the Duhamel formula, the solution  
$W_{N,j}^{\var}(t)$ of the equations (\ref{eq: hierarchyN})
with initial datum $W_{N,j}^{\var}(0)$
can be written as
\begin{eqnarray}\label{eq: expFORwig}
W_{N,j}^{\var}(t)&&=\Phi_{j}^{(N)}(t)W_{N,j}^{\var}(0)+\nonumber\\
&&+\sum_{n=1}^{N-j}\int_{0\leq t_n\leq \dots \leq t_1\leq t}\ud  
t_n\dots\ud t_1\  
\Phi_j^{(N)}(t-t_1)\(\frac{N-j}{N}\)C_{j,j+1}^\var\dots\nonumber\\
&&\qquad \quad\dots \(\frac{N-j-n+1}{N}\)C_{j+n-1,j+n}^\var\  
\Phi_{j+n}^{(N)}(t_n)W_{N,j+n}^{\var}(0).
\end{eqnarray}
where $\Phi_{j}^{(N)}$ is the flow associated with the $j$-particle  
operator $-V_j\cdot \nabla_{X_j}+T_{N,j}^\var$.

\subsection{The Hartree dynamics in the Wigner formalism}
\setcounter{equation}{0}
\def\theequation{4.3.\arabic{equation}}

This section is devoted to the description of the Hartree dynamics  
discussed in Chapter 2 in terms of the Wigner formalism. \\

By applying the Wigner transform (\ref{eq: WignerUNAconRHO}) to the  
Hartree equation (in the Heisenberg form) (\ref{eq: quant15}) we find
\begin{equation}\label{eq: WigLIOUhartree}
\left(\pa_t+v\cdot \nabla_x\right)f^{\var}(t)=T_{f^{\var}}^{\var}f^{\var}(t),
\end{equation}
where $f^{\var}(t):=f^{\var}(x, v;t)$ is the Wigner function  
describing the state of the system (namely, the Wigner transform of  
the density matrix $\hat{\rho}_{t}$ solving the Hartree equation  
(\ref{eq: quant15})).

For any fixed $g$, the operator $T^\var_g$ acts as follows:
\begin{equation}\label{eq: operatorT}
T_g^{\var} f^{\var}(x, v)=(2\pi)^{-3}i\int_{-1/2}^{1/2}\ud  
\lambda\int_{\R^3}\ud k \hat{\phi}(k)\hat{\rho}_g(k) e^{i \ k\cdot  
x}(k\cdot \nabla_v)f^{\var}(x, v+\var\lambda k),
\end{equation}
where
\begin{equation}\label{eq: rho}
\rho_g(x)=\int_{\R^3}\ud v \ g(x, v),
\end{equation}
and $\hat{\rho}_g$ is the Fourier transform of $\rho_g$, namely:
\begin{equation}\label{eq: fourier}
\hat{\rho}_g(k)=\int_{\R^3}\ud x\ e^{-i\ k\cdot x} \rho_g(x).
\end{equation}

We observe that equation (\ref{eq: WigLIOUhartree}) is nonlinear \ (as  
we can see by (\ref{eq: operatorT}) replacing $g$ with $f^{\var}$ )  
because it arises from a nonlinear Heisenberg equation. Thus in the  
following we will refer to (\ref{eq: WigLIOUhartree}) as ''(Hartree)  
nonlinear Wigner-Liouville equation''.
Furthermore, we note that (\ref{eq: WigLIOUhartree}) is the analogue  
of the classical Vlasov equation (\ref{eq: Vlasov}) and, roughly  
speaking, by setting ''$\var=0$'' in (\ref{eq: operatorT}) we obtain  
precisely the Vlasov operator appearing in (\ref{eq: Vlasov}).

By the analysis we did in the previous section, we know that the  
linear equation (\ref{eq: propDENS5}) preserves the $L^2$-norm (see  
(\ref{eq: propDENS8})). The same holds for the nonlinear equation  
(\ref{eq: WigLIOUhartree}) and, by assuming the potential to be  
sufficiently smooth, it can be proved that the $H_s$-norm is  
controlled for any $s>0$. Indeed we have the following

\begin{prop}\label{PROPH_s}
Let $f^\var(t)$ be the solution of the nonlinear Wigner-Liouville  
equation (\ref{eq: WigLIOUhartree})
whit initial datum $f_0^\var\in H_s\(\R^3\times\R^3\)$ with $s\in \mathbb{N}$.
Assuming the potential $\phi$ to satisfy
\begin{eqnarray}\label{eq: semicEXP2}
&&\int \ud k \ \hat{\phi}(k)\vert k\vert^n<+\infty \ \ \ \forall\  
n=1,2,\dots,s
\end{eqnarray}
we find that
\begin{eqnarray}\label{eq: semicEXP3}
&&
\left\|f^\var(t)\right\|_{H_s\(\R^3\times\R^3\)}\leq  
e^{Ct}\left\|f_0^\var\right\|_{H_s\(\R^3\times\R^3\)},
\end{eqnarray}
where $C$ is a positive constant depending on $s$ and on $\phi$ but  
not on $\var$. For $s=0$ we have $C=0$ and $(3.3)$
becomes an equality (conservation of the $L^2$-norm).
\end{prop}
\begin{dem}
For any multi index $\alpha=\{\alpha_1,\alpha_2,\alpha_3\}$, we use  
the standard notation
\begin{eqnarray}\label{eq: semicEXP4}
&& D^\alpha_x=\frac{\pa^{\vert  
\alpha\vert}}{\pa^{\alpha_1}x_1\pa^{\alpha_2}x_2\pa^{\alpha_3}x_3},
\end{eqnarray}
where $\vert \alpha\vert=\alpha_1+\alpha_2+\alpha_3$. Analogously we set
\begin{eqnarray}\label{eq: semicEXP5}
&& D^\alpha_v=\frac{\pa^{\vert  
\alpha\vert}}{\pa^{\alpha_1}v_1\pa^{\alpha_2}v_2\pa^{\alpha_3}v_3}.
\end{eqnarray}
It is well known that $H_s\(\R^3\times\R^3\)$ equipped with the scalar product
\begin{eqnarray}\label{eq: semicEXP6}
&& \(f,g\)_{s}=\sum_{\substack{\alpha,\beta\in \mathbb{N}:\\ \vert  
\alpha\vert+\vert \beta\vert\leq s}}\(D^\alpha_v D^\beta_x  
f,D^\alpha_v D^\beta_x g\)_{L^2\(\R^3\times\R^3\)}
\end{eqnarray}
is an Hilbert space and the corresponding norm is  
$\left\|g\right\|_s:=\left\|g\right\|_{H_s\(\R^3\times\R^3\)}=\sqrt{\(g,g\)_{s}}$.
In order to estimate $\left\|f^\var(t)\right\|_s$, we compute the time  
derivative $\pa_t D^\alpha_v D^\beta_x f^\var(t)$ with $\vert  
\alpha\vert+\vert \beta\vert\leq s$. By (\ref{eq: WigLIOUhartree}) we  
find:
\begin{eqnarray}\label{eq: semicEXP7}
&& \pa_t D^\alpha_v D^\beta_x f^\var(t)=D^\alpha_v D^\beta_x\(-v\cdot  
\nabla_x +T^\var_{f^\var}\)f^\var(t)=\(-v\cdot \nabla_x  
+T^\var_{f^\var}\)D^\alpha_v D^\beta_x f^\var(t)+\nonumber\\
&&\nonumber\\
&&+ \sum_{\substack{\alpha'<\alpha:\\ \vert  
\alpha'\vert=1}}C_{\alpha,\alpha'}D^{\alpha'}_v v\cdot \nabla_x  
D^{\alpha-\alpha'}_v D^\beta_x f^\var(t)+\nonumber\\
&&+\sum_{\substack{\beta'<\beta:\\ \vert \beta'\vert\geq 1}}\frac{i\  
C_{\beta,\beta'}}{(2\pi)^{3}}\int_{-1/2}^{1/2}\ud \lambda\int \ud k \  
\hat{\phi}(k)\hat{\rho}^\var(k;t)D^{\beta'}_x e^{i\ k\cdot  
x}\(k\cdot\nabla_v\) D^\alpha_v D^{\beta-\beta'}_x  
f^\var(x,v+\var\lambda k;t)\nonumber\\
&&
\end{eqnarray}
where $C_{\alpha,\alpha'}$, $C_{\beta,\beta'}$ are suitable  
combinatorial coefficients, $\alpha'<\alpha$, $\beta'<\beta$ mean  
$\alpha'_j<\alpha_j$,  $\beta'_j<\beta_j$ (for $j=1,2,3$) respectively  
and finally $\alpha-\alpha'=\{\alpha_j - \alpha'_j\}_{j=1}^{3}$,  
$\beta-\beta'=\{\beta_j - \beta'_j\}_{j=1}^{3}$.

We observe now that, by virtue of the antisymmetry  of the operators  
$v\cdot\nabla_x$ and $T^\var_g$ (for any function $g$), we have
\begin{eqnarray}\label{eq: semicEXP8}
&& \(h,v\cdot \nabla_x h\)_{L^2\(\R^3\times\R^3\)}=\(h, T^\var_g  
h\)_{L^2\(\R^3\times\R^3\)}=0,
\end{eqnarray}
for any $g$ and for each $h$ smooth enough. Moreover, reminding that  
$f^\var(t)\in \R$ for all $t$, if $s>0$, for any $\alpha,\beta:\  
0<\vert \alpha\vert+\vert \beta\vert\leq s$, we find:
\begin{eqnarray}\label{eq: semicEXP9}
&& \frac{1}{2}\frac{d}{d t}\(D^\alpha_v D^{\beta}_x  
f^\var(t),D^\alpha_v D^{\beta}_x  
f^\var(t)\)_{L^2\(\R^3\times\R^3\)}=\(D^\alpha_v D^{\beta}_x  
f^\var(t),\pa_t D^\alpha_v D^{\beta}_x  
f^\var(t)\)_{L^2\(\R^3\times\R^3\)},\nonumber\\
&&
\end{eqnarray}
which for $s=0$ (namely $\vert \alpha\vert=\vert \beta\vert=0$) becomes:
\begin{eqnarray}\label{eq: semicEXP9s=0}
&& \frac{1}{2}\frac{d}{d t}\( f^\var(t),  
f^\var(t)\)_{L^2\(\R^3\times\R^3\)}=\( f^\var(t),\pa_t  
f^\var(t)\)_{L^2\(\R^3\times\R^3\)}.
\end{eqnarray}

Inserting (\ref{eq: semicEXP7}) in the right hand side of (\ref{eq:  
semicEXP9s=0}), by virtue of (\ref{eq: semicEXP8}) we find:
\begin{eqnarray}\label{eq: semicCONSl^2}
&& \frac{1}{2}\frac{d}{d t}\( f^\var(t),  
f^\var(t)\)_{L^2\(\R^3\times\R^3\)}=\frac{d}{d t}\left\|  
f^\var(t)\right\|_{L^2\(\R^3\times\R^3\)}=0,
\end{eqnarray}
namely, the $L^2$-norm is conserved.

On the contrary, for $s>0$, we insert (\ref{eq: semicEXP7}) in the  
right hand side of (\ref{eq: semicEXP9}). We find the term involving  
$D^\alpha_v D^{\beta}_x f^\var(t)$ does not give any contribution by  
virtue of (\ref{eq: semicEXP8}). Thus, by using the shorthand notation  
$\(\cdot,\cdot\)_{L^2\(\R^3\times\R^3\)}=\(\cdot,\cdot\)_{L^2}$, we  
obtain
\begin{eqnarray}\label{eq: semicEXP10}
&& \frac{1}{2}\frac{d}{d t}\(D^\alpha_v D^{\beta}_x  
f^\var(t),D^\alpha_v D^{\beta}_x  
f^\var(t)\)_{L^2}=\sum_{\substack{\alpha'<\alpha:\\ \vert  
\alpha'\vert=1}}C_{\alpha,\alpha'}\(D^\alpha_v D^{\beta}_x  
f^\var(t),D^{\alpha'}_v v\cdot \nabla_x D^{\alpha-\alpha'}_v D^\beta_x  
f^\var(t)\)_{L^2}+\nonumber\\
&&+\sum_{\substack{\beta'<\beta:\\ \vert \beta'\vert\geq 1}}\frac{i\  
C_{\beta,\beta'}}{(2\pi)^{3}}\int_{-1/2}^{1/2}\ud \lambda\int \ud k \  
\hat{\phi}(k)\hat{\rho}^\var(k;t)\(D^\alpha_v D^{\beta}_x  
f^\var(t),D^{\beta'}_x e^{i\ k\cdot x}\(k\cdot\nabla_v\) D^\alpha_v  
D^{\beta-\beta'}_x f^\var(x,v+\var\lambda k;t)\)_{L^2}.\nonumber\\
&&
\end{eqnarray}


We note that the first term on the right hand side of (\ref{eq:  
semicEXP10}) is absent when $\vert \alpha\vert=0$. On the contrary, if  
$\vert \alpha\vert\geq 1$, by using the Schwartz inequality we obtain:
\begin{eqnarray}\label{eq: semicEXP11}
&& \(D^\alpha_v D^{\beta}_x f^\var(t),D^{\alpha'}_v v\cdot \nabla_x  
D^{\alpha-\alpha'}_v D^\beta_x f^\var(t)\)_{L^2}\leq C  
\left\|f^\var(t)\right\|_{s}^2
\end{eqnarray}
because $\vert \alpha\vert-\vert \alpha'\vert+\vert \beta\vert+1=\vert  
\alpha\vert+\vert \beta\vert\leq s$. Analogously, we find that the  
second term in the right hand side of (\ref{eq: semicEXP10}) is  
estimated by
\begin{eqnarray}\label{eq: semicEXP12}
&&\int \ud k \ \hat{\phi}(k)\hat{\rho}^\var(k;t)\vert k\vert^{\beta'  
+1}\left\| D^{\alpha}_v D^{\beta}_x  
f^\var(t)\right\|_{L^2}\left\|\nabla_v D^{\alpha}_v D^{\beta-\beta'}_x  
f^\var(t)\right\|_{L^2}\leq\nonumber\\
&&\leq \int \ud k \ \hat{\phi}(k)\hat{\rho}^\var(k;t)\vert  
k\vert^{\beta' +1}\left\|f^\var(t)\right\|_{s}^2,
\end{eqnarray}
where we used that $\vert \alpha\vert+1+\vert \beta\vert-\vert  
\beta'\vert\leq s$. Now we remind that $\rho^\var(x;t)$ is the spatial  
density associated with the Wigner function $f^\var(t)$, namely  
$\rho^\var(x;t)\geq 0$ for all $x$ and $t$,
\begin{eqnarray}\label{eq: semicEXP13}
&&\rho^\var(x;t)=\int \ud v f^\var(x,v;t),
\end{eqnarray}
then the $L^1$-norm of $\rho^\var(t)\vert_{t=0}$ is preserved by the  
evolution and it is equal to one. Thus, the Fourier transform  
$\hat{\rho}^\var(t)$ is in $L^\infty(\R^3\times\R^3)$ for all $t$ and  
we find:
\begin{eqnarray}\label{eq: semicEXP14}
&& \int \ud k \ \hat{\phi}(k)\hat{\rho}^\var(k;t)\vert k\vert^{\beta'  
+1}\leq  
\left\|\hat{\rho}^\var(t)\right\|_{L^\infty(\R^3\times\R^3)}\int \ud k  
\ \hat{\phi}(k)\vert k\vert^{\beta' +1}<+\infty,
\end{eqnarray}
by virtue of the assumption (\ref{eq: semicEXP2}) on the pair  
interaction potential $\phi$ (we remind that $1\leq \vert \beta'\vert<  
\vert \beta\vert\leq s$). Finally, by (\ref{eq: semicEXP10}),  
(\ref{eq: semicEXP11}), (\ref{eq: semicEXP12}) and (\ref{eq:  
semicEXP14})
, it follows that:
\begin{eqnarray}\label{eq: semicEXP15}
&& \frac{1}{2}\frac{d}{d t}\sum_{\substack{\alpha,\beta:\\\vert  
\alpha\vert+\vert \beta\vert\leq s}}\(D^\alpha_v D^{\beta}_x  
f^\var(t),D^\alpha_v D^{\beta}_x  
f^\var(t)\)_{L^2}=\frac{1}{2}\frac{d}{d  
t}\left\|f^\var(t)\right\|_{s}^2\leq C  
\left\|f^\var(t)\right\|_{s}^2,\ \ \ \forall\ t
\end{eqnarray}
$C$ depending on $\phi$ and $s$ but not on $\var$. We conclude  
straightforward by observing that inequality (\ref{eq: semicEXP15}) is  
equivalent to (\ref{eq: semicEXP3}).
\end{dem}

\subsubsection{The Wigner infinite hierarchy}
Let us consider the sequence $\{f_j^{\var}(t)\}_{j\geq 1}$, where  
$f_j^{\var}(t)=f_j^{\var}(X_j, V_j;t)$ is given by:
  \begin{eqnarray}\label{eq: factJ}
&&f_j^{\var}(X_j, V_j;t)=\prod_{k=1}^{j}f^{\var}(x_k,  
v_k;t)=\(f^{\var}\)^{\otimes j}(X_j, V_j;t)
\end{eqnarray}
and $f^{\var}(t)$ is the solution of the nonlinear Wigner-Liouville  
equation (\ref{eq: WigLIOUhartree}). By differentiating in time  
(\ref{eq: factJ}) we easily deduce the following (infinite) hierarchy  
of equations:
\begin{eqnarray}\label{eq: hierarchyINF}
&&\left(\pa_t +V_j\cdot  
\nabla_{X_j}\right)f_{j}^{\var}(t)=C_{j,j+1}^{\var}f_{j+1}^{\var}(t),
\end{eqnarray}
where the operator $C_{j,j+1}^{\var}$ is the same of (\ref{eq:  
opC_j+1}). This is precisely the Hartree hierarchy (\ref{eq: quant13})  
rephrased in the Wigner formalism and it can be seen as the quantum  
analogue of the Vlasov hierarchy (\ref{eq: class17}). Here we derived  
the Hartree hierarchy by considering the $j$-particle Wigner function  
(\ref{eq: factJ}) which is a product of solution of the nonlinear  
Wigner-Liouville equation (\ref{eq: WigLIOUhartree}). Conversely, as  
we observed in Chapter 2 for the Heisenberg formalism, by starting  
from the hierarchy (\ref{eq: hierarchyINF}) and assuming the solution  
to be factorized according to a one-particle time dependent Wigner  
function $f^\var(t)$, it turns out that $f^\var(t)$ has to solve  
equation (\ref{eq: WigLIOUhartree}). \\

By using (iteratively) the Duhamel formula, the solution  
$f_{j}^{\var}(t)$ of the equations (\ref{eq: hierarchyINF})
with initial datum $f_{j}^{\var}(0)$
can be written as
\begin{eqnarray}\label{eq: expFORwigINF}
f_{j}^{\var}(t)&&=\Phi_{j}(t)f_{j}^{\var}(0)+\nonumber\\
&&+\sum_{n=1}^{N-j}\int_{0\leq t_n\leq \dots \leq t_1\leq t}\ud  
t_n\dots\ud t_1\ \Phi_j(t-t_1)C_{j,j+1}^\var\dots C_{j+n-1,j+n}^\var\  
\Phi_{j+n}(t_n)f_{j+n}^{\var}(0).\nonumber\\
&&
\end{eqnarray}
where $\Phi_{j}$ is the flow associated with the $j$-particle operator  
$-V_j\cdot \nabla_{X_j}$, namely it is the free $j$-particle flow
\begin{eqnarray}\label{eq: freeFLOW}
\Phi_{j}(t)f_j^\var(X_j,V_j)=f_j^\var(X_j-V_j t,V_j).
\end{eqnarray}

\subsection{The Limit $N\to\infty$}
\setcounter{equation}{0}
\def\theequation{4.4.\arabic{equation}}


By the analysis done in the previous section, it is quite natural to  
rephrase the mean-field result discussed in Chapter 2 in the Wigner  
formalism. This will be the subject of this section and, here and in  
the sequel, we will always assume the interaction potential to be  
sufficiently smooth.

Thanks to Theorem \ref{teoSPOHN}, we know that, for bounded  
potentials, the sequence $\hat{\rho}_{N,t}^{(j)}$ of the RDM  
associated with the $N$-particle mean-field dynamics is converging in  
trace norm, as $N\to\infty$, to the $j$-fold product  
$\hat{\rho}_t^{\otimes j}$ of solutions of the Hartree equation. By  
reminding that the space $\mathscr{L}^1\(L^2\(\R^{3j}\)\)$ of trace  
class operators on $L^2\(\R^{3j}\)$ is a subspace of the space  
$\mathscr{L}^2\(L^2\(\R^{3j}\)\)$ of Hilbert-Schmidt operators on  
$L^2\(\R^{3j}\)$, it follows that:
\begin{eqnarray}\label{eq: convergenceHS}
&&\left\|\hat{\rho}_{N,t}^{(j)} - \hat{\rho}_t^{\otimes j}  
\right\|_{\mathscr{L}^2\(L^2\(\R^{3j}\)\)}
\leq \left\|\hat{\rho}_{N,t}^{(j)} - \hat{\rho}_t^{\otimes j}  
\right\|_{\mathscr{L}^1\(L^2\(\R^{3j}\)\)}
\rightarrow 0,\ \ \ \text{as}\ \ N\to\infty.\nonumber\\
&&
\end{eqnarray}
Therefore, by virtue of the equality (\ref{eq: propDENS2}) concerning  
the Hilbert-Schmidt norm and thanks to the property (\ref{eq:  
propDENSisometry}) of the Wigner function, we can conclude that
\begin{eqnarray}\label{eq: convergenceL2STRONG}
&&\left\| W_{N,j}^{\varepsilon}(t) - \(f^\varepsilon(t)\)^{\otimes j}  
\right\|_{L^2\(\R^{3j}\times \R^{3j}\)}
\leq \left\|\hat{\rho}_{N,t}^{(j)} - \hat{\rho}_t^{\otimes j}  
\right\|_{\mathscr{L}^1\(L^2\(\R^{3j}\)\)}
\rightarrow 0,\ \ \ \text{as}\ \ N\to\infty,\nonumber\\
&&
\end{eqnarray}
where $W_{N,j}^{\varepsilon}(t)$ are the time evolved Wigner marginals  
defined in (\ref{eq: marginaliJ}) and $f^\varepsilon(t)$ is the  
solution of the (Hartree) nonlinear Wigner equation (\ref{eq:  
WigLIOUhartree}). \\
Therefore, the mean-field theorem ensuring the convergence in trace  
norm of the RDM, guarantees also the $L^2$-strong convergence of the  
corresponding Wigner marginals. Nevertheless, by looking at (\ref{eq:  
convergenceL2STRONG}), it is clear that the error in the approximation  
for large $N$ is precisely the same we saw previously, and then, it is  
depending on $\var$ and diverging as $\var\to 0$.

It turns out that, in the perspective of obtaining estimates on the  
error in the mean-field approximation which are uniform with respect  
to $\var$ or, at least, which exhibit a less singular dependence on  
$\var$, a quite natural approach is to rephrase the whole mean-field  
result discussed in Chapter 2 (by assuming the interaction to be  
sufficiently smooth) in the Wigner formalism.\\

By looking at the Wigner BBGKY hierarchy (\ref{eq: hierarchyN}) we  
observe that the operator $T_{N,j}^\var$ is of size  
$O\(\frac{j^2}{N}\)$ while the operator $C_{j,j+1}^\var$ is $O(1)$  
with respect to $N$ and it is properly the same appearing in the  
infinite hierarchy (\ref{eq: hierarchyINF}). Therefore, in analogy to  
what we did in proving Theorem \ref{teoSPOHN} one expects that the  
flow $\Phi_j^{(N)}(t)$ appearing in (\ref{eq: expFORwig}) converges in  
a suitable sense to the free flow $\Phi_j(t)$ as $N\to\infty$ so that,  
this time by using the BBGKY hierarchy, one can prove that
\begin{eqnarray}\label{eq: convergenceWIG}
W_{N,j}^\var(t)\rightarrow f_j^\var(t),\ \ \ \text{as}\ \ N\to\infty,
\end{eqnarray}
in a sense to be made precise.
\\

In Chapters 1 and 2, to show the validity of propagation of chaos, we  
considered as initial datum for the $N$-particle dynamics the  
(bosonic) factorized state (\ref{eq: quant17bis}), or equivalently,  
(\ref{eq: quant17}). We observe that the Wigner transform  
$f^\var_\rho$ defined in (\ref{eq: WignerUNAconRHO}) is linear with  
respect to the density matrix (kernel) $\rho$, thus we find that the  
Wigner transform of the factorized state  
$\hat{\rho}_{N,0}=\hat{\rho}_0^{\otimes N}$ considered in Theorem  
\ref{teoSPOHN} is also factorized, namely
\begin{eqnarray}\label{eq: WigNinit}
&&W_{N}^{\var}(X_N,V_N)=\prod_{i=1}^{N}f_0^{\var}(x_i,v_i),
\end{eqnarray}
where $f_0^{\var}$ is the Wigner transform of $\hat{\rho}_0$. Moreover, being
$\hat{\rho}_0=\vert\psi_{0}><\psi_{0}\vert$, we find
\begin{eqnarray}\label{eq: ONEpartINDAT}
f_0^{\var}=f_{\rho_0}^{\var}\leftrightarrow\hat{\rho}_0 =\vert \psi_0  
 ><\psi_0 \vert
\end{eqnarray}
and
\begin{eqnarray}\label{eq: ONEpartINDATII}
\left\|f_0^{\var}\right\|_{L^2(\R^3\times\R^3)}=\left\|\rho_0\right\|_{L^2(\R^3\times\R^3)}=\left\|\psi_0\right\|_{L^2(\R^3)}^2=1.
\end{eqnarray}

By taking the $j$-particle marginal associated with  
$W_{N}^{\var}(X_N,V_N)$ we straightforward obtain
\begin{eqnarray}\label{eq: margINDAT}
&&W_{N,j}^{\var}(X_j,V_j)=\prod_{i=1}^{j}f_0^{\var}(x_i,v_i)=\(f_0^{\var}\)^{\otimes j}(X_j,  
V_j),
\end{eqnarray}
then, by (\ref{eq: expFORwig}), the solution of the equations  
(\ref{eq: hierarchyN}) with initial datum (\ref{eq: margINDAT}) is  
given by
\begin{eqnarray}\label{eq: expFORwigFACT}
W_{N,j}^{\var}(t)&&=\Phi_{j}^{(N)}(t)\(f_0^{\var}\)^{\otimes j}+\nonumber\\
&&+\sum_{n=1}^{N-j}\int_{0\leq t_n\leq \dots \leq t_1\leq t}\ud  
t_n\dots\ud t_1\  
\Phi_j^{(N)}(t-t_1)\(\frac{N-j}{N}\)C_{j,j+1}^\var\dots\nonumber\\
&&\qquad \quad\dots \(\frac{N-j-n+1}{N}\)C_{j+n-1,j+n}^\var\  
\Phi_{j+n}^{(N)}(t_n)\(f_0^{\var}\)^{\otimes j+n},
\end{eqnarray}
while the hierarchy (\ref{eq: expFORwigINF}) with initial datum  
$\(f_0^{\var}\)^{\otimes j}$ is
\begin{eqnarray}\label{eq: expFORwigINFfact}
f_{j}^{\var}(t)&&=\Phi_{j}(t)\(f_0^{\var}\)^{\otimes j}+\nonumber\\
&&+\sum_{n=1}^{N-j}\int_{0\leq t_n\leq \dots \leq t_1\leq t}\ud  
t_n\dots\ud t_1\ \Phi_j(t-t_1)C_{j,j+1}^\var\dots C_{j+n-1,j+n}^\var\  
\Phi_{j+n}(t_n)\(f_0^{\var}\)^{\otimes j+n}.\nonumber\\
&&
\end{eqnarray}
\vspace{0.2cm}

Following the line of the proof of Theorem \ref{teoSPOHN}, to prove  
the convergence of the series (\ref{eq: expFORwigFACT}) to (\ref{eq:  
expFORwigINFfact})
we must find a norm $\left|\cdot\right|_j$ for the marginals  
$W_{N,j}^{\var}(t)$ which plays the role of the trace norm on  
$L^2\(\R^{3j}\)$ in Theorem \ref{teoSPOHN}. First of all, it has to be  
controlled by the flows $\Phi_j^{(N)}$ and $\Phi_j$ in the sense that  
for any $T>0$ and for fixed $j$
\begin{eqnarray}\label{eq: contrTHEflow}
\left|\Phi_j^{(N)}(t)W_{N,j}^{\var}\right|_j\leq  
C_{t,j}\left|W_{N,j}^{\var}\right|_j,\ \ \ \ C_{t,j}>0:  
\sup_{t\in[0,T]}C_{t,j}<+\infty,
\end{eqnarray}
and
\begin{eqnarray}\label{eq: contrTHEflowLIB}
\left|\Phi_j(t)W_{N,j}^{\var}\right|_j\leq  
C'_{t,j}\left|W_{N,j}^{\var}\right|_j,\ \ \ \ C'_{t,j}>0:  
\sup_{t\in[0,T]}C'_{t,j}<+\infty,
\end{eqnarray}
(note that for the flows $S_j^{(N)}$ and $S_j$ involved Theorem  
\ref{teoSPOHN} we had properly conservation of the trace norm,  
actually estimates of the form (\ref{eq: contrTHEflow}) and (\ref{eq:  
contrTHEflowLIB}) would have been sufficient).
Thus, by (\ref{eq: contrTHEflow}) we could have the following bound  
for the $n$-th term of the (formal) series (\ref{eq: expFORwigFACT})
\begin{eqnarray}\label{boundFORser}
\frac{t^n}{n!}j(j+1)\dots(j+n-1)\ (C_t)^n  
\left|\(f_0^{\var}\)^{\otimes j+n}\right|_{j+n},\ \ \ \  
C_t=C_t(\phi,j)>0:\ \forall\ T>0 \sup_{t\in[0,T]}C_t<+\infty\nonumber\\
&&
\end{eqnarray}
provided that the operator $C_{j,j+1}^\var$ satisfies
\begin{eqnarray}\label{eq: contrOPC}
\left|C_{j,j+1}^\var W_{N,j+1}^{\var}\right|_j\leq j \ C  
\left|W_{N,j+1}^{\var}\right|_{j+1},\ \ \ \ C=C(\phi)>0.
\end{eqnarray}
Clearly (\ref{boundFORser}) and (\ref{eq: contrOPC}) would hold even  
for the $n$-th term of the series (\ref{eq: expFORwigINFfact}) by  
virtue of (\ref{eq: contrTHEflowLIB}).\\
By (\ref{boundFORser}), it would follow that $\left|\cdot\right|_j$  
has to be such that
\begin{eqnarray}\label{eq: proprP1}
\left|\(f_0^{\var}\)^{\otimes  
j}\right|_j=\(\left|f_0^{\var}\right|_1\)^j\leq a^j \ \ \text{for any  
$j$},
\end{eqnarray}
where $a$ is some positive constant. Then
we could conclude that the $n$-th term of the (formal) series  
(\ref{eq: expFORwigFACT}) and the $n$-th term of (\ref{eq:  
expFORwigINFfact}) are bounded by:
\begin{eqnarray}\label{boundFORserII}
\frac{t^n}{n!}j(j+1)\dots(j+n-1)\ (C_t)^n a^{j+n}< t^n (C_j\  a^{j})\  
(2a C_t)^n ,\ \ \ \ C=C(\phi)>0
\end{eqnarray}
and then we would have convergence for short times $\vert t\vert <t_0$  
($t_0$ depending on $\phi$ and $a$) of (\ref{eq: expFORwigFACT}) and  
(\ref{eq: expFORwigINFfact}) with respect to the norm  
$\left|\cdot\right|_j$. This would imply that the solution of the  
infinite hierarchy (\ref{eq: expFORwigINF}) with initial datum  
$\(f_0^\var\)^{\otimes j}$ is uniquely determined up to time $t_0$,  
thus, by the analysis done in the previous section, we would know that  
it is given by $\(f^\var(t)\)^{\otimes j}$, $f^\var(t)$ solving the  
nonlinear Wigner-Liouville equation (\ref{eq: WigLIOUhartree}).  
Moreover, by (\ref{eq: ONEpartINDAT}) it would follow that
\begin{eqnarray}\label{eq: controparteWAVE}
f^\var(t)=f_{\rho_t}^\var\leftrightarrow\hat{\rho}_t =\vert \psi_t  
 ><\psi_t\vert,
\end{eqnarray}
$\psi_t$ solving the Hartree equation (\ref{eq: Hartree}) with initial  
datum $\psi_0$.

To prove convergence of (\ref{eq: expFORwigFACT}) to (\ref{eq:  
expFORwigINFfact}), the norm $\left|\cdot\right|_j$ has to be such that
\begin{eqnarray}\label{eq: contrOPT}
\left\|T_{N,j}^\var
\right\|\rightarrow  0\ \ \ \text{as}\ \ N\to\infty,
\end{eqnarray}
where $\left\|\cdot\right\|$ is the operator norm on the space of  
$j$-particle functions with finite norm $\left|\cdot\right|_j$. In  
fact, we observe that
\begin{eqnarray}\label{eq: convFLOWSNUOVA}
\left|\Phi_j^{(N)}(t)\ f_j-\Phi_{j}(t)\ f_j\right|_j\leq \int_0^t \ud  
\tau\left| \Phi_j(t-\tau)T_{N,j}^\var\  f_j(\tau)\right|_j,
\end{eqnarray}
for any $j$-particle Wigner function $f_j$. Thus,
by virtue of (\ref{eq: contrTHEflowLIB}) and (\ref{eq: contrOPT}) we  
would obtain
\begin{eqnarray}\label{eq: convFLOWS}
\lim_{N\to\infty}\left\|\Phi_j^{(N)}(t)-\Phi_{j}(t)\right\|=0,
\end{eqnarray}
implying convergence of (\ref{eq: expFORwigFACT}) to (\ref{eq:  
expFORwigINF}) with respect to the norm $\left|\cdot\right|_j$,  
namely, propagation of chaos, for short times $\vert t\vert< t_0$ in  
the sense that, for any fixed $j$,
\begin{eqnarray}\label{eq: propOFchaosSHORT}
\left|W_{N,j}^\var(t)-\(f^\var(t)\)^{\otimes j}\right|_j\rightarrow  
0,\ \ \text{as}\ \ N\to\infty \ \ \forall\ t<t_0.
\end{eqnarray}
Finally, the argument just given could be iterated to prove  
propagation of chaos for $t\in[t_0-\delta,t_0+\delta]$ (for any  
$\delta>0$) if we could prove that
\begin{eqnarray}\label{eq: NONLINwigCONTROLLAnorma}
\left|f_{j}^\var(t_0-\delta)\right|_j=\(\left|f^\var(t_0-\delta)\right|_1\)^j\leq  C a^j  
.
\ \ \ \text{for any fixed}\ j,\nonumber\\
&&
\end{eqnarray}
In fact, the bound (\ref{eq: NONLINwigCONTROLLAnorma}), together with  
the convergence proved previously up to time $t_0$, would
imply estimate (\ref{eq: proprP1}) to hold for  
$W_{N,j}^\varepsilon(t)$ where $t=t_0-\delta$. Then, by iteration we  
could conclude that propagation of chaos in the sense of (\ref{eq:  
propOFchaosSHORT}) holds for all $t$.\\

By looking at the scheme we have just presented it turns out that the  
accuracy of the mean-field approximation
would be provided by the speed of convergence of the operator norm of  
$T_{N,j}^\var$ to zero as $N\to\infty$ (see (\ref{eq: contrOPT})).
Then, if one was able to provide an estimate uniform in $\var$ for  
$\left\|T_{N,j}^\var\right\|$, the convergence (\ref{eq:  
propOFchaosSHORT}) would be also uniform with respect to $\var$ and  
then, by iteration, we would have uniformity in $\var$ for all times. \\
\subsubsection{Choice of the norm $\left|\cdot\right|$}
Since by (\ref{eq: convergenceL2STRONG}) we already know that the  
$j$-particle Wigner marginals are converging strongly in $L^2$ to the  
$j$-fold product of solutions of the (Hartree) nonlinear  
Wigner-Liouville equation, it would be reasonable to choose the  
$L^2$-norm as $\left|\cdot\right|$ to check if it is possible to  
improve the ''bad'' dependence of the error (in the limit  
$N\to\infty$) with respect to $\var$.
Moreover, on the basis of (\ref{eq: ONEpartINDATII}) one could think  
to choose the $L^2$-norm because (\ref{eq: proprP1}) would be  
satisfied (with $a=1$) and this would hold for each $t$ because the  
Hartree dynamics preserves the $L^2$-norm (see (\ref{eq:  
semicCONSl^2})). Furthermore, the flows $\Phi_j^{(N)}(t)$  and  
$\Phi_j(t)$ not only control the $L^2$-norm in the sense of (\ref{eq:  
contrTHEflow}) and (\ref{eq: contrTHEflowLIB}) but even preserve it. \\
Nevertheless, it turns out that the operator $C_{j,j+1}^\var$ is  
unbounded from $L^2\(\R^{3(j+1)}\times\R^{3(j+1)}\)$ to  
$L^2\(\R^{3j}\times\R^{3j}\)$ (it can be verified easily by looking at  
(\ref{eq: opC_j+1})), thus property (\ref{eq: contrOPC}) fails.


Actually, one can verify that, by assuming $H_s$ regularity at time  
$t=0$, it is propagated by the flow $\Phi_j^{(N)}(t)$ (see  
\cite{PULVIRENTI}), by the free flow $\Phi_j(t)$ and also by the  
nonlinear Wigner-Liouville equation (\ref{eq: WigLIOUhartree})  
(according to Proposition \ref{PROPH_s}). So, this choice could be  
appropriate for the preservation in time of property (\ref{eq:  
proprP1}) but, as for the $L^2$-norm, the boundness of  
$C_{j,j+1}^\var$ fails.



By taking into account the (formal) analogy between the $N$-particle  
Wigner-Liouville equation (\ref{eq: WigLIOUhartreeNsec3}) and the  
Liouville equation (\ref{eq: class4}) one could think to use the  
$L^1$-norm. Indeed it is easy to check that the operators  
$T_{N,j}^\var$ and $C_{j,j+1}^\var$ are bounded in $L^1$ and it can be  
also verified that the flow $\Phi_j^{(N)}(t)$ controls the $L^1$-norm  
in the sense of (\ref{eq: contrTHEflow}). The free flow $\Phi_j(t)$  
clearly preserves the $L^1$-norm. Furthermore, by assuming property  
(\ref{eq: proprP1}) to hold, namely $f_0^\var\in  
L^1\(\R^3\times\R^3\)$, it is easy to check that it is verified for  
all $t$ because the $L^1$-norm is controlled by the Hartree dynamics.  
In other words, property (\ref{eq: NONLINwigCONTROLLAnorma}) would be  
satisfied and we could iterate the procedure presented above to prove  
convergence for all times. Therefore the $L^1$-norm could seem a good  
choice but the point is that Wigner functions are, in general, not in  
$L^1$. More precisely, by only knowing that $f_0^\var$ is the Wigner  
transform of a wave function $\psi_0\in L^2\(\R^3\)$
(as in the present situation), we are not guaranteed that $f_0^\var\in  
L^1\(\R^3\times\R^3\)$. Indeed, in general
the $L^1$-norm of Wigner functions is not related to any norm of the  
wave functions from which they arise.
We will come back on this topic in Remark \ref{RemarkSUl^1}.

It turns out that a fruitful approach is to use a norm which, from on  
side, is ''good'' for estimating $C_{j,j+1}^\var$, it is controlled by  
$\Phi_j^{(N)}(t)$, $\Phi_j(t)$ and even by the Hartree dynamics, and,  
on the other side, it somehow ''relates'' Wigner functions to the wave  
functions from which they arise. \\

Let us to consider the Fourier transform $\mathscr{F}_x$ of the  
$N$-particle Wigner function with respect to position variables, namely
\begin{eqnarray}\label{eq: normaNUOVA}
&&\(\mathscr{F}_x W_N^\var\)(P_N,V_N):=\tilde{W}_N^\var(P_N,V_N)=\int  
\ud X_N\ e^{-i\ P_N\cdot X_N} W_N^\var(X_N,V_N),
\end{eqnarray}
with $P_N=(p_1,\dots, p_N)\in \R^{3N}$
and let us define the $\tilde{L}^1$-norm as
\begin{eqnarray}\label{eq: normaNUOVA1}
&&\left\|W_N^\var\right\|_{\tilde{L}^1(\R^{3N}\times\R^{3N})}:=\left\|\tilde{W}_N^\var\right\|_{L^1(\R^{3N}\times\R^{3N})}=\int \ud P_N\int \ud V_N\ \vert  
\tilde{W}_N^\var(P_N,V_N)\vert.
\end{eqnarray}
We can verify that the operator  
$T_{N,j}^\var:\tilde{L}^1(\R^{3j}\times\R^{3j})\to\tilde{L}^1(\R^{3j}\times\R^{3j})$ is bounded under the assumption $\left\|\hat{\phi}\right\|_{L^1\(\R^{3}\)}<+\infty$. Indeed, by  
computing
\begin{eqnarray}\label{eq: opT_jTRASF}
&&\mathscr{F}_x\(T_{N,j}^{\var}W_{N,j}^{\var}\)(P_j,V_j)
=\int \ud P_j\   e^{-i\ P_j\cdot X_j}\  
T_{N,j}^{\var}W_{N,j}^{\var}(X_j,V_j),\nonumber\\
&&
\end{eqnarray}
by manipulating (\ref{eq: opT_j}) we find
\begin{eqnarray}\label{eq: opT_jTRASF1}
&&\mathscr{F}_x\(T_{N,j}^{\var}W_{N,j}^{\var}\)(P_j,V_j):=
\(\tilde{T}_{N,j}^{\var}\tilde{W}_{N,j}^{\var}\)(P_j,V_j)=\frac{i(2\pi)^{-3N}}{\var N}\sum_{ l\neq r}^{j}\ \sum_{\sigma=\pm 1} \sigma \int \ud k\   
\hat{\phi}(k)\nonumber\\
&&\quad \tilde{W}_{N,j}^{\var}\(p_1,\dots,p_l - k
,\dots,p_j,\ v_1,\dots,v_l +\frac{\sigma\var k}{2},\dots
, v_j\),\nonumber\\
&&
\end{eqnarray}
then
\begin{eqnarray}\label{eq: opT_jBOUND}
&&\left\|T_{N,j}^{\var}W_{N,j}^{\var}\right\|_{\tilde{L}^1\(\R^{3j}\times\R^{3j}\)}=\left\|\tilde{T}_{N,j}^{\var}\tilde{W}_{N,j}^{\var}\right\|_{L^1\(\R^{3j}\times\R^{3j}\)}\leq
  \frac{1}{(2\pi)^{3N}}\frac{2 j^2}{\var N}  
\left\|\hat{\phi}\right\|_{L^1\(\R^{3}\)}\left\|W_{N,j}^{\var}\right\|_{\tilde{L}^1\(\R^{3j}\times\R^{3j}\)}.\nonumber\\
  &&
\end{eqnarray}
In a similar way, we verify that the operator  
$C_{j,j+1}^\var:\tilde{L}^1(\R^{3(j+1)}\times\R^{3(j+1)})\to\tilde{L}^1(\R^{3j}\times\R^{3j})$ is  
bounded
  under the assumption  
$\left\|\hat{\phi}\right\|_{L^\infty\(\R^{3}\)}<+\infty$. In fact we  
compute
\begin{eqnarray}\label{eq: opC_j+1TRASF}
&&\mathscr{F}_x\(C_{j,j+1}^{\var}W_{N,j+1}^{\var}\)(P_{j},V_{j})
=\int \ud P_j\   e^{-i\ P_j\cdot X_j}\  
\(C_{j,j+1}^{\var}W_{N,j+1}^{\var}\)(X_j,V_j),\nonumber\\
&&
\end{eqnarray}
obtaining by (\ref{eq: opC_j+1}) that
\begin{eqnarray}\label{eq: opC_j+1TRASF1}
\mathscr{F}_x\(C_{j,j+1}^{\var}W_{N,j+1}^{\var}\)(P_j,V_j)&&:=
\(\tilde{C}_{j,j+1}^{\var}\tilde{W}_{N,j+1}^{\var}\)(P_j,V_j)=\nonumber\\
&&=\frac{i(2\pi)^{-3N}}{\var}\(\frac{N-j}{N}\)\sum_{l=1}^j\  
\sum_{\sigma=\pm 1} \sigma \int \ud v_{j+1}\int \ud k\   
\hat{\phi}(k)\nonumber\\
&&\quad \tilde{W}_{N,j+1}^{\var}\(p_1,\dots,p_l -  k,\dots,p_j,k,\  
v_1,\dots,v_l +\frac{\sigma\var k}{2},\dots, v_{j+1}\),\nonumber\\
&&
\end{eqnarray}
then
\begin{eqnarray}\label{eq: opC_j+1BOUND}
\left\|C_{j,j+1}^{\var}W_{N,j+1}^{\var}\right\|_{\tilde{L}^1\(\R^{3j}\times\R^{3j}\)}&&=\left\|\tilde{C}_{j,j+1}^{\var}\tilde{W}_{N,j+1}^{\var}\right\|_{L^1\(\R^{3j}\times\R^{3j}\)}\leq\nonumber\\
&&\leq
(2\pi)^{-3N}\frac{(2j)}{\var}\(\frac{N-j}{N}\)  
\left\|\hat{\phi}\right\|_{L^\infty\(\R^{3}\)}\left\|W_{N,j+1}^{\var}\right\|_{\tilde{L}^1\(\R^{3(j+1)}\times\R^{3(j+1)}\)}.\nonumber\\
&&
  \end{eqnarray}

Furthermore, concerning the initial datum $\(f_0^\var\)^{\otimes j}$,  
by (\ref{eq: ONEpartINDAT}) we have
\begin{eqnarray}\label{eq: propP1perL^1tilde}
\left\|\(f_0^{\var}\)^{\otimes  
j}\right\|_{\tilde{L}^1\(\R^{3j}\times\R^{3j}\)}=\(\left\|f_0^{\var}  
\right\|_{\tilde{L}^1\(\R^{3}\times\R^{3}\)}\)^j\leq C  
\(\left\|\hat{\psi}_0\right\|_{L^1\(\R^{3}\)}\)^{2j}\leq C  
\(\left\|\psi_0\right\|_{H_s\(\R^{3}\)}\)^{2j},\ \ s>3/2\nonumber\\
&&
\end{eqnarray}
where the last inequalities are simply obtained by explicit  
computations. For any $t>0$, by (\ref{eq: controparteWAVE}) we have
\begin{eqnarray}\label{eq: propP1perL^1tildeITER}
\left\|f^{\var}(t) \right\|_{\tilde{L}^1\(\R^{3}\times\R^{3}\)}\leq C  
\left\|\hat{\psi}_t\right\|_{L^1\(\R^{3}\)}^2\leq C  
\left\|\psi_t\right\|_{H_s\(\R^{3}\)}^2,\ \ s>3/2,
\end{eqnarray}
and by using standard energy methods it is easy to check that, under  
suitable smoothness assumption on the potential $\phi$, the $H_s$-norm  
of $\psi_t$ is controlled by the $H_s$-norm of $\psi_0$ for any $s$.  
Furthermore, even by looking at the (Hartree) nonlinear  
Wigner-Liouville equation (\ref{eq: WigLIOUhartree}), it is easy to  
check that the $\tilde{L}^1$-norm of $f^{\var}(t)$ is controlled by  
the $\tilde{L}^1$-norm of $f_0^{\var}$.

Finally, by virtue of (\ref{eq: opT_jBOUND}), (\ref{eq:  
opC_j+1BOUND}), (\ref{eq: propP1perL^1tilde}) and (\ref{eq:  
propP1perL^1tildeITER}), it follows that by setting
\begin{eqnarray}\label{eq: defNORM}
\left|\cdot\right|_{j}:=\left\|\cdot\right\|_{\tilde{L}^1\(\R^{3j}\times\R^{3j}\)},
\end{eqnarray}
and by assuming $\psi_0\in H_s\(\R^3\)$ with $s>3/2$ and the potential  
$\phi$ to be sufficiently smooth (in order to make all constants  
appearing in the estimates finite), we have
\begin{eqnarray}\label{eq: propOFchaos}
\left|W_{N,j}^\var(t)-\(f^\var(t)\)^{\otimes j}\right|_j\rightarrow  
0,\ \ \text{as}\ \ N\to\infty \ \ \forall\ t.
\end{eqnarray}
\vspace{0.3cm}

Therefore, for smooth potentials, we can show propagation of chaos in  
the Wigner formulation by following the same strategy of Theorem  
\ref{teoSPOHN}. Nonetheless, we note that the error in the  
approximation (\ref{eq: propOFchaos}) is still not uniform with  
respect to $\var$ and diverging when $\var\to 0$ because from  
(\ref{eq: opT_jBOUND}) we see that the operator norm of $T_{N,j}^\var$  
is of order $1/\var$ (as in Theorem \ref{teoSPOHN}).\\

We conclude the present analysis by observing that (\ref{eq:  
propOFchaos}) implies straightforward that
\begin{eqnarray}\label{eq: propOFchaosII}
\int_{\R^{3j}}\ud V_j\  
\sup_{X_j}\left|W_{N,j}^\var(X_j,V_j;t)-\(f^\var(t)\)^{\otimes  
j}(X_j,V_j)\right| \rightarrow 0,\ \ \text{as}\ \ N\to\infty \ \  
\forall\ t,
\end{eqnarray}
namely
\begin{eqnarray}\label{eq: propOFchaosIII}
\left\|W_{N,j}^\var(t)-\(f^\var(t)\)^{\otimes  
j}\right\|_{L^\infty\(\R^{3j}_{X_j}\)\cap L^1\(\R^{3j}_{V_j}\)}  
\rightarrow 0,\ \ \text{as}\ \ N\to\infty \ \ \forall\ t.
\end{eqnarray}
Despite the fact that (\ref{eq: propOFchaosII}) is a quite ''strong''  
convergence, it is not related to any convergence for the reduced  
density matrices and it does not imply any convergence for the  
expected value of $j$-particle observables (namely, it does not  
provide informations about macroscopic values of physically  
interesting quantities).

Nevertheless, one can verifies that the convergence (\ref{eq:  
propOFchaos}) and the uniform bounds
\begin{eqnarray}\label{eq: unifBOUNDL^2}
\left\|W_{N,j}^\var(t)\right\|_{L^2\(\R^{3j}\times \R^{3j}\)}\leq 1,\  
\ \ \left\|f^\var(t)\right\|_{L^2\(\R^{3}\times \R^{3}\)}\leq 1,
\end{eqnarray}
imply
\begin{eqnarray}\label{eq: propOFchaosINL^2weak}
W_{N,j}^\var(t)\rightarrow \(f^\var(t)\)^{\otimes j},\ \ \text{as}\ \  
N\to\infty \ \ \forall\ t,\ \ \ L^2-\text{weakly}.
\end{eqnarray}
By virtue of property (\ref{eq: WignerUNAerd2}), (\ref{eq:  
propOFchaosINL^2weak}) ensures the convergence of expected values of  
suitable observables. More precisely, (\ref{eq: propOFchaosINL^2weak})  
allows to compute ''macroscopic'' (or ''effective'') expected value of  
$j$-particle observables $O_j$ whose phase space representations  
(symbols) are in $L^2\(\R^{3j}\times\R^{3j}\)$ (see also \cite{FO}).
Indeed, for any $j$-particle observable $O_j$ with symbol  
$O_j(X_j,V_j)$, we have the following estimate
\begin{eqnarray}\label{eq: distrINEQinL^2weak}
\(O_j,W_{N,j}^\var(t)\)_{L^2}\approx\(O_j,\(f^\var(t)\)^{\otimes  
j}\)_{L^2}+\frac{C_j(\var)}{N},\ \ \ \forall\ \ t,\ \ \text{as}\ \  
N\to\infty,
\end{eqnarray}
where $C_j(\var)\to \infty\ \ \text{as}\ \var\to 0$.\\

%\vspace{0.3cm}
\begin{rem}\label{RemarkSUl^1}
\textnormal{We observe that all estimates we did by using the  
$\tilde{L}^1$-norm would be also valid for the $L^1$-norm. Thus, by  
assuming $f_0^\var\in L^1\(\R^3\times\R^3\)$ and following exactly the  
same strategy leading to the $\tilde{L}^1$-convergence (\ref{eq:  
propOFchaos}), we can prove that
\begin{eqnarray}\label{eq: propOFchaosINL^1}
\left\|W_{N,j}^\var(t)-\(f^\var(t)\)^{\otimes  
j}\right\|_{L^1\(\R^{3j}\times\R^{3j}\)} \rightarrow 0,\ \ \text{as}\  
\ N\to\infty \ \ \forall\ t,
\end{eqnarray}
and, as for the $\tilde{L}^1$-convergence, it can be verified that  
(\ref{eq: propOFchaosINL^1}) together with the uniform bounds  
(\ref{eq: unifBOUNDL^2}) leads to the $L^2$-weak convergence (\ref{eq:  
propOFchaosINL^2weak}) and, in particular, to the estimate (\ref{eq:  
distrINEQinL^2weak}). Then, apparently, there is no reason for  
considering the $\tilde{L}^1$-norm instead of the $L^1$-norm. In fact,  
in both cases we can realize the limit $N\to\infty$ in the $L^2$-weak  
sense and in both cases we find that the error in the mean-field  
approximation is not uniform with respect to $\var$, indeed diverging  
as $\var\to 0$ (by looking at the constant $C_j(\var)$ in (\ref{eq:  
distrINEQinL^2weak})). Nevertheless,
as we have already noticed, the crucial point is: which assumptions  
one has to do on the wave function $\psi_0$ to ensure that its Wigner  
transform $f_0^\var$ is in $L^1\(\R^{3}\times\R^{3}\)$?
We remind that
\begin{eqnarray}\label{eq: noNORMAL^1}
\int\ud x\int \ud v\ f_0^\var(x,v)=\int \ud x\vert  
\psi_0(x)\vert^2=\texttt{Tr}\hat{\rho}_0=1\ \ \ \ \ \text{with}\ \  
\hat{\rho}_0=\vert. \psi_0><\psi_0\vert
\end{eqnarray}
We know that the integral on the phase space of $f_0^\var(x,v)$ does  
not correspond to its $L^1$-norm being $f_0^\var$ not positive in  
general. But, by considering a wave function $\psi_0$ such that  
$f_0^\var(x,v)\geq 0$ for any $x,v$ we could identify the $L^2$-norm  
of $\psi_0$ (which is taken equal to one) with the $L^1$-norm of  
$f_0^\var$ and we are guaranteed that property (\ref{eq: proprP1}) is  
verified (with $a=1$). The only way for having a positive Wigner  
function is to choose $\psi_0\approx e^{-x^2}$ (see for example  
\cite{FO}), in particular we can consider coherent states of the form  
$\psi(x)=N_\var e^{-\frac{(x-x_0)^2}{\var}}e^{i\frac{v_0 x}{\var}}$,\  
for some\ $x_0,v_0\in \R^3$. }\\

\textnormal{In the end, we found that propagation of chaos in the  
mean-field limit by using the Wigner formalism can be proven, for  
smooth potentials, in the $L^2$-norm, directly by the mean-field  
result for the RDM (the use of the BBGKY hierarchy is prevented by the  
unboundedness of the operators involved).
On the other side, by treating the Wigner BBGKY hierarchy, it can be  
proven in the $L^1$-norm by choosing initial gaussian states, and, in  
the $\tilde{L}^1$-norm, by choosing initial wave functions in $H_s$,  
$s>3/2$ (if the dimension of the system is assumed to be equal to 3;  
in general, in any dimension $d$,
we have $s>d/2$). In each of the three cases we obtain convergence of  
expected values of $j$-particle observables $O_j$ with symbol in  
$L^2\(\R^{3j}\times\R^{3j}\)$.
Furthermore, in each of these cases the error in the mean-field  
approximation is not uniform with respect to $\var$ and it is  
diverging as $\var\to 0$.}

\end{rem}
\vspace{0.2cm}

\subsection{Alternative approaches}
\setcounter{equation}{0}
\def\theequation{4.5.\arabic{equation}}


The validity of propagation of chaos in the mean-field limit has been  
established also in \cite{FGS} by using the ''second-quantization  
formalism''. For fixed $\var$, the authors provide an alternative  
proof of the emergence of the Hartree dynamics for bounded potential  
$\phi$ and, even if obtained by using a different formalism, the  
general strategy of the proof is analogous to that of \cite{SPOHN} and  
the result can be formulated in terms of convergence of reduced  
density matrices to products of solutions of the Hartree equation.  
Then, by passing to the Wigner formalism, for a restricted class of  
two-body interactions the following (distributional) estimate in  
$\mathcal{S}'\(\R^{3j}\times\R^{3j}\)$ is proven
\begin{eqnarray}\label{eq: distrINEQfro}
W_{N,j}^\var(t)\approx\(f^\var(t)\)^{\otimes  
j}+\frac{C_j}{N}+O\(e^{-1/\sqrt{\left\|\phi\right\|_\infty t}}\),\ \ \  
\forall\ t,\ \ \text{as}\ \ \ N\to\infty,
\end{eqnarray}
where  
$\left\|\phi\right\|_\infty:=\left\|\phi\right\|_{L^\infty\(\R^3\)}$,  
$C_j$ is a positive constant only depending on $j$ and
$W_{N,j}^\var(t)$ and $f^\var(t)$ are defined as in (\ref{eq:  
distrINEQinL^2weak}). It turns out that the error in approximating the  
$N$-particle evolution with the Hartree dynamics is indeed uniform  
with respect to $\var$ but the exponential remainder appearing in  
(\ref{eq: distrINEQfro}) is small only if $\left\|\phi\right\|_\infty  
t <<1$, namely, by looking at very short times or by considering an  
interaction potential having very small $L^\infty$-norm.\\

\subsubsection{Joint limit $N\to\infty$ and $\var\to 0$}

In looking at the connection between mean-field limit and  
semiclassical approximation,
a joint limit $N\to\infty$ and $\var\to 0$ can be considered. Indeed,  
there are systems in which this kind of limit arises quite naturally  
by the scaling properties of the Hamiltonian.
%

A remarkable example is provided by the model considered in  
\cite{fermioni} and, previously, in \cite{NS} (with a somewhat  
different interpretation). The model considered in \cite{fermioni} is
a system of $N$ fermions interacting by the mean-field potential  
(\ref{eq: quant1}) with initial data localized in a cube of size of
order one and at energy comparable with the ground state
energy of the system.
The Hamiltonian of the system is given by (\ref{eq: quant2}), thus all  
the potential energy arises from the interaction term (\ref{eq:  
quant1}) and it follows straightforward that the potential energy per  
particle is of order one. As regard to the kinetic energy, it
%is well-known
can be verified that the
kinetic energy per particle of $N$ fermions, i.e.,  
$-\frac{1}{2}\var^2\Delta_{x_k}\ (k=1,\dots,N)$, in a cube of size one  
scales like $\var^2 N^{2/3}$ in the ground state. Therefore, in order  
to look at the limit $N\to\infty$ keeping the kinetic energy per  
particle of order one, one has to multiply the kinetic energy in  
(\ref{eq: quant2}) by $N^{-2/3}$. Then, by defining the ''effective  
Planck constant'' $\var_{eff}$ such that
$\var_{eff}=\var N^{-1/3}$,
the Hamiltonian of the systems becomes
\begin{equation}\label{eq: quant2fermi}
H_{N,eff}^{Q}=-\sum_{k=1}^{N}\frac{\var_{eff}^2\Delta_k}{2}+U^{Q}(X_N),\ \ \  
\text{$\var_{eff}\approx N^{-1/3}\to 0$ as $N\to\infty$}.
\end{equation}
Therefore the kinetic and the
potential energy per particle in the Hamiltonian $H_{N,eff}^Q$ are  
comparable and, as we already observed in introducing the mean-field  
model, this is the basic physical criterion to obtain a non trivial  
limiting dynamics (as $N\to\infty$) that captures the nonlinear effect  
of the interaction. Clearly, the limit $N\to\infty$ for the system  
whose Hamiltonian is $H_{N,eff}^{Q}$ entails the limit $\var_{eff}\to  
0$ which is a semiclassical limit for (\ref{eq: quant2}). Thus, one  
expects to find a limiting dynamics which is ruled by a classical  
equation. On the other side, it is known (and it is validated by  
numerous applications) that the equation governing the macroscopic  
(physically observable) dynamics of a Fermi gas in states close to the  
ground state is the Hartree-Fock equation:
\begin{eqnarray}\label{eq: hartreeFock}
&&i\var\pa_t\hat{\rho}_{t}=\[-\frac{\var^2}{2}  
\Delta,\hat{\rho}_{t}\]+\texttt{Tr}_{2}\left\{\[\phi(x-x_{2})
%+\phi(x_{2}-x)
,\hat{\rho}_{t}\otimes\hat{\rho}_{t}\]\right\} - \int \ud z  
\[\phi(x-z)-\phi(y-z)\]\rho_t(x,z)\rho_t(z,y).\nonumber\\
&&
\end{eqnarray}
Equation (\ref{eq: hartreeFock}) differs from the Hartree equation  
(\ref{eq: quant15}) because of the presence of the so called  
''exchange term'' which is the main effect of the correlations induced  
by the Fermi-Dirac statistics (see (\ref{eq: quant4})).
In \cite{fermioni} it has been proven that there exists a fixed time  
$T>0$ such that the difference, in a suitable weak sense,
between the $j$-particle marginal associated with the $N$-particle  
Wigner function of this system and the solution of the (Hartree)  
nonlinear Wigner-Liouville equation (\ref{eq: WigLIOUhartree})
is of order $N^{-1}\approx \var^3$ for any time $t \leq T$, provided that
the potential $\phi$ is real analytic.
In other words, all $\var^2$ corrections come from the difference between
the Vlasov equation (\ref{eq: Vlasov}) and the Hartree equation  
(\ref{eq: quant15}); hence they are related to the accuracy of
the semiclassical approximation in the one-body theory. In particular  
it is proven that all correlation
effects (the main of them is precisely the exchange term) are of order  
at most $O(\var^3)$.\\

We observe that
the case of undistinguishable particles (in the sense specified by  
(\ref{eq: quant3NEW}) and (\ref{eq: quant3NEWwig})) and even the  
bosonic case are
crucially different from the fermionic case discussed above. Indeed in  
these situations the
kinetic energy per particle
, i.e., $-\frac{1}{2}\var^2\Delta_{x_k}\ (k=1,\dots,N)$, in a cube of  
size one scales like $\var^2 $ in the ground state. Thus the  
Hamiltonian of an $N$-particle system interacting by the potential  
(\ref{eq: quant1}) is precisely (\ref{eq: quant2}) because no further  
scaling is needed. Therefore, there is no reason for considering a  
joint limit and the problem of realizing the (mean-field) limit $N\to  
\infty$ uniformly in $\var$ arises quite naturally.

On the other side, even
for undistinguishable particles there are models in which the scaling  
of the potential
somewhat leads to define a rescaled Hamiltonian which exhibits an  
effective Planck constant going to $0$ as $N\to\infty$ (as in the  
fermionic case).
These kind of systems are taken into account in \cite{Graffi} (and  
previously in \cite{NS} with the specific scaling $\var\approx  
N^{-1/3}$) and an example is provided by systems interacting by Kac  
potentials defined below.\\

\emph{Example: the Kac potential}. Consider a system of $N$ identical  
bosons of mass
$m = 1$ interacting through the (Kac) potential
\begin{equation}\label{eq: Kac}
\phi_\lambda(x)=\frac{1}{\lambda}
\phi\(\frac{x}{\lambda}\),
\end{equation}
where $\lambda$ is a large parameter of the same order of $N$ and  
$\phi$ is a given smooth
potential. The Hamiltonian is:
\begin{equation}\label{eq: KacHAM}
H_{N}=-\frac{\var^2}{2}\sum_{k=1}^{N}\Delta_{x_k}+\sum_{1\leq k<l\leq  
N}\phi_\lambda(x_k - x_l).
\end{equation}
After the rescaling $x = \lambda q$ the Hamiltonian becomes:
\begin{equation}\label{eq: KacHAMresc}
H_{N}=-\frac{1}{2}\(\frac{\var}{\lambda}\)^2\sum_{k=1}^{N}\Delta_{q_k}  
+\frac{1}{\lambda}\sum_{1\leq k<l\leq N}\phi(q_k - q_l).
\end{equation}
Setting $\lambda= N$, and $\var_{sc}=\frac{\var}{\lambda}=\frac{\var}{N}$
we finally get:
\begin{equation}\label{eq: KacHAMrescII}
H_{N}^{sc}=-\frac{\var_{sc}^2}{2}\sum_{k=1}^{N}\Delta_{q_k}  
+\frac{1}{N}\sum_{1\leq k<l\leq N}\phi(q_k - q_l),
\end{equation}
where $\var_{sc}\approx 1/N\to 0$ as $N\to\infty$.\\

In \cite{Graffi} it has been proven that in all situations in which
$N \to\infty$ entails $\var\to 0$ (as the case of the Kac potential),  
which, roughly speaking, are ''asymptotically classical'', the Vlasov
equation is indeed recovered in the limit $N\to\infty$ even when  
$\var\to 0$ according to an
arbitrary law. For WKB states of the form $\psi(x)=a(x)e^{iS(x)/\var}$  
the result, formulated in terms of weak convergence of $j$-particles  
Wigner marginals, is local in time (as one expects from WKB analysis),  
while by considering suitable mixtures of WKB states, the result holds  
globally in time.
The potential is assumed to be in $\mathcal{C}_b^2\(\R^3\)$
and the explicit rate of convergence is computed
by means of a constructive method.

In \cite{NS} the same result had been proven for the specific scaling  
$\var\approx N^{-1/3}$ by considering more general initial data but  
assuming the potential to be analytic. In \cite{NS} it was also proven  
that the solution of the (Hartree) nonlinear Wigner-Liouville equation  
(\ref{eq: WigLIOUhartree}) converges in  
$\mathcal{S}'\(\R^3\times\R^3\)$ to the solution of the Vlasov  
equation (\ref{eq: Vlasov}) as $\var\to 0$.


\section{Mean-field limit and Semiclassical Expansion}
\setcounter{equation}{0}
\def\theequation{5.\arabic{equation}}

In this chapter we describe a different approach in investigating the  
$\var$ dependence of the error in the mean-field approximation, by  
using the Wigner formalism. It consists in looking at the  
semiclassical expansion of the $N$-particle system and proving that  
each term of the expansion agrees, in the limit
$N\to\infty$, with the corresponding one associated with the Hartree  
equation (see \cite{MIO}).

This idea is motivated by the following argument. \\

In Chapter 1 we recalled what is established by classical mean-field  
theory, namely, under suitable assumption on the potential, for any  
fixed $j$ we have
\begin{eqnarray}\label{eq: CLASSconv}
F_N^{(j)}(t)\rightarrow \(f(t)\)^{\otimes j}, \ \ \ \text{as}\ \ N\to\infty,
\end{eqnarray}
in the weak topology of probability measures, where $F_N^{(j)}(t)$ are  
the $j$-particle marginals associated with the solution $F_N(t)$ of  
the Liouville equation (\ref{eq: class4}) with a factorized initial  
datum $f_0^{\otimes N}$ and $f(t)$ is the solution of the Vlasov  
equation (\ref{eq: Vlasov}) with initial datum $f_0$. Thus, (\ref{eq:  
CLASSconv}) means that propagation of chaos holds for the classical  
mean-field model.
On the other side, in Chapter 3 we proved that, under suitable  
assumptions on the potential and on the initial datum, the following  
quantum mean-field limit result holds
\begin{eqnarray}\label{eq: QUANTconvINS'}
W_{N,j}^\var(t)\rightarrow \(f^\var(t)\)^{\otimes j}, \ \ \ \text{as}\  
\ N\to\infty, \ \ \ L^2-\text{weakly},
\end{eqnarray}
where $W_{N,j}^\var(t)$ are the $j$-particle Wigner marginals  
associated with the solution $W_N^\var(t)$ of the $N$-particle  
Wigner-Liouville equation (\ref{eq: WigLIOUhartreeNsec3}) with a  
factorized initial datum $\(f_0^\var\)^{\otimes N}$ and $f^\var(t)$ is  
the solution of the (Hartree) nonlinear Wigner-Liouville equation  
(\ref{eq: WigLIOUhartree}) with initial datum $f_0^\var$. Furthermore,  
we found that the error in approximating $W_{N,j}^\var(t)$ with $  
\(f^\var(t)\)^{\otimes j}$ is not uniform with respect to $\var$ and  
diverging when $\var\to 0$.

As regard to the semiclassical limit $\var\to 0$, it is known that the  
$N$-particle quantum (mean-field) dynamics for sufficiently smooth  
potentials converges (for fixed $N$) in a suitable sense to the the  
$N$-particle classical (mean-field) evolution (see for example  
\cite{MarkowichN}, \cite{PULVIRENTI}). Moreover, it has also been  
proven that the Hartree dynamics with smooth interaction, rephrased in  
the Wigner formalism, is approximated in a suitable sense by the  
classical Vlasov evolution (see for example \cite{NS},  
\cite{Markowich}).

So that, it seems natural to consider the solution $W_{N}^{\var}(t)$  
of the $N$-particle Wigner-Liouville equation (\ref{eq:  
WigLIOUhartreeNsec3}) with a suitable factorized initial datum  
$\(f_0^\var\)^{\otimes N}$ and to look for an asymptotic expansion as
\begin{equation}\label{eq: INIZIALE1}
f_0^\var=f_0^{(0)}+\var f_0^{(1)}+\var^2 f_0^{(2)}+\dots
\end{equation}
implying, for the initial $j$-particle marginals,
\begin{equation}\label{eq: INIZIALEexp}
W_{N,j}^{\var}\left(0\right)=\(f_0^\var\)^{\otimes  
j}=\(f_0^{(0)}\)^{\otimes j}+\var W_{N,j}^{(1)}\left(0\right)+\var^2  
W_{N,j}^{(2)}\left(0\right)+\dots
\end{equation}
Then, for the time evolved marginals we expect to find an analogous  
expansion as
\begin{equation}\label{eq: WignerJNexpcap4}
W_{N,j}^{\var}\left(t\right)=W_{N,j}^{(0)}\left(t\right)+\var  
W_{N,j}^{(1)}\left(t\right)+\var^2 W_{N,j}^{(2)}\left(t\right)+\dots
\end{equation}
where the zero order term $W_{N,j}^{(0)}\left(t\right)$ is expected to  
be equal to the (classical) marginals associated with the solution of  
the Liouville equation with initial datum $\(f_0^{(0)}\)^{\otimes N}$  
determined by (\ref{eq: INIZIALE1}).

In a similar way, we consider the $j$-fold product  
$\(f^\var(t)\)^{\otimes j}$ of solutions of the (Hartree) nonlinear  
Wigner-Liouville equation (\ref{eq: WigLIOUhartree}) with initial  
datum $f_0^\var$ and we look for an expansion as
\begin{equation}\label{eq: WignerJlimexpcap4}
\left(f^\var(t)\right)^{\otimes j}=
f_j^{(0)}\left(t\right)
+\var f_j^{(1)}\left(t\right)+\var^2 f_j^{(2)}\left(t\right)+\dots
\end{equation}
where the zero order term $f_j^{(0)}$
is expected to be equal to
the $j$-fold product $\(f^{(0)}(t)\)^{\otimes j}$, where $f^{(0)}(t)$  
solves the Vlasov equation with initial datum $f_0^{(0)}$ given by  
(\ref{eq: INIZIALE1}).
%
Therefore, by recognizing that at zero order in $\var$ we find the  
classical quantities, by (\ref{eq: CLASSconv}) we know that  
$W_{N,j}^{(0)}(t)$ converges to $f_j^{(0)}(t)=\(f^{(0)}(t)\)^{\otimes  
j}$ in the weak topology of probability measures.
Then, it looks natural to ask if the following convergence holds
\begin{equation}\label{eq: convergenceCAP4}
W_{N,j}^{(k)}\left(t\right)\to f_j^{(k)}\left(t\right),\ \ \text{as  
$N\to\infty$, for any}\ k>0
\end{equation}
in a suitable sense.
This is what we are going to show in the present chapter and it is  
contained in our recent paper \cite{MIO}. \\

Note that the term by term convergence (\ref{eq: convergenceCAP4})  
does not provide the uniformity in $\var$ of the limit $N\to\infty$  
because this would require a control of the remainder of the expansion  
(\ref{eq: WignerJNexpcap4}), and for the moment we are not able to do  
it. On the other side, in proving (\ref{eq: convergenceCAP4}),
we provide quantum corrections to the classical mean-field limit  
result and, by characterizing explicitly both coefficients  
$W_{N,j}^{(k)}\left(t\right)$ and $f_j^{(k)}\left(t\right)$, we prove  
that those corrections are given in terms of the classical Liouville  
flow and, in particular, of suitable derivatives of the classical  
trajectories.

We note that to prove (\ref{eq: convergenceCAP4}) we make use of  
coherent states (see Section 5.5) and in that framework it is somewhat  
expected to find that quantum corrections to the classical dynamics  
can be expressed in terms of derivatives of the classical trajectories  
(see for example \cite{HEPP}, \cite{HAG}).\\

\subsection{Semiclassical expansion for the Hartree dynamics}
\setcounter{equation}{0}
\def\theequation{5.1.\arabic{equation}}



We want to determine an expansion in power series of $\var$ of the  
solution $f^{\var}(x,v;t)$ of the (Hartree) nonlinear Wigner-Liouville  
equation (\ref{eq: WigLIOUhartree}) for a given initial datum  
$f_0^\var(x,v)$, namely:
\begin{equation}\label{eq: fEXP}
f^{\var}(t)=f^{(0)}(t)+\var f^{(1)}(t)+\var^2 f^{(2)}(t)+\dots
\end{equation}
by knowing that the initial datum $f_0^\var$ is expanded as follows
\begin{equation}\label{eq: fEXPzeroTEMPO}
f_0^{\var}(x,v)=f_0^{(0)}(x,v)+\var f_0^{(1)}(x,v)+\var^2 f_0^{(2)}(x,v)+\dots
\end{equation}

Indeed an expansion like (\ref{eq: fEXPzeroTEMPO}) holds for general  
quantum states. For example, in \cite{PULVIRENTI} the semiclassical  
expansion for various kinds of states is presented, both gently  
varying with respect to $\var$ (such as pure states whose wave  
function is not depending on $\var$) and singularly behaving as  
$\var\to 0$ (such as states of semiclassical type: WKB and coherent  
states). In the first situation we find an expansion of the form  
(\ref{eq: fEXPzeroTEMPO}) where the coefficients $f_0^{(k)}$ are  
smooth, on the contrary, for WKB and coherent states we find  
distributional coefficients (precisely Dirac $\delta$-functions and  
suitable derivatives of it) which apparently are more difficult to  
treat (with respect to the smooth case). Nevertheless, by manipulating  
them in a suitable way, such kinds of ''singular'' expansions can be  
very useful to deal with problems of semiclassical approximation (see  
Section 5.5 and Chapter 6).

Following \cite{PULVIRENTI}, for a fixed $g$, the operator  
$T_g^{\var}$ appearing in equation (\ref{eq: WigLIOUhartree}) can be  
expanded as
\begin{equation}\label{eq: operatorTexp}
T_g^{\var}=T^{(0)}_g+\var T^{(1)}_g+\var^2 T^{(2)}_g+\dots
\end{equation}
where
\begin{equation}\label{eq: operatorT^nEVEN}
T^{(n)}_g =c_n(2\pi)^{-3}i\int_{\R^3}\ud k  
\hat{\phi}(k)\hat{\rho}_g(k) e^{i \ k\cdot x}(k\cdot \nabla_v)^{n+1},
\end{equation}
\begin{equation}\label{eq: operatorT^nCOSTANTE}
c_n=\frac{1}{2^n (n+1)!},
\end{equation}
for $n$ even and
\begin{equation}\label{eq: operatorT^nODD}
T^{(n)}_g =0,
\end{equation}
for $n$ odd. The operator $T^{(n)}_g$, for $n$ even, can be also written as
\begin{equation}\label{eq: operatorT^nEVEN1}
T^{(n)}_g =(-1)^{n/2}c_n\left(D_x^{n+1}\phi\ast g\right)\cdot D_v^{n+1},
\end{equation}
where, as in (\ref{eq: Vlasov}), $\ast$ denotes the convolution with  
respect to both $x$ and $v$ and we used the notation:
\begin{eqnarray}\label{eq: derivate PRODOTTO}
&&D_x^{n}\nu\cdot D_v^{n}\zeta=\sum_{\substack{n_1,n_2,n_3\in  
\mathbb{N}:\\\sum_j n_j=n}}\frac{\pa^n  
\nu}{\pa^{n_1}{x^1}\pa^{n_2}{x^2}\pa^{n_3}{x^3}}\frac{\pa^n  
\zeta}{\pa^{n_1}{v^1}\pa^{n_2}{v^2}\pa^{n_3}{v^3}},\\
&&\text{with}\ \ x=(x^1,x^2,x^3)\in\R^3\ \text{and}\  
v=(v^1,v^2,v^3)\in\R^3\nonumber
\end{eqnarray}
for the one-particle functions $\nu$ and $\zeta$.

Inserting (\ref{eq: fEXP}) in (\ref{eq: operatorTexp}) and setting:
\begin{equation}\label{eq: opTnot}
T_k^{(n)}=T_{f^{(k)}(t)}^{(n)},
\end{equation}
we readily arrive to the following sequence of problems for the  
coefficients $ f^{(k)}(t)$ of the expansion (\ref{eq: fEXP}):
\begin{equation}\label{eq: WigLIOUseqk=0}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)f^{(0)}(t)=T_{0}^{(0)}f^{(0)}(t),\\
&\left.f^{(0)}(x,v;t)\right\vert_{t=0}=f_0^{(0)}(x,v),
\end{aligned}
\right.
\end{equation}
and
\begin{equation}\label{eq: WigLIOUseq}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot  
\nabla_x\right)f^{(k)}(t)=L(f^{(0)}(t))f^{(k)}(t)+\Theta^{(k)}(t),\\
&\left.f^{(k)}(x,v;t)\right\vert_{t=0}=f_0^{(k)}(x,v),
\end{aligned}
\right.
\end{equation}
for $k\geq1$, where
\begin{equation}\label{eq: opL}
L(h)f=T_{h}^{(0)}f+T_{f}^{(0)}h=\left(\nabla_x\phi\ast h\right)\cdot  
\nabla_v f+\left(\nabla_x\phi\ast f\right)\cdot \nabla_v h,
\end{equation}
and
\begin{equation}\label{eq: Teta}
\Theta^{(k)}(t)=\sum_{\substack{l,p,r:\\l+p+r=k\\l<k,r<k}}T_r^{(p)}f^{(l)}(t).
\end{equation}
Note that, as we expected, equation (\ref{eq: WigLIOUseqk=0}) we found  
at zeroth order in $\var$ is precisely the classical Vlasov equation  
(\ref{eq: Vlasov})) associated with the interaction $\phi$. Thus we  
need to assume $\phi\in \mathcal{C}_b^2\(\R^3\)$
to guarantee that the Vlasov flow is well-defined and, as we recalled  
in Chapter 1, the Vlasov equation can be solved by means of  
characteristics and fixed point. Moreover, the problems (\ref{eq:  
WigLIOUseq}) are linear and can be solved by a recursive argument
by observing that the source terms $\Theta^{(k)}(t)$ involve only  
those coefficients $f^{(n)}(t)$ with $n<k$, so that they are known by  
the previous steps. Clearly we shall state suitable smoothness  
assumptions on $\phi$ because the operators $T_r^{(p)}$ appearing in  
$\Theta^{(k)}(t)$ involve derivatives of $\phi$ of order possibly  
higher than $2$, depending on $k$.
\\

{\bf Hypotheses H}:\
\textnormal{Here and henceforth we will assume that is spherically  
symmetric ($\phi(x)=\phi(\vert x\vert), \forall\ x\in \R^3$) and  
$\phi\in C_b^\infty(\R^3)$}\footnote{Here and henceforth we denote by  
$C_b^{\infty}(\R^d)$ the space of infinitely differentiable functions  
on $\R^d$ with uniformly bounded derivatives}.\\

Actually, in some of the results we are going to show less regularity  
is needed on $\phi$, but for the sake of simplicity we state here  
''maximal'' hypotheses under which we can deal both with semiclassical  
expansions and with the term by term convergence of the $N$-particle  
expansion (see Section 4.2 and 4.6 below). Furthermore, under the same  
hypotheses on $\phi$, by taking the initial datum as specified in  
Section 5.5, we are ensured about the validity of the result proven in  
Section 4.4 concerning the mean-field limit for Wigner functions. \\

In order to simplify the notation, from now on we will denote the time  
evolved coefficients $f^{(k)}(t)=f^{(k)}(x,v;t)$ solving (\ref{eq:  
WigLIOUseq}) by $f^{(k)}$ and the source terms  
$\Theta^{(k)}(x,v;t)=\Theta^{(k)}(t)$ by $\Theta^{(k)}$. Moreover, we  
will denote the initial coefficients $f_0^{(k)}(x,v)$ by $f_0^{(k)}$.  
We will specify the dependence on time and on the phase space  
variables just in case it is not clear from the context. \\

The crucial tool we shall use to give a sense to the solutions  
$f^{(k)}$, for $k\geq 1$, of problems (\ref{eq: WigLIOUseq}) is the  
following proposition whose proof will be given in Appendix A (see  
also \cite{MIO}).
\begin{prop}\label{PROP6.1}
Consider the initial value problem:
\begin{equation}\label{eq: Prop6.1}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)\gamma=L(h)\gamma+\Theta,\\
&\left. \gamma(x,v;t)\right\vert_{t=0}=\gamma_{0}(x,v),
\end{aligned}
\right.
\end{equation}
where $\gamma_{0}\in L^1(\R^3\times\R^3)$, $h=h(x,v;t)$ is such that  
$\left \vert \nabla_v h\right\vert \in  
\mathcal{C}^0\(L^1(\R^3\times\R^3), \R^{+} \)$, $\Theta=\Theta(x,v;t)$  
is such that $\Theta\in \mathcal{C}^0\(L^1(\R^3\times\R^3),\R^{+}\)$.\\
Then
there exists a unique solution $\gamma=\gamma(x,v;t)$ of (\ref{eq:  
Prop6.1}), such that $\gamma\in \mathcal{C}^0\(L^1(\R^3\times\R^3),  
\R^{+} \)$, given by an explicit series expansion.\\
Furthermore, denoting by $\Sigma_h$ the flow generated by $L(h)$, we  
have that $\Sigma_h(t,0)\gamma_0\in \mathcal{C}^{d}\(\R^3\times  
\R^3\)$ provided that $\nabla_v h\in \mathcal{C}^{d}\(\R^3\times \R^3\)$
and $\gamma_0 \in \mathcal{C}^{d}\(\R^3\times \R^3\)$.
\end{prop}

By looking at the problems (\ref{eq: WigLIOUseq}), we realize that the  
coefficients $f^{(k)}$, for $k\geq 1$, play the role of  
$\gamma(x,v;t)$ in Proposition \ref{PROP6.1}, while the time evolved  
zero-order coefficient $f^{(0)}$ and the source term $\Theta^{(k)}$  
play the role of the functions $h(x,v;t)$ and $\Theta(x,v;t)$  
respectively. Clearly the initial datum $\gamma_0(x,v)$ in this case  
is given by $f_0^{(k)}(x,v)$, $k\geq 1$.

Therefore,
by applying Proposition \ref{PROP6.1} to identify the coefficients  
$f^{(k)}$, $k\geq 1$, as the unique solutions of (\ref{eq:  
WigLIOUseq}) in $\mathcal{C}^0\(L^1(\R^3\times\R^3), \R^{+} \)$,
we have to consider an initial Wigner function $f^\var_0(x,v)$ such  
that $f_0^{(k)}\in L^1(\R^3\times\R^3)$ for $k\geq 1$ and $f_0^{(0)}$  
is sufficiently smooth to ensure $f^{(0)}, \vert \nabla_v f^{(0)}\vert  
\in \mathcal{C}^0\(L^1(\R^3\times\R^3), \R^{+} \)$.
Concerning the source terms $\{\Theta^{(k)}\}_{k\geq1}$, we have to  
prove that, with our choice of the initial datum, $\Theta^{(k)}\in  
\mathcal{C}^0\(L^1(\R^3\times\R^3), \R^{+} \)$ for $k\geq 1$.

We take the initial Wigner function $f_0^\var(x,v)$ in such a way that
\begin{equation}\label{eq: INDATinS}
\text{$f_0^{(k)}\in \mathcal{S}\(\R^3\times \R^3\)$  for any $k\geq 0$,}
\end{equation}
in particular $f_0^{(k)}\in L^1\(\R^3\times \R^3\)$ for any $k$. In  
Section 4.5 and in Chapter 5 we will give explicit examples of Wigner  
functions verifying this property (see also \cite{MIO}). Then, by  
(\ref{eq: INDATinS}) we find
\begin{equation}\label{eq: caso0}
f^{(0)}\in \mathcal{C}^{0}\(\mathcal{S}\(\R^3\times \R^3\),\R^+\)
\end{equation}
(because of the smoothness of the Vlasov flow as discussed in Section  
1.2) and, in particular
\begin{equation}\label{eq: caso0derINv}
\vert \nabla_v f^{(0)}\vert \in \mathcal{C}^0\(\mathcal{C}^\infty \cap  
L^1(\R^3\times\R^3), \R^{+} \).
\end{equation}
  As a consequence, from Proposition \ref{PROP6.1} we find
\begin{equation}\label{eq: C^0L^1intC^dpres}
\Sigma_{f^{(0)}}(t,s):\mathcal{C}^0\(\mathcal{C}^\infty\cap  
L^1\(\R^3\times \R^3\),\R^+\)\to\mathcal{C}^0\(\mathcal{C}^\infty\cap  
L^1\(\R^3\times \R^3\),\R^+\)\ \ \forall\ s\in [0,t].
\end{equation}
Moreover, by looking at the series expansion associated with the  
solution of the homogeneous version of problem (\ref{eq: Prop6.1})  
(see Appendix B below and \cite{MIO}), we realize that
\begin{equation}\label{eq: Spres}
\Sigma_{f^{(0)}}(t,s):\mathcal{C}^0\(\mathcal{S}\(\R^3\times  
\R^3\),\R^+\)\to\mathcal{C}^0\(\mathcal{S}\(\R^3\times \R^3\),\R^+\)\  
\ \forall\ s\in [0,t],
\end{equation}
provided that $f^{(0)}(t)\in \mathcal{S}\(\R^3\times \R^3\)$ for each  
$t$ and that suitable smoothness assumptions on the potential $\phi$  
are satisfied. In particular, under our assumptions,
we are guaranteed that (\ref{eq: Spres}) holds.

As regard to the source terms, from (\ref{eq: Teta}) it is easy to  
check that $\Theta^{(1)}=0$ then  
$f^{(1)}(t)=\Sigma_{f^{(0)}}(t,0)f_0^{(1)}\in L^1(\R^3\times\R^3)$ for  
each $t$ by virtue of the fact that  
$f_0^{(1)}\in\mathcal{S}(\R^3\times\R^3)\subset L^1(\R^3\times\R^3)$  
and thanks to Proposition \ref{PROP6.1}.
Moreover, by (\ref{eq: Spres}) we have also
\begin{eqnarray}\label{eq: caso1}
f^{(1)}\in  \mathcal{C}^0\(\mathcal{S}(\R^3\times\R^3),\R^+\).
\end{eqnarray}
On the contrary, for $k\geq 2$ we have $\Theta^{(k)}\neq 0$ and by
(\ref{eq: Teta}), (\ref{eq: operatorT^nODD}) and (\ref{eq:  
operatorT^nEVEN1}) we find that the norm of $\Theta^{(k)}$ ($k\geq 2$)  
in $\mathcal{C}^0\(L^1\(\R^3\times \R^3\),\R^+\)$  
($\mathcal{C}^0\(L^1\)$-norm in the sequel) is bounded by
\begin{eqnarray}\label{eq: TetaSTIMAl1}
&&\sup_{\tau\in[0,t]}\left\|\Theta^{(k)}(\tau)\right\|_{L^1(\R^3\times\R^3)}=\sup_{\tau\in[0,t]}\sum_{\substack{l,p,r:\\l+p+r=k\\l<k,r<k}}\left\|T_r^{(p)}f^{(l)}(\tau)\right\|_{L^1(\R^3\times\R^3)}\leq\nonumber\\
&&\leq \sum_{\substack{r,p,l:\\l+p+r=k\\l<k,r<k}}c_p  
\sup_{\tau\in[0,t]}\left\{\left\|D_x^{p+1}\phi\ast  
f^{(r)}(\tau)\right\|_{L^\infty(\R^3\times\R^3)}
\left\|D_v^{p+1}f^{(l)}(\tau)\right\|_{L^1(\R^3\times\R^3)}\right\}\leq  
\nonumber\\
&&\leq
\sum_{\substack{l,p,r:\\l+p+r=k\\r<k,l<k}}c_p  
\left\|D_x^{p+1}\phi\right\|_{L^\infty(\R^3\times\R^3)}\sup_{\tau\in[0,t]}\left\{\left\|f^{(r)}(\tau)\right\|_{L^1(\R^3\times\R^3)}
\left\|D_v^{p+1}f^{(l)}(\tau)\right\|_{L^1(\R^3\times\R^3)}\right\}.\nonumber\\
&&
\end{eqnarray}

Let us check if it is possible to use a recursive argument to prove  
that $\Theta^{(k)}\in\mathcal{C}^0\(L^1(\R^3\times\R^3),\R^+\)$ (for  
$k\geq 2$).
By (\ref{eq: TetaSTIMAl1}) it follows that if we knew that
\begin{equation}\label{induzPASSO}
\sup_{\tau\in[0,t]}\left\{\left\|f^{(r)}(\tau)\right\|_{L^1(\R^3\times\R^3)}\left\|D_v^{p+1}f^{(l)}(\tau)\right\|_{L^1(\R^3\times\R^3)}\right\}<+\infty
\end{equation}
for each $l,p,r: l+p+r=k$ and $l<k,r<k$, being the potential as in  
Hypotheses H,
we would find that the $\mathcal{C}^0\(L^1\)$-norm of $\Theta^{(k)}$  
is finite and we could conclude that for each $T>0$:
\begin{eqnarray}\label{eq: claim}
\sup_{t\in[0,T]}\left\|f^{(k)}(t)\right\|_{L^1(\R^3\times\R^3)}<+\infty\ \ \    
\forall\ \  k\geq 2.
\end{eqnarray}
Thus, we note that a recursive argument is not ''well-posed'' because  
it is not possible to provide a uniform bound for the  
$\mathcal{C}^0\(L^1\)$-norm of $f^{(k)}$, by assuming the same to hold  
for $f^{(n)}$ with $n<k$. Indeed, by (\ref{induzPASSO}), we see that  
we would need to assume even that the $L^1$-norm of any derivative of  
$f^{(n)}$ with $n<k$ is bounded uniformly in time.

On the contrary, we realize that we can use an induction procedure in  
$\mathcal{C}^0\(\mathcal{S}(\R^3\times\R^3),\R^+\)$.
Indeed, by the expression of $\Theta^{(k)}$ and by the regularity of  
$\phi$ we know that $\Theta^{(k)}\in  
\mathcal{C}^0\(\mathcal{S}(\R^3\times\R^3),\R^+\)$ if all coefficients  
$f^{(n)}$ up to $n=k-1$ are in  
$\mathcal{C}^0\(\mathcal{S}(\R^3\times\R^3),\R^+\)$. Then, by  
(\ref{eq: Spres}), we obtain
\begin{eqnarray}\label{eq: induzPASSO3}
&& f^{(n)}\in\mathcal{C}^0\(\mathcal{S}(\R^3\times\R^3),\R^+\)
\ \ \text{for all $n<k$}\nonumber\\
&&\Downarrow\nonumber\\
&& \Theta^{(k)}\in  
\mathcal{C}^0\(\mathcal{S}(\R^3\times\R^3),\R^+\)\nonumber\\
&&\Downarrow\nonumber\\
&&f^{(k)}\in\mathcal{C}^0\(\mathcal{S}(\R^3\times\R^3),\R^+\),
\end{eqnarray}
and thanks to (\ref{eq: caso0}) and (\ref{eq: caso1}) the induction  
procedure is ''closed''. Therefore, we have well-posedeness of  
problems (\ref{eq: WigLIOUseq}) in $L^1(\R^3\times\R^3)$ and, in  
particular, the coefficients $f^{(k)}(t)$ are also in  
$\mathcal{S}(\R^3\times\R^3)$ for all $t$. Actually, we could relax  
our assumption on the initial datum $f_0^\var$ obtaining less smooth  
coefficients, but in the present context this not an issue because we  
want to focus on the structure of the expansion and not on minimal  
regularity properties of the solutions.


\subsection{Semiclassical expansion for the $N$-particle dynamics }
\setcounter{equation}{0}
\def\theequation{5.2.\arabic{equation}}

In this section we determine an expansion in power series of $\var$ of  
the solution $W_N^{\var}(t)=W_N^{\var}(X_N,V_N;t)$ of the $N$-particle  
Wigner-Liouville equation equation (\ref{eq: WigLIOUhartreeNsec3}) for  
the factorized initial datum
\begin{equation}\label{eq: meanFIELDinCOND}
W_{N,0}^{\var}(X_N,V_N)=\prod_{j=1}^{N}f_0^{\var}(x_j,v_j),
\end{equation}
where $f_0^\var$ is the same one-particle Wigner function we chose as  
initial datum for the nonlinear Wigner-Liouville equation (\ref{eq:  
WigLIOUhartree}). Thus, we know that $f_0^\var$ is expanded as in  
(\ref{eq: fEXPzeroTEMPO}) and it turns out that, for the $N$-fold  
product $\(f_0^\var\)^{\otimes N}$, we find
\begin{equation}\label{eq: fEXPzeroTEMPON}
\(f_0^{\var}\)^{\otimes N}= W_{N,0}^{(0)}
+\var W_{N,0}^{(1)}+\var^2 W_{N,0}^{(2)}+\dots
\end{equation}
with
\begin{eqnarray}
&&W_{N,0}^{(0)}(X_N,V_N)=\prod_{j=1}^{N}f_0^{(0)}(x_j, v_j),\label{eq:  
W_Nindatcoeff0}\\
&& W_{N,0}^{(k)}(X_N,V_N)=\sum_{\substack{s_1\dots s_N\\0\leq s_j\leq  
k\\ \sum_j s_j=k}}\prod_{j=1}^{N}f_0^{(s_j)}(x_j, v_j)\ \ \   
\text{for}\ k\geq 1.\label{eq: W_Nindatcoeffk}
\end{eqnarray}
Note that $W_{N,0}^{(k)}$ is factorized only for $k=0$. \\

For the sake of simplicity, here and henceforth we will make explicit  
the dependence on time and on the phase space variables only if not  
clear from the context.\\

Following \cite{PULVIRENTI}, the operator $T_N^\var$ appearing in  
(\ref{eq: WigLIOUhartreeNsec3}) can be expanded as
\begin{equation}\label{eq: expT_N}
T_N^{\var}=T_N^{(0)}+\var T_N^{(1)}+\var^2 T_N^{(2)}+\dots
\end{equation}
where, for $n$ even we have
\begin{equation}\label{eq: T_NcoeffEVEN}
T_N^{(n)}=i(2\pi)^{-3N}C_n\int_{\R^{3N}}\ud K_N  
\hat{U}(K_N)e^{iK_N\cdot X_N}\left(K_N \cdot \nabla_{V_N}\right)^{n+1},
\end{equation}
$C_n$ being constants depending on $n$, and $\hat{U}$ being the  
Fourier transform of the potential in (\ref{eq: quant1}) (to simplify  
the notations we omit the superscript ''$Q$''). For $n$ odd, we find
\begin{equation}\label{eq: T_NcoeffODD}
T_N^{(n)}=0.
\end{equation}

Looking for a semiclassical expansion
\begin{equation}\label{eq: expW_N}
W_N^{\var}(t)=W_N^{(0)}(t)+\var W_N^{(1)}(t)+\var^2 W_N^{(2)}(t)+\dots,
\end{equation}
by (\ref{eq: fEXPzeroTEMPON}), (\ref{eq: T_NcoeffEVEN}) and (\ref{eq:  
T_NcoeffODD}) we arrive to the sequence of problems:
\begin{equation}\label{eq: WigLiouNk=0}
\left\{
\begin{aligned}
&\left(\pa_t+V_N\cdot  
\nabla_{X_N}\right)W_N^{(0)}(t)=T_{N}^{(0)}W_N^{(0)}(t),\\
&\left.W_N^{(0)}(X_N,V_N;t)\right\vert_{t=0}=W_{N,0}^{(0)}(X_N,V_N),
\end{aligned}
\right.
\end{equation}
and
\begin{equation}\label{eq: WigLiouN}
\left\{
\begin{aligned}
&\left(\pa_t+V_N\cdot  
\nabla_{X_N}\right)W_N^{(k)}(t)=T_N^{(0)}W_N^{(k)}(t)+\Theta_N^{(k)}(t),\\
&\left.W_N^{(k)}(X_N,V_N;t)\right\vert_{t=0}=W_{N,0}^{(k)}(X_N,V_N),
\end{aligned}
\right.
\end{equation}
for $k\geq1$, where
\begin{equation}\label{eq: Teta_N}
\Theta_N^{(k)}(t)=\sum_{\substack{0\leq l<k}}T_N^{(k-l)}W_N^{(l)}(t).
\end{equation}
Note that $T_N^{(0)}=\nabla_{X_N}U_N\cdot  
\nabla_{V_N}=\frac{1}{N}\sum_{i\neq j }^{N}\nabla_{x_i}\phi(x_i -  
x_j)\cdot\nabla_{v_i}$ is the classical Liouville operator, while the  
source terms $\Theta_N^{(k)}(t)$, at each order $k$, are known by the  
previous steps. As we recalled in Chapter 1, under smoothness  
assumptions on the interaction potential $\phi$, equation (\ref{eq:  
WigLiouNk=0}) can be solved by considering the Hamiltonian flow  
$\Phi^t(X_N, V_N)$ associated with the Newton equations (\ref{eq:  
class3}). Thus we find
\begin{equation}\label{eq: Liouville}
W_N^{(0)}(X_N,V_N;t)=S_N(t)W_{N,0}^{(0)}(X_N,V_N)=W_{N,0}^{(0)}\left(\Phi^{-t}\left(X_N,V_N\right)\right),
\end{equation}
where, from now on, we denote by $S_N$ the flow generated by the  
Liouville operator $T_N^{(0)}$. On the other side,
equations (\ref{eq: WigLiouN}) can be solved by recurrence thanks to  
the Duhamel formula:
\begin{equation}\label{eq: DuhamelN}
W_N^{(k)}(t)=S_N(t)W_{N,0}^{(k)}+\int_{0}^{t}\ud t_1\  
S_N(t-t_1)\Theta_N^{(k)}(t_1).
\end{equation}

We conclude this section by expressing the operators $T_N^{(n)}$ ($n$  
even) in terms of the variables $X_N, V_N$. From (\ref{eq:  
T_NcoeffEVEN}), we find that:
\begin{equation}\label{eq: restoOPt}
T_N^{(n)}=\hat{T}_N^{(n)}+R_N^{(n)},
\end{equation}
where
\begin{equation}\label{eq: diagOPt}
\hat{T}_N^{(n)}=c_n\frac{(-1)^{n/2}}{N}\sum_{ l\neq  
j}^{N}D_x^{n+1}\phi(x_l - x_j)\cdot D_{v_l}^{n+1},
\end{equation}
where $c_n$ is the same of (\ref{eq: operatorT^nCOSTANTE}), and
\begin{eqnarray}\label{eq: OFFdiagOPt}
&&R_N^{(n)}=\frac{1}{N}\sum_{l\neq j}^{N}
\sum_{\substack{k_1,k_2\in \mathbb{N}^{3}\\ \vert k_1\vert+\vert  
k_2\vert=n+1}}C_{k_1, k_2}\frac{\pa^{n+1}}{\pa_{x_l}^{\vert  
k_1\vert}\pa_{x_j}^{\vert k_2\vert}}\phi(x_l -  
x_j)\cdot\frac{\pa^{n+1}}{\pa_{v_l}^{\vert k_1\vert}\pa_{v_j}^{\vert  
k_2\vert}},
\end{eqnarray}
where,\ for $i=1,2$, $k_i=\(k_{i,1},k_{i,2},k_{i,3}\)$, $\vert  
k_i\vert=k_{i,1}+k_{i,2}+k_{i,3}$ , and
\begin{eqnarray}\label{eq: DICOder}
&&\frac{\pa^{n+1}}{\pa_{x_l}^{\vert k_1\vert}\pa_{x_j}^{\vert  
k_2\vert}}=\frac{\pa^{\vert k_1\vert}}{\pa^{ k_{1,1}}x_l^1\pa^{  
k_{1,2}}x_l^2\pa^{k_{1,3}} x_l^3}\ \frac{\pa^{\vert k_2\vert  
}}{\pa^{k_{2,1}}x_j^1\pa^{ k_{2,2}}x_j^2\pa^{ k_{2,3}}x_j^3}, \\
&&\text{with}\ x_j=\(x_{j}^1,x_{j}^2,x_{j}^3\), \  
x_l=\(x_{l}^1,x_{l}^2,x_{l}^3\),\nonumber
\end{eqnarray}
while $C_{k_1,k_2}$ are suitable coefficients. The same holds for the  
derivatives with respect to the velocities.

We observe that, by the expression (\ref{eq: OFFdiagOPt}), we mean  
that the derivative of order $\vert k_1\vert$ is distributed over the  
three components of $x_l$ in the same way in which it is distributed  
over the three components of $v_l$, and the same holds for the  
derivative of order $\vert k_2\vert$.

\subsection{Structure of the $j$-particles limiting marginals}
\setcounter{equation}{0}
\def\theequation{5.3.\arabic{equation}}

Let us consider the sequence $\{f_j^{\var}(t)\}_{j\geq 1}$, where  
$f_j^{\var}(t)=f_j^{\var}(X_j, V_j;t)$ is given by
  \begin{eqnarray}\label{eq: factJcap4}
&&f_j^{\var}(t)=
\(f^{\var}(t)\)^{\otimes j}
\end{eqnarray}
and $f^{\var}(t)$ is the solution of the nonlinear Wigner-Liouville  
equation (\ref{eq: WigLIOUhartree}) with initial datum $f_0^\var$
%being $f_0^\var$
chosen as in Section 4.1. By the one-particle expansion (\ref{eq:  
fEXP}), it turns out that
\begin{eqnarray}\label{eq: EXPperFATT}
&&f_j^{\var}(t)=
\(f^{\var}(t)\)^{\otimes j}=f_j^{(0)}(t)
+\var f_j^{(1)}(t)+\var^2 f_j^{(2)}(t)+\dots,
\end{eqnarray}
with
\begin{eqnarray}
&&f_j^{(0)}(X_j,V_j;t)=\prod_{i=1}^{j}f^{(0)}(x_i,v_i;t)
\label{eq: INFindatcoeff0}\\
&&f_{j}^{(k)}(X_j, V_j;t)=\sum_{\substack{s_1\dots s_j:\\  0\leq  
s_r\leq k\\ \sum_r s_r=k}}\prod_{r=1}^{j}f^{(s_r)}(x_r, v_r;t)\ \ \ \  
\text{for $k\geq 1$},\label{eq: coeffHAR}
\end{eqnarray}
where the one-particle functions $f^{(s_r)}(x_r, v_r;t)$ solve  
equation (\ref{eq: WigLIOUseqk=0}), if $s_r=0$, and (\ref{eq:  
WigLIOUseq}), for $s_r>0$. Note that
$f_{j}^{(k)}(t)$ is factorized only for $k=0$.

Thus by (\ref{eq: INFindatcoeff0})
we find that, as expected, the zero order term of the expansion of  
$\(f^{\var}(t)\)^{\otimes j}$ is given by the $j$-fold product  
$\(f^{(0)}(t)\)^{\otimes j}$ of solutions of the Vlasov initial value  
problem (\ref{eq: WigLIOUseqk=0}).\\


By the analysis done in Chapter 3, we know that  
$\{f_j^{\var}(t)\}_{j\geq 1}$ solves the infinite (Hartree) hierarchy  
(\ref{eq: hierarchyINF}) with factorized initial datum  
$\{\(f_0^{\var}\)^{\otimes j}\}_{j\geq 1}$. On the other hand, the  
sequence of $j$-particle marginals $\{W_{N,j}^\var(t)\}_{j=1}^N$  
associated with the solution of the $N$-particle Wigner-Liouville  
equation (\ref{eq: WigLIOUhartreeNsec3}) with factorized initial datum  
$\(f_0^{\var}\)^{\otimes N}$, solves the Wigner BBGKY hierarchy  
(\ref{eq: hierarchyN}) with initial datum $\{\(f_0^{\var}\)^{\otimes  
j}\}_{j=1}^N$. Moreover, in Chapter 3 we proved also that, for any  
$j$, $W_{N,j}^\var(t)\to \(f^{\var}(t)\)^{\otimes j}$ $L^2$-weakly
as $N\to\infty$ and the error in approximating the $N$-particle  
dynamics with the limiting one is not uniform with respect to $\var$  
and diverging as $\var\to 0$.
We recall that the reason for that arises from the fact that the  
operator $T_{N,j}^\var$ involved in the BBGKY hierarchy (\ref{eq:  
hierarchyN}) is bounded in the norm appropriate to study the  
convergence (namely $\tilde{L}^1\(\R^{3j}\times\R^{3j}\)$ or  
$L^1\(\R^{3j}\times\R^{3j}\)$), but its norm is diverging when $\var$  
goes to zero (in particular it is $O(1/\var)$).
This suggests to consider the semiclassical equations described in  
Sections 4.1 and 4.2. In this way, considering equations at each order  
in $\var$ and analyzing the hierarchies associated with each of those  
equation, we have to deal with operators which are clearly independent  
of $\var$ (e.g. $T_N^{(n)}$), %for the $N$-particle case, and  
$T_k^{(n)}$ for the expansion associated with the Hartree dynamics)
, and we have to investigate only the limit $N\to\infty$ without any  
dependence on $\var$. The price we have to pay is that now those  
operators are unbounded, as it comes out for the classical mean-field  
limit we faced in Chapter 1.

Thus, if we want to prove that the coefficient of order $\var^k$ of  
the expansion of the $j$-particle marginals $W_{N,j}^{\var}(t)$, namely:
\begin{eqnarray}\label{eq: coeffN}
&&W_{N,j}^{(k)}(X_j, V_j;t)=\int_{\R^{3(N-j)}\times \R^{3(N-j)}}\ud  
X_{N-j}\ud V_{N-j}W_{N}^{(k)}(X_j,X_{N-j}, V_j,V_{N-j};t),\nonumber\\
&&
\end{eqnarray}
converges to the corresponding object $f_j^{(k)}(t)$ arising from the  
Hartree dynamics (i.e (\ref{eq: coeffHAR})), the use of the hierarchy  
solved by $W_{N,j}^{(k)}(t)$ does not seem a good idea. In fact, even  
at level zero, when we have to deal with the classical mean-field  
limit, the hierarchy is very difficult to handle with (see Section  
1.4) because it involves derivation operators which are clearly  
unbounded, unless to make them act on analytic functions (see  
(\ref{eq: class14})-(\ref{eq: class16})). The obstacle which occurs in  
facing the higher order terms is precisely the same. \\

However, as we saw in Chapter 1,
%remarked
in the classical case we can treat the convergence in a more natural  
way, avoiding to use the hierarchy. Indeed we can control the  
$j$-particle marginals associated with the Liouville equation  
(\ref{eq: class4}) in terms of the expectation of the $j$-fold product  
of empirical measures with respect to the initial $N$-particle  
probability distribution (see Section 1.4).
Then, to use this strategy to establish the convergence of  
$W_{N,j}^{(0)}(t)$ to $\(f^{(0)}(t)\)^{\otimes j}$, we have to choose  
the one-particle initial Wigner function $f_0^\var$ in such a way that  
the zeroth order coefficient $f_0^{(0)}$ is a one-particle probability  
distribution. As a consequence, the (factorized) zeroth order  
coefficient $W_{N,0}^{(0)}$ (\ref{eq: W_Nindatcoeff0}) of the  
$N$-particle expansion is also a probability distribution (we will  
discuss this choice in Section 4.5). Then, we will follow a similar  
strategy in dealing with the convergence of the higher order terms of  
the expansion. More precisely, we will express $W_{N,j}^{(k)}(t)$ in  
terms of the expectation, with respect to $W_{N,0}^{(0)}$, of suitable  
(derivation) operators acting on empirical measures. The control of  
these objects will be obtained thanks to some estimates of the  
derivatives of the classical flow with respect to the initial data  
(see Proposition \ref{DERIVATE1}).



\subsection{Idea of the proof}
\setcounter{equation}{0}
\def\theequation{5.4.\arabic{equation}}

As we already noticed in the previous section, the convergence of the  
$j$-particle marginal at zeroth order in $\var$ is ensured by our  
assumption on the initial datum (to be specified in Section 4.5) and  
by the classical mean-field theory.

Thus, the first non-trivial term is that of order one in $\var$. By  
looking at (\ref{eq: WigLIOUseq}) for $k=1$, we realize that the first  
correction to the Vlasov equation in the Hartree dynamics satisfies
\begin{equation}\label{eq: WigLIOUseqONE}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)f^{(1)}=L(f^{(0)})f^{(1)},\\
&\left.f^{(1)}(x,v;t)\right\vert_{t=0}=f_0^{(1)}(x,v),
\end{aligned}
\right.
\end{equation}
(looking at the expression (\ref{eq: Teta}) for the source terms  
$\Theta^{(k)}$, we straightforward verify that $\Theta^{(1)}\equiv  
0$). As we shall see in detail in the following section, our choice  
for the initial one-particle datum is a mixture of coherent states  
such that each coefficient of the expansion is given by suitable  
derivatives of the zeroth order term which, as we already observed, is  
a probability distribution. In particular, the explicit form for  
$f_0^{(1)}$ is:
\begin{equation}\label{eq: PRIMOcoef}
f_0^{(1)}(x, v)=D^2_G f_0^{(0)}(x, v),
\end{equation}
where $D_G^2$ is a suitable second order derivation operator (see  
formula (\ref{eq: operatorD_G}) below in the case $k=2$)
%involving derivatives
with respect to the variable $z\in \R^6$ (we recall the notation  
$z=(x,v)\in \R^3\times\R^3$ introduced in Chapter 1).
%initial variables $z_1,\dots, z_N$ (where $z_i=(x_i,v_i)$ for $i=1,\dots,N$.

As regard to the $N$-particle dynamics, looking at (\ref{eq:  
W_Nindatcoeffk}) in the case $k=1$, we know that the initial datum for  
the coefficient of order one in $\var$
%, namely $W_{N,0}^{(1)}$,
is:
\begin{eqnarray}\label{eq: PRIMOcoefN}
W_{N,0}^{(1)}(Z_N)&&=\sum_{j=1}^{N}f_0^{(1)}(z_j)\prod_{l\neq  
j}^{N}f_0^{(0)}(z_l)=\mathcal{D}^2W_{N,0}^{(0)}(Z_N),
\end{eqnarray}
where
\begin{eqnarray}\label{eq: opDstorto}
\mathcal{D}^2=\sum_{j=1}^{N}D_{G,j}^2,
\end{eqnarray}
and $D_{G,j}^2$ is the operator $D_G^2$ acting on the variable $z_j\in  
\R^6$ . Let us consider the time evolved empirical measure $\mu_N(t)$  
(see Section 1.3) associated with the flow generated by the Newton  
equations (\ref{eq: class3}) and let us define $\mathcal{D}^2\mu_N(t)$  
as the distribution acting on a test function $u$
%$\in C_b^{\infty}(\R^6)$
in the following way:
\begin{eqnarray}\label{eq: opDstortoMu}
\(u,\mathcal{D}^2\mu_N(t)\)&&=\mathcal{D}^2\(\frac{1}{N}\sum_{l=1}^{N}u(z_l(t))\)=\frac{1}{N}\sum_{l,j=1}^{N}D_{G,j}^2  
u(z_l(t)).
\end{eqnarray}
We know that the operators $D_{G,j}^2$ involve derivatives with  
respect to the initial variables $z_j$, $j=1,\dots,N$,
% $z_1,\dots,z_N$,
thus, if at time $t=0$ we have $\mu_N\to f_0^{(0)}$ when $N\to\infty$
%a.e with respect to the (factorized) measure  
$W_{N,0}^{(0)}=\(f_0^{(0)}\)^{\otimes \infty}$
in the weak sense of probability measures,
%(see (\ref{eq: class11})),
it follows that:
\begin{eqnarray}\label{eq: opDstortoMut=0}
\(u,\mathcal{D}^2\mu_N\)&&=\mathcal{D}^2\frac{1}{N}\sum_{l=1}^{N}u(z_l)=  
\frac{1}{N}\sum_{l,j=1}^{N}D_{G,j}^2  
u(z_l)=\frac{1}{N}\sum_{j=1}^{N}D_{G,j}^2 u(z_j)=\nonumber\\
&& =\(D_{G}^2 u,\mu_N \)\ \rightarrow\ \(D_{G}^2 u,f_0^{(0)}\)=  
\(u,D_{G}^2f_0^{(0)}\)=\(u,f_0^{(1)}\)\nonumber\\
&&
\end{eqnarray}
as $N\to\infty$. By the Strong Law of Large Numbers (\ref{eq:  
class11}) we know that the convergence (\ref{eq: opDstortoMut=0}) holds
a.e with respect to the product measure $\(f_0^{(0)}\)^{\otimes  
\infty}$, then, by (\ref{eq: PRIMOcoefN}) and (\ref{eq:  
opDstortoMut=0}), we can conclude that:
\begin{eqnarray}\label{eq: marg1ord1t=0}
\(u,W_{N,1}^{(1)}(t)\vert_{t=0}\)=\(u,\mathbb{E}_N\left[\mathcal{D}^2\mu_N\right]\) \ \rightarrow\  \(u,f_0^{(1)}\)\ \text{as}\  
N\to\infty,\nonumber\\
&&
\end{eqnarray}
where $\mathbb{E}_N\left[\cdot\]$ denotes the expectation with respect  
to the $N$-particle probability distribution  
$W_{N,0}^{(0)}=\(f_0^{(0)}\)^{\otimes N}$ (see (\ref{eq:  
W_Nindatcoeff0})).\\

In the sequel, as in Chapter 1, we will say that a configuration $Z_N$  
is ''typical'' with respect to the probability measure $f_0^{(0)}$, if  
the corresponding empirical measure $\mu_N(z|Z_N)$ converges to  
$f_0^{(0)}$ in the weak topology of probability measures.\\

By equation (\ref{eq: WigLiouN}) for $k=1$, we have:
\begin{eqnarray}\label{eq: W_N^1eq}
&&\left(\pa_t +V_N\cdot  
\nabla_{X_N}\right)W_{N}^{(1)}=\nabla_{X_N}U\cdot  
\nabla_{V_N}W_{N}^{(1)},\nonumber\\
&&W_{N}^{(1)}(Z_N;t)\vert_{t=0}=W_{N,0}^{(1)}(Z_N),
\end{eqnarray}
namely, the classical Liouville equation (\ref{eq: class4}), where, to  
simplify the notation, we omitted the superscript ''$cl$''. Therefore:
\begin{eqnarray}\label{eq: W_N^1espress}
&&W_{N}^{(1)}(Z_N;t)=S_N(t)W_{N,0}^{(1)}(Z_N).
\end{eqnarray}
Finally,
%putting (\ref{eq: W_N^1espress}) in (\ref{eq: W_N,1sufunztest}) and
by virtue of (\ref{eq: W_N^1espress}) and (\ref{eq: PRIMOcoefN}), we obtain
\begin{eqnarray}\label{eq: W_N,1sufunztestA}
\(u,W_{N,1}^{(1)}(t)\)&&=\int_{\R^{3N}\times \R^{3N}}\ud  
Z_{N}S_N(t)W_{N,0}^{(1)}(Z_N)\(u,\mu_N\)=\nonumber\\
&&=\int_{\R^{3N}\times \R^{3N}}\ud  
Z_{N}W_{N,0}^{(1)}(Z_N)\(u,\mu_N(t)\)=\nonumber\\
&&=\int_{\R^{3N}\times \R^{3N}}\ud  
Z_{N}\mathcal{D}^2W_{N,0}^{(0)}(Z_N)\(u,\mu_N(t)\)=\nonumber\\
&&=\int_{\R^{3N}\times \R^{3N}}\ud  
Z_{N}W_{N,0}^{(0)}(Z_N)\(u,\mathcal{D}^2\mu_N(t)\)=\nonumber\\
&&=\(u,\mathbb{E}_N\left[\mathcal{D}^2\mu_N(t)\right]\).
%\ \ \text{for any}\ t\geq 0.\nonumber\\
%&&
\end{eqnarray}
Therefore, the behavior of $W_{N,1}^{(1)}(t)$ is determined by that of  
$\mathcal{D}^2\mu_N(t)$ for any initial configuration $Z_N$ which is  
typical with respect to $f_0^{(0)}$.
%the $N$-particle probability measure $W_{N,0}^{(0)}$.
Finally, since $\mu_N(t)$ solves the Vlasov equation in the weak form  
(see Chapter 1):
\begin{equation}\label{eq: VlasovWEAK}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)\mu_N(t)=\left(\nabla \phi \ast  
\mu_N(t)\right)\cdot \nabla_v \mu_N(t)\\
&\left. \mu_N(t)\right\vert_{t=0}=\mu_N,
\end{aligned}
\right.
\end{equation}
applying $\mathcal{D}^2$, we get:
\begin{equation}\label{eq: VlasovWEAKconD^2}
\left\{
\begin{aligned}
&\left(\pa_t + v\cdot \nabla_x  
\right)\mathcal{D}^2\mu_N(t)=L\left(\mu_N(t)\right)  
\mathcal{D}^2\mu_N(t)+R_N,\\
&\left. \mathcal{D}^2\mu_N(t)\right\vert_{t=0}=\mathcal{D}^2\mu_N,
\end{aligned}
\right.
\end{equation}
where $R_N$ is a term involving objects of the form $\sum_j  
\(D_{G,j}\mu_N(t)\)\( D_{G,j}\mu_N(t)\)$ which, as we shall see later,  
are of order $1/N$ when tested versus smooth functions.
The equation (\ref{eq: VlasovWEAKconD^2}) is similar to (\ref{eq:  
WigLIOUseqONE}), except for the presence of the term $R_N$ and for the  
fact that we have $L\left(\mu_N(t)\right) $ instead of  
$L\left(f^{(0)}\right)$. Therefore, the proof of the convergence of  
$W_{N,1}^{(1)}(t)$ to $f^{(1)}(t)$ reduces to that of a stability  
property for the solution of (\ref{eq: WigLIOUseqONE}) with respect to  
suitable weak topologies. Proposition \ref{PROP6.2} in the forthcoming  
Section 4.5 will provide us such property.

The general case $k>1$ is only technically more complicated because of  
the presence of source terms, but the main ideas are those presented  
here.

\begin{rem}\label{REMARKsuTIPOdiCONV}
\textnormal{By looking at the strategy of the proof for $k=1$ we  
realize that the basic idea is to apply the classical mean-field  
theory at suitable derivatives of the empirical measure. Therefore, if  
in the classical framework we have to deal with convergence with  
respect to the weak topology of probability measures, namely, with  
continuous and uniformly bounded test functions, now we need to deal  
with test functions whose derivatives are also continuous and  
uniformly bounded (e.g (\ref{eq: opDstortoMut=0})). Therefore we can  
argue that we will establish the term by term convergence in a  
suitable distributional sense.}
\end{rem}

We conclude by establishing a Proposition controlling the size of the  
derivatives of the Hamilton flow associated with (\ref{eq: class3})  
with respect to the initial data.

 From now on we shall denote by $C$ a positive constant, independent  
of $N$, possibly changing from line to line.

\begin{prop}\label{DERIVATE1}
Let $z_i(t)=\left(x_i(t),v_i(t)\right),\ i=1,\dots,N$ be the solution  
of equations (\ref{eq: class3}) with initial datum  
$z_i=\left(x_i,v_i\right),\ i=1,\dots,N$. %given at a certain time  
$t_0<t$.
Let $z_{i}^{\beta}\ \forall\ \beta=1,\dots,6$ be the $\beta$-th  
component of $z_i\in \R^6$. If the pair interaction potential $\phi$  
is assumed to satisfy Hypotheses H,
then, for each $k\in \mathbb{N}$:
\begin{equation}\label{eq: derivateK}
\left\vert\frac{\pa^k z_i^{\beta}(t)}{\pa z_{j_1}^{\alpha_{1}}\dots  
\pa z_{j_k}^{\alpha_{k}}} \right\vert\leq
\frac{C}{N^{d_k^{(i)}}},
\end{equation}
where $I:=(j_1,\dots, j_k)$ is any sequence of possibly repeated  
indices and $d_k^{(i)}$ is the number of different indices in $I$  
which are also different from \ $i$.
\end{prop}
The physical significance of (\ref{eq: derivateK}) is obvious. In the  
mean-field context, the quantity $z_i(t)$ depends weakly on $z_j$ if  
$j\neq i$ for each $t>0$. Actually $\frac{\pa z_i^\beta(t)}{\pa  
z_j^\alpha}=O\(\frac{1}{N}\)$ while $\frac{\pa z_i^\beta(t)}{\pa  
z_i^\alpha}=O\(1\)$ and these two estimates give rise to (\ref{eq:  
derivateK}) in the case $k=1$. Estimate (\ref{eq: derivateK}) says  
that for each derivative of any order with respect to some $z_j$ of
$z_i(t)$ , we gain a factor $1/N$.
We have also the following corollary whose straightforward proof will  
be omitted.

\begin{cor}\label{PROP5.2}
Let $\mathcal{U}=\mathcal{U}(Z_N(t))$ be a function of the time  
evolved configuration $Z_N(t)$ of the form:
$$
\mathcal{U}(Z_N(t))=\frac{1}{N}\sum_{i=1}^{N}u(z_i(t)),
$$
where $u\in C_b^{\infty}(\R^3\times\R^3)$. Then, if the pair  
interaction potential $\phi$ satisfies Hypotheses H, the following  
estimate holds:
\begin{equation}\label{eq: corollario}
\left\vert\frac{\pa^k \mathcal{U}(Z_N(t))}{\pa  
z_{j_1}^{\alpha_{1}}\dots \pa z_{j_k}^{\alpha_{k}}} \right\vert\leq
\frac{C}{N^{d_k}},
\end{equation}
where $d_k$ is the number of different indices in the sequence  
$I=(j_1,\dots, j_k)$.
\end{cor}
The proof of Proposition \ref{DERIVATE1} will be given in Appendix A.
\subsection{Results and technical preliminaries}
\setcounter{equation}{0}
\def\theequation{5.5.\arabic{equation}}

We choose, as initial condition for the one-particle Wigner function,  
a mixture of coherent states. The Wigner function associated with a  
pure coherent state centered at the point $(x_0, v_0)$ is given by:
\begin{equation}\label{eq: coeherent}
w(x,v\vert  
x_0,v_0)=\frac{1}{(\pi\var)^{3}}e^{-\frac{(x-x_0)^{2}}{\var}}e^{-\frac{(v-v_0)^{2}}{\var}}.
\end{equation}

Let now $g=g(x,v)$ be a smooth probability density on the one-particle  
phase space independent of $\var$
(see Hypotheses $\text{H}^1$ below) . Then we define:
\begin{equation} \label{eq: mix}
f_{0}^\var(x, v)=\int_{\R^3\times\R^3}\ud x_0 \ud v_0 \ w(x, v\vert  
x_0,v_0)g(x_0, v_0).
\end{equation}
Using the standard notation $z=(x,v)$ and $z_0=(x_0,v_0)$, (\ref{eq:  
mix}) is equivalent to:
\begin{eqnarray} \label{eq: mix2}
f_{0}^\var(z)&&=\frac{1}{(\pi\var)^{3}}\int_{\R^6}\ud z_0\   
e^{-\frac{(z-z_0)^{2}}{\var}} g(z_0)=\nonumber\\
&&=\frac{1}{(\pi)^{3}}\int_{\R^6}\ud \zeta \ e^{- \zeta^{2}}  
g(z-\sqrt{\var}\zeta).
\end{eqnarray}
Expanding
\begin{eqnarray} \label{eq: expG}
  g(z-\sqrt{\var}\zeta)&&=g(z)-\left(\zeta \cdot \nabla_z  
\right)g(z)\sqrt{\var}+\left(\zeta \cdot \nabla_z  
\right)^2g(z)\frac{(\sqrt{\var})^2}{2}+\dots\nonumber\\
&& \dots  -\left(\zeta \cdot \nabla_z  
\right)^{2n-1}g(z)\frac{(\sqrt{\var})^{2n-1}}{(2n-1)!}+\left(\zeta  
\cdot \nabla_z  
\right)^{2n}g(z)\frac{(\sqrt{\var})^{2n}}{(2n)!}+\dots,\nonumber\\
&&
\end{eqnarray}
and performing the gaussian integrations (which cancels the terms with  
the odd powers of $\sqrt{\var}$), we readily arrive to the following  
expansion for the Wigner function $f_0^\var$:
\begin{eqnarray} \label{eq: expf_0}
&& f_0^\var=f_0^{(0)}+\var f_0^{(1)}+\dots +\var^n f_0^{(n)}+\dots,
\end{eqnarray}
where
\begin{eqnarray}
&& f_0^{(0)}=g,\label{eq: expf_0COEFF}\\
&& f_0^{(n)}=D_G^{2n}f_0^{(0)}\ \ \text{for}\ n\geq 1,\label{eq: expf_0COEFFk}
\end{eqnarray}
and $D_G^{k}$ ($G$ stands for ''Gaussian''), for each $k>0$, is the  
following derivation operator with respect to the variable $z=(x,v)$:
\begin{eqnarray} \label{eq: operatorD_G}
&&D_G^{k}=\sum_{\substack{\alpha_1\dots\alpha_k:\\ \alpha_j=1,\dots,6}}\
\sum_{\substack{s_1\dots s_k:\\ 0\leq s_j\leq k\\s_1+\dots +s_k=k}}
C_G(\alpha_1\dots \alpha_k)\frac{\pa^k}{\pa z^{\alpha_1}\dots \pa  
z^{\alpha_k}},
\end{eqnarray}
where
\begin{eqnarray} \label{eq: operatorD_Gcost}
&&C_G(\alpha_1\dots \alpha_k)=\frac{1}{k!}\int_{\R^6}\ud \zeta\  
e^{-\zeta^2} \prod_{j=1}^{k} \zeta^{\alpha_j}.
\end{eqnarray}
Therefore, $C_G(\alpha_1\dots \alpha_k)$ is equal to zero for each  
sequence $\alpha_1\dots \alpha_k$ in which at least one index appears  
an odd number of times.\\
\\
{\bf Hypotheses $\text{H}^1$}:

We assume that
$g=f_0^{(0)}\in \mathcal{S}(\R^3\times\R^3)$, thus (\ref{eq:  
expf_0COEFFk}) make sense for any $n\geq 1$ and, in particular,  
$f_0^{(n)}\in \mathcal{S}(\R^3\times\R^3)$ for any $n$. By the  
analysis done in Section 4.1, this allows to identify the time-evolved  
coefficients $f^{(n)}(t)$, $n\geq 1$, as the unique solutions of the  
initial value problems (\ref{eq: WigLIOUseq}).

\begin{rem} \label{Remark6.1}

\textnormal{Here we consider a completely factorized $N$-particle  
initial state (see (\ref{eq: meanFIELDinCOND})), then property  
(\ref{eq: quant3NEWwig}) is satisfied. Furthermore the one-particle  
state is a mixture and this automatically excludes the Bose statistics.}
\end{rem}
\begin{rem} \label{Remark6.2}

\textnormal{We made the choice to expand fully the initial state  
$f_0^\var$ according to equation (\ref{eq: expf_0}). Another  
possibility is to assume the ($\var$ dependent) state $f_0^\var$  
(which is a probability measure in the present case) as initial  
condition for the Vlasov problem and, consequently, $f_0^{(k)}=0$ for  
the problems (\ref{eq: WigLIOUseq}). Now the coefficients $f^{(k)}(t)$  
are $\var$ dependent but this does not change deeply our analysis  
because $f_0^\var$ is smooth, uniformly in $\var$}.
\end{rem}
As we explained at the beginning of the present Chapter, our goal is  
to compare the $j$-particle semiclassical expansion associated with  
the $N$-particle flow, namely $W_{N,j}^{(k)}(t)$, $k=0,1,2,\dots$,  
with the corresponding coefficients $f_j^{(k)}(t)$ of the expansion:
\begin{equation}\label{eq: expHartreeJ}
f_j^{\var}(t)=f_j^{(0)}(t)+\var f_j^{(1)}(t)+\dots +\var^k f_j^{(k)}(t)+\dots,
\end{equation}
where $f_j^{(k)}(t)$ is given by (\ref{eq: coeffHAR}). The main result  
is the following.

\begin{teo}\label{THEO6.1}
Let us consider the (Hartree) nonlinear Wigner-Liouville equation  
(\ref{eq: WigLIOUhartree}) as in (\ref{eq: mix}) where the probability  
distribution $g$ satisfies Hypotheses $\text{H}^1$. Moreover, let us  
consider the $N$-particle Wigner-Liouville equation (\ref{eq:  
WigLIOUhartreeNsec3}) with factorized initial datum as in (\ref{eq:  
meanFIELDinCOND}). If the pair interaction potential $\phi$ is assumed  
to verify Hypotheses H, for all $t> 0$, for any integers $k$ and $j$,  
the following limit holds in $\mathcal{S}'(\R^{3j}\times\R^{3j})$:
\begin{equation}\label{eq: Th6.1}
W_{N,j}^{(k)}(t)\rightarrow f_j^{(k)}(t).
\end{equation}
as $N\to \infty$.
\end{teo}

\begin{rem} \label{Remark6.3}

\textnormal{As we shall see in the sequel, the convergence (\ref{eq:  
Th6.1}) is slightly stronger than the convergence in  
$\mathcal{S}'(\R^{3j}\times\R^{3j})$. Indeed, the sequence  
$W_{N,j}^{(k)}(t)$ converges also when it is tested on functions in  
$\mathcal{C}_b^{\infty}(\R^{3j}\times\R^{3j})$.
Such kind of convergence, which is natural in the present context,  
will be called $\mathcal{C}_b^{\infty}$-weak convergence}.
\end{rem}




A crucial tool in proving Theorem \ref{THEO6.1} is provided by the following

\begin{prop}\label{PROP6.2}
Let $\gamma_N(x,v;t)$ be a sequence in  
$\mathcal{S}'(\R^{3}\times\R^{3})$ (for each $t$) satisfying:
\begin{equation}\label{eq: Prop6.2}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)\gamma_N=L(h_N)\gamma_N+\Theta_N,\\
&\left. \gamma_N(x,v;t)\right\vert_{t=0}=\gamma_{N,0}(x,v),
\end{aligned}
\right.
\end{equation}
where $\gamma_{N,0}$, $\Theta_N$ are sequences in  
$\mathcal{S}'(\R^{3}\times\R^{3})$. We assume that:\\
{\bf i)}\ $h_N(x,v;t)$ is a sequence of probability measures  
converging, as $N\to\infty$, to a measure $h(t)\ud x\ud v$ with a  
density $h(t)\in \mathcal{C}^{\infty}_b\(\R^3\times\R^3\)$ and such that
$\left\vert \nabla_v h\right\vert \in \mathcal{C}^0\(  
L^1(\R^{3}\times\R^{3}),\R^+\)$.\\
{\bf ii)}\ for all $u_1,u_2$ in  
$\mathcal{C}_b^{\infty}(\R^{3}\times\R^{3})$ , there exists a constant  
$C=C(u_1, u_2)>0$, not depending on $N$, such that:
\begin{equation}\label{eq: Prop6.2hyp}
\left\|  u_1\ast \(u_2\gamma_N\)\right\|_{L^{\infty}(\R^3\times\R^3)}<  
C<+\infty\ \ \ for \ any \  t.
\end{equation}
{\bf iii)}\ $\gamma_{N,0}\rightarrow \gamma_0\text{,\ as}\ \  
N\to\infty$,\ \  $\mathcal{C}_b^{\infty}$-weakly ,  
$\gamma_0=\gamma_0(x,v)$ is a function belonging to  
$L^1(\R^{3}\times\R^{3})$.
\\
{\bf iv)}\ $\Theta_N \rightarrow \Theta\text{,\ as}\ \ N\to\infty$,\   
\ $\mathcal{C}_b^{\infty}$-weakly , $\Theta=\Theta(x,v;t)$ is a  
function belonging to \ $\mathcal{C}^0\(L^1(\R^{3}\times\R^{3}),  
\R^+\)$.
\\
\emph{Then:}
\begin{eqnarray}\label{eq: Prop6.2thesis}
&&\text{$\gamma_N \rightarrow\gamma\text{,\ as}\ \ N\to\infty$ \ \  
$\mathcal{C}_b^{\infty}$-weakly},\nonumber\\
&&
\end{eqnarray}
where $\gamma$ is the unique solution of the problem (\ref{eq:  
Prop6.1}) in $\mathcal{C}^0\(L^1\(\R^{3}\times\R^{3}\),\R^+\)$.
\end{prop}
For the proof, see Appendix B.

\subsection{Convergence}
\setcounter{equation}{0}
\def\theequation{5.6.\arabic{equation}}

This section is devoted to the proof of Theorem \ref{THEO6.1}.\\

By (\ref{eq: DuhamelN}) and (\ref{eq: Teta_N}), for $k\geq 0$ we have:
\begin{eqnarray}\label{eq: expW_N^k}
W_N^{(k)}(Z_N;t)&&=\sum_{n\geq  
0}\sum_{r=0}^{k}\sum_{\substack{r_1\dots r_n:\\r_j>0\\\sum  
r_j=k-r}}\int_{0}^{t}\ud t_1\int_{0}^{t_1}\ud t_2\dots  
\int_{0}^{t_{n-1}}\ud t_n \nonumber\\
&&\quad \quad \quad S_N(t-t_1)T_N^{(r_1)}S_N(t_1-t_2)\dots  
T_N^{(r_n)}S_N(t_n)W_{N,0}^{(r)}(Z_N).
\end{eqnarray}

It is useful to remind that, the only non-vanishing terms in (\ref{eq:  
expW_N^k}) are those for which all $r_1,\dots,r_n$ are even (because  
the odd terms in the expansion for the operator $T_N^\var$ appearing  
in (\ref{eq: WigLIOUhartreeNsec3}) are vanishing (see (\ref{eq:  
T_NcoeffODD})).

According to (\ref{eq: W_Nindatcoeffk}) and (\ref{eq: expf_0COEFFk}),
\begin{eqnarray}\label{eq: W_NindatcoeffkSEC7}
&&W_{N,0}^{(r)}(Z_N)=\sum_{\substack{s_1\dots s_N\\0\leq s_j\leq r\\  
\sum_j s_j=r}}\prod_{j=1}^{N}\(D_{G,j}^{2 s_j}f_0^{(0)}(z_j)\),
\end{eqnarray}
where $D_{G,j}^{k}$ is defined in (\ref{eq: operatorD_G}) and the  
extra symbol $j$ means that this operator acts on the variable $z_j$.
Defining the operator $\mathcal{D}^{2r}$ as:
\begin{eqnarray}\label{eq: opDstortoGEN}
&&\mathcal{D}^{0}=\texttt{1},\nonumber\\
&&\mathcal{D}^{2r}=\sum_{\substack{s_1\dots s_N:\\0\leq s_j\leq  
r\\\sum_j s_j=r}}\prod_{j=1}^{N}D_{G,j}^{2s_j},\ \ \ r\geq 1,
\end{eqnarray}
we have:
\begin{eqnarray}\label{eq: W_N,0^r}
W_{N,0}^{(r)}(Z_N)=\mathcal{D}^{2r}W_{N,0}^{(0)}(Z_N)\ \ \forall\ r\geq 0.
\end{eqnarray}

In order to investigate the behavior of the $j$-particle functions  
$W_{N,j}^{(k)}(Z_j;t)$ when $N\to\infty$, we consider the following  
object, for a given configuration $Z'_j=(z'_1\dots z'_j)$:
\begin{eqnarray}\label{eq: W_N,j^k}
&&\omega_{N,j}^{(k)}(Z'_j;t)=\int_{\R^{6N}}\ud Z_N\  
W_{N}^{(k)}(Z_N;t)\mu_N(z'_1\vert Z_N)\dots \mu_N(z'_j\vert Z_N).
\end{eqnarray}
In the end of the section, we will show that (\ref{eq: W_N,j^k}) is  
asymptotically equivalent to $W_{N,j}^{(k)}(Z'_j;t)$.

 From (\ref{eq: expW_N^k}), (\ref{eq: W_N,0^r}) and (\ref{eq:  
W_N,j^k}), it follows that:
\begin{eqnarray}\label{eq: expW_N^kBIS}
&&\omega_{N,j}^{(k)}(Z'_j;t)=\sum_{n\geq  
0}\sum_{r=0}^{k}\sum_{\substack{r_1\dots r_n:\\r_j>0\\\sum  
r_j=k-r}}\int_{0}^{t}\ud t_1\int_{0}^{t_1}\ud t_2\dots  
\int_{0}^{t_{n-1}}\ud t_n \int_{\R^{6N}}\ud Z_N \mu_{N,j}(Z'_j\vert  
Z_N)\nonumber\\
&&\qquad \quad
S_N(t-t_1)T_N^{(r_1)}S_N(t_1-t_2)\dots  
T_N^{(r_n)}S_N(t_n)\mathcal{D}^{2r}W_{N,0}^{(0)}(Z_N),
\end{eqnarray}
where
\begin{eqnarray}\label{eq: muperJ}
&&\mu_{N,j}(Z'_j\vert Z_N)=\mu_{N}(z'_1\vert Z_N)\dots \mu_{N}(z'_j\vert Z_N).
\end{eqnarray}
Integrating by parts,
reminding that each $r_j$ is even and that each $T_N^{(r_j)}$ involves  
derivatives of order $r_j +1$,
we have:
\begin{eqnarray}\label{eq: expW_N^kTRIS}
\omega_{N,j}^{(k)}(Z'_j;t)&&=\sum_{n\geq 0}(-1)^n\sum_{r=0}^{k}\  
\sum_{\substack{ {\bf \underline{r}}_n:\ r_j>0\\\vert {\bf  
\underline{r}}_n\vert =k-r}}\ \int_{ord}^{t}\ud  {\bf  
\underline{t}}_n\nonumber\\
&&\qquad  
\mathbb{E}_N\left[\mathcal{D}^{2r}T_N^{(r_n)}(t_n)T_N^{(r_{n-1})}(t_{n-1})\dots T_N^{(r_{1})}(t_{1}) \mu_{N,j}(Z'_j\vert  
Z_N(t))\right],
\end{eqnarray}
where ${\bf \underline{r}}_n$ is the sequence of positive integers  
$r_1,\dots ,r_n$, $\vert {\bf \underline{r}}_n\vert=\sum_{j=1}^{n}r_j$  
and $Z_N(t)$ is the Hamiltonian flow associated with (\ref{eq:  
class3}). Moreover ${\bf \underline{t}}_n=t_1\dots t_n$ and  
$\int_{ord}^{t}\ud  {\bf \underline{t}}_n$ denotes the integral over  
he simplex $0<t_n<t_{n-1}<\dots <t_1<t$. Finally, $\mathbb{E}_N$  
stands for the expectation with respect to the $N$-particle density  
$W_{N,0}^{(0)}$ and
\begin{eqnarray}\label{eq: opTtempo}
T_N^{(r)}(t)=S_N(-t)T_N^{(r)}S_N(t).
\end{eqnarray}
Therefore, the objects we have to investigate in the limit $N\to\infty$ are:
\begin{eqnarray}\label{eq: senza MEDIA}
\nu_j^{(k)}(Z'_j;t)&&=\sum_{n\geq 0}(-1)^n\sum_{r=0}^{k}\  
\sum_{\substack{{\bf \underline{r}}_n:\ r_j>0\\\vert {\bf  
\underline{r}}_n\vert =k-r}}\ \int_{ord}^{t}\ud  {\bf  
\underline{t}}_n\eta_j(Z'_j;t,r,{\bf \underline{r}}_n,{\bf  
\underline{t}}_n,Z_N),\nonumber\\
&&
\end{eqnarray}
(for any configuration $Z_N$, typical with respect to $f_0^{(0)}$),
where $\eta_j$ is given by:
\begin{eqnarray}\label{eq: eta_j}
&&\eta_j(Z'_j;t,r,{\bf \underline{r}}_n,{\bf  
\underline{t}}_n,Z_N)=\mathcal{D}^{2r}T_N^{(r_n)}(t_n)T_N^{(r_{n-1})}(t_{n-1})\dots T_N^{(r_{1})}(t_{1}) \mu_{N,j}(Z'_j\vert  
Z_N(t)).\nonumber\\
&&
\end{eqnarray}

Note that:
\begin{eqnarray}\label{eq: senza MEDIA k=0}
\nu_j^{(0)}(Z'_j;t)=
\mu_{N,j}\(Z'_j\vert Z_N(t)\).
\nonumber\\
&&
\end{eqnarray}

We start by analyzing the behavior of $\nu_{j}^{(k)}$ in the cases  
$j=1,2$, thus we are lead to consider:
\begin{eqnarray}
&&\eta_1(z'_1;t,r,{\bf \underline{r}}_n,{\bf  
\underline{t}}_n,Z_N)=\mathcal{D}^{2r} T_N^{(r_n)}(t_n)  
T_N^{(r_{n-1})}(t_{n-1})\dots T_N^{(r_{1})}(t_{1}) \mu_{N}(z'_1\vert  
Z_N(t)),\label{eq: aUnCorpo}\nonumber\\
&&
\end{eqnarray}
and
\begin{eqnarray}
&&\eta_2(z'_1,z'_2;t,r,{\bf \underline{r}}_n,{\bf  
\underline{t}}_n,Z_N)=\mathcal{D}^{2r}T_N^{(r_n)}(t_n)T_N^{(r_{n-1})}(t_{n-1})\dots T_N^{(r_{1})}(t_{1}) \mu_{N,2}\(Z'_2\vert  
Z_N(t)\).\nonumber\\
&&\label{eq: aDueCorpi}
\end{eqnarray}

It is useful to stress that the operators $T_N^{(r_j)}(t_j)$   
($j=1,\dots,n$) and $\mathcal{D}^{2r}$ act as suitable distributional  
derivatives with respect to the variables $Z_N$. To evaluate $\eta_1$,  
let us first analyze the action of $T_N^{(r)}(\tau)$. By (\ref{eq:  
opTtempo}) and (\ref{eq: restoOPt}), for any function $G=G(Z_N)$, we  
have:
\begin{eqnarray}\label{eq: opTtemposuG}
&&\left(T_N^{(r)}\(\tau\)G\right)\(Z_N\)=
%S_N\(-\tau\)T_N^{(r)}S_N\(\tau\)G\(Z_N\)=\nonumber\\
S_N\(-\tau\)\(\hat{T}_N^{(r)}
%\left(S_N\(-\tau\)G\right)\(Z_N\(\tau\)\)
+R_N^{(r)}\)\(S_N\(\tau\)G\)\(Z_N\)=\nonumber\\
&&=(-1)^{r/2}\frac{c_r}{N}\sum_{j,l}S_N\(-\tau\)D_x^{r+1}\phi(x_j-x_l)\cdot  
D_{v_j}^{r+1}\(S_N\(\tau\)G\)\(Z_N\)+\nonumber\\
&&+\frac{1}{N}\sum_{l,j=1}^{N}\sum_{\substack{k_1,k_2\in  
\mathbb{N}^{3}\\ \vert k_1\vert+\vert k_2\vert=r+1}}C_{k_1,  
k_2}S_N\(-\tau\)\frac{\pa^{r+1}}{\pa_{x_l}^{\vert  
k_1\vert}\pa_{x_j}^{\vert k_2\vert}}\phi(x_l -  
x_j)\cdot\frac{\pa^{r+1}}{\pa_{v_l}^{\vert k_1\vert}\pa_{v_j}^{\vert  
k_2\vert}}\(S_N\(\tau\)G\)\(Z_N\).\nonumber\\
&&
\end{eqnarray}
Note that the derivatives involved here are done with respect to the  
variables at time $t=0$.

Denoting by $D_{z_j}^{r}$ any derivative of order $r$ with respect to  
a variable $z_j$ at time $t=0$, we observe that:
\begin{eqnarray}\label{eq: derTEMPsec7}
&&  
S_N(-t)D_{z_j}^{r}G(Z_N)=\(D_{z_j}^{r}G\)(Z_N(t))=D_{z_j}^{r}(t)\(S_N(-t)G\)(Z_N),
\end{eqnarray}
where, by $D_{z_j}^{r}(t)$, we denote the same derivative of order $r$  
with respect to the variable $z_j(t)$. Then, by (\ref{eq:  
derTEMPsec7}) and (\ref{eq: opTtemposuG}):
\begin{eqnarray}\label{eq: opTtemposuGbis}
&&\left(T_N^{(r)}\(\tau\)G\right)\(Z_N\)=S_N\(-\tau\)\(\hat{T}_N^{(r)}
%\left(S_N\(-\tau\)G\right)\(Z_N\(\tau\)\)
+R_N^{(r)}\)S_N\(\tau\)G\(Z_N\)=\nonumber\\
&&=(-1)^{r/2}\frac{c_r}{N}\sum_{j,l}\(D_x^{r+1}\phi\)(x_j(\tau)-x_l(\tau))\cdot  
D_{v_j}^{r+1}(\tau)G\(Z_N\)+\nonumber\\
&&+\frac{1}{N}\sum_{l,j=1}^{N}\sum_{\substack{k_1,k_2\in  
\mathbb{N}^{3}\\ \vert k_1\vert+\vert k_2\vert=r+1}}C_{k_1,  
k_2}\(\frac{\pa^{r+1}}{\pa_{x_l}^{\vert k_1\vert}\pa_{x_j}^{\vert  
k_2\vert}}\phi\)(x_l(\tau) -  
x_j(\tau))\cdot\frac{\pa^{r+1}}{\pa_{v_l}^{\vert  
k_1\vert}\pa_{v_j}^{\vert k_2\vert}}(\tau)G\(Z_N\).\nonumber\\
&&
\end{eqnarray}
Therefore, in computing the action of $T_N^{(r)}(\tau)$, we have to  
consider derivatives with respect to the variables at time $\tau$.
As a consequence, we have to deal with a complicated function of the  
configuration $Z_N$ which, however, we do not need to make explicit,  
as we shall see in a moment.

On the basis of the previous considerations,
we compute the time derivative of $\eta_1$ by applying the operators  
$\mathcal{D}^{2r}T_N^{(r_n)}(t_n)T_N^{(r_{n-1})}(t_{n-1})\dots  
T_N^{(r_{1})}(t_{1}) $ to the Vlasov equation:
\begin{eqnarray}\label{eq: VlasovWEAKcap7}
&&\left(\pa_t+v'_1\cdot  
\nabla_{x'_1}\right)\mu_N(t)=\(\nabla_{x'_1}\phi\ast\mu_N(t)\)\cdot\nabla_{v'_1}\mu_N(t).
\end{eqnarray}
In doing this we have to compute
\begin{eqnarray}\label{eq: VlasovWEAKcap7BIS}
&&\mathcal{D}^{2r}T_N^{(r_{n})}(t_{n})T_N^{(r_{n-1})}(t_{n-1})  \dots  
T_N^{(r_{1})}(t_{1}) \mu_N(z'_1\vert Z_N(t))\mu_N(z'_2\vert Z_N(t)).
\end{eqnarray}
Now we select the contribution in which each  
$T_N^{(r_{\ell})}(t_{\ell})$ and $\mathcal{D}^{2r}$ apply either on $  
\mu_N(z'_1\vert Z_N(t))$ or to $\mu_N(z'_2\vert Z_N(t))$. The other  
contribution involves terms in which are present products of  
derivatives with respect to the same variable. By Proposition  
\ref{DERIVATE1} and Corollary \ref{PROP5.2} we expect those terms to  
be negligible (in the $\mathcal{C}^{\infty}_{b}$-weak sense) in the  
limit $N\to\infty$.
Therefore we obtain the following equation:
\begin{eqnarray}\label{eq: eqPEReta}
&&\left(\pa_t+v'_1\cdot \nabla_{x'_1}\right)\eta_1(z'_1,t,r,{\bf  
\underline{r}}_n,{\bf  
\underline{t}}_n,Z_N)=L(\mu_N(t))\eta_1(z'_1,t,r,{\bf  
\underline{r}}_n,{\bf \underline{t}}_n,Z_N)+\nonumber\\
&&\quad +\sum_{0\leq \ell\leq r}\ \sum_{0\leq m\leq  
n}\sum_{\substack{I\subset I_n:\\ \vert I\vert=m,\\0<\vert{\bf   
\underline{r}}_I\vert+\ell<k}}\(\nabla_{x'_1}\phi\ast\eta_1(\cdot,t,\ell,{\bf  
\underline{r}}_I,{\bf  
\underline{t}}_I,Z_N)\)\cdot\nabla_{v'_1}\eta_1(z'_1,t,r-\ell,{\bf  
\underline{r}}_{I_n\setminus I},{\bf \underline{t}}_{I_n\setminus  
I},Z_N)+\nonumber\\
&&\quad+E_N^1,
\end{eqnarray}
where $E_N^1$ is an error term which will be proven to be negligible  
in the limit $N\to\infty$ in Appendix C.
In (\ref{eq: eqPEReta}) we used the notations:
\begin{eqnarray}\label{eq: NOTAZIONI}
&& I_n=\{1,2,\dots,n\},\ I\ \text{is any subset of}\ I_n,\  {\bf  
\underline{r}}_I=\{r_j\}_{j\in I}, \  {\bf  
\underline{t}}_I=\{t_j\}_{j\in I}.\nonumber\\
&&
\end{eqnarray}

Next, we compute the time derivative of $\nu_1^{(k)}$. We have:
\begin{eqnarray}\label{eq: eqPERnu}
&&\left(\pa_t+v'_1\cdot \nabla_{x'_1}\right)\nu_1^{(k)}=\sum_{n\geq  
0}(-1)^n\sum_{r=0}^{k}\sum_{\substack{\vert {\bf  
\underline{r}}_n\vert:\\r_j>0\\\vert {\bf \underline{r}}_n\vert  
=k-r}}\left.\int_{0}^t \ud t_2\int_0^{t_2}\ud t_3\dots  
\int_0^{t_{n-1}}\ud t_n \eta_1\(z'_1;t,r,{\bf \underline{r}}_n,{\bf  
\underline{t}}_n,Z_N\)\right \vert_{t_1=t}+\nonumber\\
&&+\sum_{n\geq 0}(-1)^n\sum_{r=0}^{k}\sum_{\substack{\vert {\bf  
\underline{r}}_n\vert:\\r_j>0\\ \vert {\bf \underline{r}}_n\vert  
=k-r}}\int_{ord}^{t}\ud  {\bf \underline{t}}_n \left(\pa_t+v'_1\cdot  
\nabla_{x'_1}\right)\eta_1(z'_1;t,r,{\bf \underline{r}}_n,{\bf  
\underline{t}}_n,Z_N).
\end{eqnarray}
In evaluating the first term on the right hand side of (\ref{eq:  
eqPERnu}), we are lead to consider $\eta_1$ evaluated in $t=t_1$.  
Thus, according to the expression of $\eta_1$ (see (\ref{eq:  
aUnCorpo})), we have to deal with:
\begin{eqnarray}\label{eq: sorgPERnu}
T_N^{(r_1)}(t)\mu_N(z'_1\vert Z_N(t))=S_N(-t)T_N^{(r_1)}\mu_N(z'_1\vert Z_N).
\end{eqnarray}
Therefore:
\begin{eqnarray}\label{eq: sorgPERnuCONTO}
&&T_N^{(r_1)}(t)\mu_N(z'_1\vert Z_N(t))
= (-1)^{r_1/2}c_{r_1}\left(D_{x'_1}^{r_1  
+1}\phi\ast\mu_N(t)\right)(x'_1)\cdot D_{v'_1}^{r_1 +1}\mu_N(z'_1\vert  
Z_N(t))=\nonumber\\
&& = (-1)^{r_1/2}c_{r_1}\int
\ud x'_2\ \ud v'_2\ D_{x'_1}^{r_1 +1}\phi(x'_1-x'_2)\cdot  
D_{v'_1}^{r_1 +1}\mu_N(x'_1,v'_1\vert Z_N(t))\mu_N(x'_2,v'_2\vert  
Z_N(t)),\nonumber\\
&&
\end{eqnarray}
where the term involving off-diagonal derivatives, namely  
$R_N^{(r_1)}$ (see (\ref{eq: OFFdiagOPt})),
disappears because both the derivatives and the empirical distribution  
are evaluated at time $t$.
Hence we compute $\eta_1$ in $t=t_1$ and, inserting it in the first  
term of the right hand side of (\ref{eq: eqPERnu}), we obtain:
\begin{eqnarray}\label{eq: sorgPERnuRESULT}
&& \sum_{n\geq 0}(-1)^n\sum_{r=0}^{k}\ \sum_{\substack{ {\bf  
\underline{r}}_n:\ r_j>0\\\vert {\bf \underline{r}}_n\vert =k-r}}\  
\left.\int_{0}^{t}\ud  t_2\int_{0}^{t_2}\ud t_3 \ud t_3\dots  
\int_{0}^{t_{n-1}}\ud t_n \eta_1\(z'_1;t,r,{\bf \underline{r}}_n,{\bf  
\underline{t}}_n,Z_N\)\right \vert_{t_1=t}=\nonumber\\
&&=\sum_{ \substack{0<r_1\leq k\\r_1 \text{\ even}}}\ (-1)^{r_1/2}c_{r_1}\int
%_{\R^3\times\R^3}
\ud x'_2\ \ud v'_2\ D_{x'_1}^{r_1 +1}\phi(x'_1-x'_2)\cdot  
D_{v'_1}^{r_1 +1}\nu_2^{(k-r_1)}(x'_1,v'_1,x'_2,v'_2;t).\nonumber\\
&&
\end{eqnarray}

Let us come back now to equation (\ref{eq: eqPERnu}). It is useful to  
observe that:
\begin{eqnarray}\label{eq: integraliINtempo}
\int_{ord}^t \ud{\bf \underline{t}}_{n} \sum_{\substack{I\subset  
I_n:\\|I|=m}}=
\int_{ord}^t \ud{\bf \underline{t}}_I \int_{ord}^t \ud{\bf  
\underline{t}}_{I_n\setminus I}.
\end{eqnarray}
Then, putting together (\ref{eq: eqPERnu}), (\ref{eq:  
sorgPERnuRESULT}), (\ref{eq: eqPEReta}) and (\ref{eq:  
integraliINtempo}), we obtain the following equation for $\nu_1^{(k)}$:
\begin{eqnarray}\label{eq: eqPERnuDEF}
&&\left(\pa_t+v'_1\cdot  
\nabla_{x'_1}\right)\nu_1^{(k)}(x'_1,v'_1;t)=L(\mu_N(t))\nu_1^{(k)}(x'_1,v'_1;t)+\nonumber\\
&&\qquad +\sum_{\substack{ 0<r_1\leq k\\r_1 \text{\  
even}}}(-1)^{r_1/2}c_{r_1}\int
\ud x'_2\ \ud v'_2\ D_{x'_1}^{r_1 +1}\phi(x'_1-x'_2)\cdot  
D_{v'_1}^{r_1 +1}\nu_2^{(k-r_1)}(x'_1,v'_1,x'_2,v'_2;t)+\nonumber\\
&&\qquad+\sum_{0<\ell<k}\(\nabla_{x'_{1}}\phi\ast\nu_1^{(\ell)}(t)\)\cdot\nabla_{v'_{1}}\nu_1^{(k-\ell)}(t)+  
E_N^2,
\end{eqnarray}
with initial datum given by:
\begin{eqnarray}\label{eq: eqPERnuINDAT}
\nu_1^{(k)}(x'_1,v'_1;t)\vert_{t=0}=\eta_1((z'_1;0,k,{\bf  
\underline{r}}_0,{\bf  
\underline{t}}_0,Z_N)=\mathcal{D}^{2k}\mu_{N}(z'_1\vert Z_N).
\end{eqnarray}
Here $E_N^2$ arises from $E_N^1$ (see (\ref{eq: eqPEReta})).
Now, we want to prove that:
\begin{eqnarray}\label{eq:convergenzaQO}
&&\nu_{1}^{(k)}(t)\rightarrow f^{(k)}(t),\ \text{as}\ \ N\to\infty,\ \  
\mathcal{C}_b^{\infty}-\text{weakly},
\end{eqnarray}
and
\begin{eqnarray}\label{eq:convergenzaQOadue}
&&\nu_{2}^{(k)}(t)\rightarrow f_2^{(k)}(t),\ \text{as}\ \ N\to\infty,\  
\ \mathcal{C}_b^{\infty}-\text{weakly},
\end{eqnarray}
for any configuration $Z_N$ such that $\mu_N\to f_0^{(0)}$ in the weak  
sense of probability measure
(namely, for any $Z_N$ typical with respect to $f_0^{(0)}$ (see Section 4.4)).
As a consequence, reminding that $\nu_{1}^{(k)}(t)$ and  
$\nu_{2}^{(k)}(t)$ are equal to $\omega_{N,1}^{(k)}(t)$ and  
$\omega_{N,2}^{(k)}(t)$ respectively, a.e. with respect to  
$W_{N,0}^{(0)}$,
(\ref{eq:convergenzaQO}) and (\ref{eq:convergenzaQOadue}) are equivalent to:
\begin{eqnarray}\label{eq:convergenza}
&&\omega_{N,1}^{(k)}(t)\rightarrow f^{(k)}(t),\ \text{as}\ \  
N\to\infty,\ \ \mathcal{C}_b^{\infty}-\text{weakly},
\end{eqnarray}
and
\begin{eqnarray}\label{eq:convergenzaadue}
&&\omega_{N,2}^{(k)}(t)\rightarrow f_2^{(k)}(t),\ \text{as}\ \  
N\to\infty,\ \ \mathcal{C}_b^{\infty}-\text{weakly}.
\end{eqnarray}

As we already remarked, the $\mathcal{C}_b^{\infty}$-weak convergence  
implies the convergence in $\mathcal{S}'$, therefore,  
(\ref{eq:convergenza}) and (\ref{eq:convergenzaadue}) imply the  
convergence of $\omega_{N,1}^{(k)}(t)$ to $f^{(k)}(t)$ in  
$\mathcal{S}'(\R^3\times\R^3)$ and of $\omega_{N,2}^{(k)}(t)$ to  
$f_2^{(k)}(t)$ in $\mathcal{S}'(\R^{6}\times\R^6)$.
\\\\
\subsubsection{ One and two-particle convergence} In evaluating the  
behavior of $\nu_1^k(t)$ when $N\to\infty$, we note that it solves the  
initial value problem (\ref{eq: eqPERnuDEF})-(\ref{eq: eqPERnuINDAT})  
for which we want to use Proposition \ref{PROP6.2}. First, however, we  
have to verify the assumptions. The first one, namely {\bf i)}, is  
verified as follows by our choice of the initial datum which ensures  
$f_0^{(0)}$ to be a smooth probability measure (see Section 4.5) and  
by the classical mean-field theory recalled in Chapter 1.

Now, we have to check that assumption {\bf ii)} is satisfied, namely,  
we have to prove that
\begin{eqnarray}\label{eq:uniformBOUNDNESS}
&&\forall\ u_1,\ u_2\ \text{in\ }\  
\mathcal{C}_b^{\infty}(\R^3\times\R^3),\nonumber\\
&& \text{there exists a constant $C=C(u_1,u_2)>0$, independent of $N$,  
such that:}\nonumber\\
&&\left\|  u_1 \ast \ \(u_2\ \nu_{1}^{(k)}(t)\)  
\right\|_{L^{\infty}\(\R^3\times\R^3\)}<C\ \ \ \text{for any} \  t.
\end{eqnarray}
We have:
\begin{eqnarray}\label{eq:uniformBOUNDNESSproof}
&& \left\|  u_1 \ast \ \(u_2\ \nu_{1}^{(k)}(t)\)  
\right\|_{L^{\infty}\(\R^3\times\R^3\)}
=\sup_{x'_1,v'_1}\left\vert\int \ud y\ud w\ u_1(x'_1-y,  
v'_1-w)u_2(y,w)\nu_{1}^{(k)}(y, w; t)\right\vert\leq\nonumber\\
&&\leq\sum_{n\geq 0}\sum_{r=0}^{k}\ \sum_{\substack{ {\bf  
\underline{r}}_n:\ r_j>0\\\vert {\bf \underline{r}}_n\vert =k-r}}\  
\int_{ord}^{t}\ud  {\bf \underline{t}}_n\sup_{x'_1,v'_1}\left\vert\int  
\ud y\ud w\ u_1(x'_1-y, v'_1-w)u_2(y,w)\eta_{1}(y, w; t;r,{\bf  
\underline{r}}_n ,{\bf \underline{t}}_n ,Z_N)\right\vert=\nonumber\\
&& =\sum_{n\geq 0}\sum_{r=0}^{k}\ \sum_{\substack{{\bf  
\underline{r}}_n:\ r_j>0\\\vert {\bf \underline{r}}_n\vert =k-r}}\  
\int_{ord}^{t}\ud  {\bf \underline{t}}_n\nonumber\\
&&\ \ \ \sup_{x'_1,v'_1}\left\vert\int \ud y\ud w\ \left(u_1(x'_1-y,  
v'_1-w)u_2(y,w)\right)\mathcal{D}^{2r}T_N^{(r_n)}(t_n)\dots  
T_N^{(r_1)}(t_1)\mu_N(y, w\vert Z_N(t))\right\vert=\nonumber\\
&& =\sum_{n\geq 0}\sum_{r=0}^{k}\ \sum_{\substack{{\bf  
\underline{r}}_n:\ r_j>0\\\vert {\bf \underline{r}}_n\vert =k-r}}\  
\int_{ord}^{t}\ud  {\bf \underline{t}}_n\nonumber\\
&&\ \ \ \sup_{x'_1,v'_1}\left\vert\int \ud y\ud w\  
g(x'_1,v'_1,y,w)\mathcal{D}^{2r}T_N^{(r_n)}(t_n)\dots  
T_N^{(r_1)}(t_1)\mu_N(y, w\vert Z_N(t))\right\vert,\nonumber\\
&&
\end{eqnarray}
where we used the notation $g(x'_1,v'_1,y,w):=u_1(x'_1-y,  
v'_1-w)u_2(y,w)$ and,
%in order to emphasize that, in this context, $x$ and $v$ play the  
role of parameters and the variables are indeed $y$ and $w$.
clearly, we have $g(x'_1,v'_1,\cdot,\cdot)\in  
\mathcal{C}^{\infty}_b(\R^3\times\R^3)$ for any $x'_1$ and $v'_1$ and  
$g(\cdot,\cdot,y,w)\in \mathcal{C}^{\infty}_b(\R^3\times\R^3)$ for any  
$y$ and $w$.
By some estimates which will be proven in Appendix C (see Lemma C.2),  
we are guaranteed that, applying the operator  
$\mathcal{D}^{2r}T_N^{(r_n)}(t_n)\dots T_N^{(r_1)}(t_1)$ on the  
empirical measure $\mu_N(t)$ and integrating versus a function in  
$\mathcal{C}^{\infty}_b(\R^3\times\R^3)$ we obtain a quantity  
uniformly bounded in $N$. This feature, by virtue of the good  
properties of the function $g$
ensures that
(\ref{eq:uniformBOUNDNESSproof}) is finite.

Let us now look at the initial datum for $\nu_1^{(k)}(t)$, in order to  
verify assumption {\bf iii)}.

 From (\ref{eq: eqPERnuINDAT}) we know that  
$\nu_1^{(k)}(0)=\mathcal{D}^{2k}\mu_N\in  
\mathcal{S}'(\R^{3}\times\R^{3})$. As regard to its limiting behavior,  
we find that:
\begin{eqnarray}\label{eq: convINDATstruttura}
&&\left. \nu_{1}^{(k)}(t)\right\vert_{t=0}=\mathcal{D}^{2k}\mu_N=
\sum_{n=1}^{N}\ \sum_{\substack{I\subset I_N\\ \vert I\vert=n}}\  
\sum_{\substack{s_j: j\in I\\1\leq s_j\leq k\\\sum_{j}  
s_j=k}}\prod_{j\in I}D_{G,j}^{2 s_j}\ \mu_N,
\end{eqnarray}
where $I_N=\{1,\dots,N\}$.
For our convenience, we have written the action of the operator  
$\mathcal{D}^{2k}$ in a equivalent and slightly different way from  
that we used in (\ref{eq: opDstortoGEN}).

We realize that the only surviving term in the sum (\ref{eq:  
convINDATstruttura}) is that with $n=1$.
Hence:
\begin{eqnarray}\label{eq: convINDATstruttura3}
&& \ \left. \nu_1^k(t)\right\vert_{t=0}=\sum_{j=1}^N  D_{G,j}^{2  
k}\mu_N=\frac{1}{N}\sum_{j=1}^{N}D_{G,j}^{2  
k}\delta(z'_1-z_j)=D_{G}^{2 k}\mu_N.
\end{eqnarray}
Therefore we can conclude, by using the mean-field limit:
\begin{eqnarray}\label{eq: convINDAT}
\left(u, \nu_{1}^{(k)}(t)\vert_{t=0}\right)&&=\left(u,D_{G}^{2  
k}\mu_N\right)=\nonumber\\
&&=\left( D_{G}^{2 k}u,\mu_N\right)\rightarrow \left(D_{G}^{2 k}  
u,f_0^{(0)}\right)=\(u,D_{G}^{2  
k}f_0^{(0)}\right)=\(u,f_0^{(k)}\right),\ \text{as}\ \ N\to\infty,\ \  
\nonumber\\
&&\ \forall\ u\ \text{in}\ \  
\mathcal{C}_{b}^{\infty}\left(\R^3\times\R^3\right).
\end{eqnarray}
Thus, $f_0^{(k)}$ plays the role of $\gamma_0$ in Proposition  
\ref{PROP6.2} and it is in $L^1\(\R^3\times \R^3\)$ because  
$f_0^{(0)}\in \mathcal{S}\(\R^3\times \R^3\)$.

We conclude the convergence proof (for the one and two-particle  
functions) by induction.
For $k=0$ we know that, for any configuration $Z_N$ which is typical  
with respect to $f_0^{(0)}$,
we have:
\begin{eqnarray}\label{eq:convergenzaQOclas}
&&\nu_{1}^{(0)}(t)=\mu_N(t)\rightarrow f^{(0)}(t),\ \ \text{as}\ \ N\to\infty
\end{eqnarray}
in the weak sense of probability measures (see (\ref{eq: class11}) and  
(\ref{eq: class12})) and, as a consequence,
the convergence holds $\mathcal{C}_{b}^{\infty}-\text{weakly}$. Moreover
\begin{eqnarray}\label{eq:convergenzaQOclasAdue}
&&\nu_{2}^{(0)}(t)=\mu_N(t)\otimes\mu_N(t)\rightarrow  
f_2^{(0)}(t)=f^{(0)}(t)\otimes f^{(0)}(t),\ \text{as}\ \ N\to\infty,
\end{eqnarray}
in the weak sense of probability measures
and, as previously,
the convergence holds $\mathcal{C}_{b}^{\infty}-\text{weakly}$.

We make the following inductive assumptions for all $h<k$:
\begin{eqnarray}\label{eq:convergenzaQOinduz}
&&\nu_{1}^{(h)}(t)\rightarrow f^{(h)}(t),\ \text{as}\ \ N\to\infty,\ \  
\  \mathcal{C}_{b}^{\infty}-\text{weakly},
\end{eqnarray}
for any configuration $Z_N$ which is typical with respect to $f_0^{(0)}$,
and
\begin{eqnarray}\label{eq:convergenzaQOinduzAdue}
&&\nu_{2}^{(h)}(t)\rightarrow f_2^{(h)}(t)=\sum_{0\leq q\leq  
h}f^{(q)}(t)f^{(h-q)}(t),\ \text{as}\ \ N\to\infty,\ \   
\mathcal{C}_{b}^{\infty}-\text{weakly},
\end{eqnarray}
for any configuration $Z_N$ which is typical with respect to $f_0^{(0)}$.

Now we want to prove that (\ref{eq:convergenzaQOinduz}) and  
(\ref{eq:convergenzaQOinduzAdue})
hold also for $h=k$.\\
Thanks to (\ref{eq:convergenzaQOinduz}), we can affirm that:
\begin{eqnarray}\label{eq:convergenzaQOinduzFINE1}
&&\sum_{0<\ell<k}\left(\nabla_{x'_1} \phi \ast \nu_{1}^{(\ell)}\right)  
\cdot \nabla_{v'_1}  
\nu_{1}^{(k-\ell)}\to\sum_{0<\ell<k}\left(\nabla_{x'_1} \phi \ast  
f^{(\ell)}\right) \cdot \nabla_{v'_1}  
f^{(k-\ell)}=\sum_{0<\ell<k}T^{(0)}_{\ell}f^{(k-\ell)},\nonumber\\
&&\nonumber\\
&&\mathcal{C}_{b}^{\infty}-\text{weakly},
\end{eqnarray}
and, thanks to (\ref{eq:convergenzaQOinduzAdue}), have:
\begin{eqnarray}\label{eq:convergenzaQOinduzFINE2}
&& \sum_{\substack{0<r_1 \leq k\\r_1\  
\text{even}}}(-1)^{r_1/2}c_{r_1}\int \ud x'_2  \ud v'_2    
D_{x'_1}^{r_1 +1}
\phi (x'_1-x'_2)  \cdot D_{v'_1}^{r_1 +1}
\nu_{2}^{(k-r_1)}(x'_1,v'_1,x'_2,v'_2;t)\nonumber\\
&&\nonumber\\
&&\downarrow\ \ \ \mathcal{C}_{b}^{\infty}-\text{weakly}\nonumber\\
&&\nonumber\\
&&  \sum_{\substack{0<r_1 \leq k\\r_1\  
\text{even}}}(-1)^{r_1/2}c_{r_1}\int \ud x'_2  \ud v'_2    
D_{x'_1}^{r_1 +1}
\phi (x'_1-x'_2)  \cdot D_{v'_1}^{r_1 +1}
f_{2}^{(k-r_1)}(x'_1,v'_1,x'_2,v'_2;t)=\nonumber\\
&&=\sum_{\substack{0<r_1 \leq k\\r_1\ \text{even}}}\ \sum_{0\leq q\leq  
k-r_{1}}(-1)^{r_1/2}c_{r_1}\int \ud x'_2 \ud v'_2  D_{x'_1}^{r_1 +1}
\phi (x'_1-x'_2)f^{(k-r_{1})}(x'_2,v'_2;t) \cdot  D_{v'_1}^{r_1 +1}
f^{(q)}(x'_1,v'_1;t)=\nonumber\\
&&=\sum_{\substack{0<r_1\leq k\\r_1\ \text{even}}}\ \sum_{0\leq q\leq  
k-r_{1}}T_{q}^{(r_1)}f^{(k-r_1-q)}(t).\nonumber\\
&&
\end{eqnarray}

At the end, putting together (\ref{eq:convergenzaQOinduzFINE1}) and  
(\ref{eq:convergenzaQOinduzFINE2}), we find that the sum of the source  
terms in equation (\ref{eq: eqPERnuDEF}) converges  
$\mathcal{C}_{b}^{\infty}$-weakly to:
\begin{eqnarray}\label{eq:sorgenteNUOVA}
&&\sum_{\substack{0<\ell<k}}T^{(0)}_{\ell}f^{(k-\ell)}+\sum_{\substack{  
0<r_{1}\leq k\\ 0 \leq q\leq k-r_{1}}}T_{q}^{(r_1)}f^{(k-r_1-q)},
\end{eqnarray}
which plays the role of $\Theta$ in Proposition \ref{PROP6.2} and it  
is easy to verify that it is in $\mathcal{C}^0\(L^1(\R^3\times\R^3),  
\mathbb{R}^+\)$.
Therefore, we can apply Proposition \ref{PROP6.2} claiming that, for  
any typical configuration $Z_N$ with respect to $f_0^{(0)}$,
$\nu_1^{(k)}(t)$ converges $\mathcal{C}_{b}^{\infty}$-weakly to the  
solution of the problem (\ref{eq: Prop6.1}).
Looking at (\ref{eq: WigLIOUseq}) and (\ref{eq: Teta}), we realize  
that we obtained the equation satisfied by $f^{(k)}(t)$.

In order to ''close'' the recurrence procedure, it remains to show the  
two-particle convergence at order $k$. It follows from the  
one-particle analysis and from the following computation
(see (\ref{eq: aDueCorpi})):
\begin{eqnarray}\label{eq: eta2BIS}
&&\eta_{2}(z'_1,z'_2;t,r,{\bf \underline{r}}_n,{\bf  
\underline{t}}_n,Z_N)=\nonumber\\
&&=\sum_{0\leq \ell\leq k}\sum_{0\leq m\leq  
n}\sum_{\substack{I:I\subseteq I_n\\ \vert  
I\vert=m}}\eta_{1}(z'_1;t,\ell,{\bf \underline{r}}_I,{\bf  
\underline{t}}_I,Z_N)\eta_{1}(z'_2;t, k-\ell,{\bf  
\underline{r}}_{I_n\setminus I},{\bf \underline{t}}_{I_n\setminus  
I},Z_N)+R_N^2,
\end{eqnarray}
where $R_N^2$ is a remainder arising from the action of the operator  
$\mathcal{D}^{2r}T_N^{(r_n)}(t_n)\dots T_N^{(r_1)}(t_1)$ on a product  
of two empirical measures $\mu_N(t)$. In Appendix C we will see that  
it is vanishing in the limit.
As a consequence, $\nu_{2}^{(k)}$ (see (\ref{eq: senza MEDIA}) for  
$j=2$) is such that:
\begin{eqnarray}\label{eq:kORDdue2TRIS}
&&\nu_{2}^{(k)}(t)= \sum_{0\leq q\leq  
k}\nu_{1}^{(q)}(t)\nu_{1}^{(k-q)}(t)+ o(1),
\end{eqnarray}
in the limit $N\to\infty$. Therefore, from the inductive assumption  
(\ref{eq:convergenzaQOinduz}) and from the one-particle convergence at  
order $k$, we conclude that:
\begin{eqnarray}\label{eq:convAdueFINALE}
&&\nu_{2}^{(k)}(t)\rightarrow \sum_{0\leq q\leq  
k}f^{(q)}(t)f^{(k-q)}(t)=f_2^{(k)}(t),\ \text{as}\ \ N\to\infty,\   
\mathcal{C}_{b}^{\infty}-\text{weakly},
\end{eqnarray}
for any configuration $Z_N$ which is typical with respect to $f_{0}^{(0)}$.
Thus, we have just proven the convergence of $\omega_{N,j}^{(k)}$ in  
the cases $j=1$, $j=2$.
\subsubsection{ $j$-particle convergence}
As for $j=2$, the $j$-particle convergence can be reduced by the  
one-particle control. Indeed by (\ref{eq: senza MEDIA}) and (\ref{eq:  
eta_j}) we have:
\begin{eqnarray}\label{eq:kORDj}
&&\nu_{j}^{(k)}(t)= \sum_{\substack{s_1\dots s_j\\0\leq s_m\leq  
k\\\sum_m s_m=k}}\prod_{m=1}^{j}\nu_{1}^{(s_m)}(t)+ R_N^j,\\
&&\text{ with $R_N^j\to 0$ when $N\to\infty$.}\nonumber
\end{eqnarray}
Again the error term $R_N^j$ arises from the presence of products of  
derivatives with respect to the same variable.
In conclusion, the result we proved for $\nu_1^{(k)}(t)$, together  
with the estimates proven in Appendix C, is sufficient to guarantee  
the $\mathcal{C}^{\infty}_b$-weak convergence of $\nu_{j}^{(k)}(t)$ to  
$f_j^{(k)}(t)$ for any $j$ (for any typical configuration $Z_N$ with  
respect to
$f_0^{(0)}$), and, as a consequence, the $\mathcal{C}^{\infty}_b$-weak  
convergence of $\omega_{N,j}^{(k)}(t)$ is $f_j^{(k)}(t)$, for any $j$.

The final step is to realize that this convergence does imply that for  
the coefficients $W_{N,j}^{(k)}(t)$, namely what is established by  
Theorem \ref{THEO6.1}.

First of all, we observe that, for any test function $u$ we have:
\begin{eqnarray}\label{eq: W_N,1sufunztestSEC7}
\(u,W_{N,1}^{(k)}(t)\)&&=\int_{\R^6}\ud z_1  
W_{N,1}^{(k)}(z_1;t)u(z_1)=\nonumber\\
&&=\int_{\R^{3N}\times \R^{3N}}\ud Z_{N}W_{N}^{(k)}(Z_N;t)u(z_1)=\nonumber\\
&&=\int_{\R^{3N}\times \R^{3N}}\ud  
Z_{N}W_{N}^{(k)}(Z_N;t)\frac{1}{N}\sum_{l=1}^{N}u(z_l)=\nonumber\\
&&=\int_{\R^{3N}\times \R^{3N}}\ud  
Z_{N}W_{N}^{(k)}(Z_N;t)\(u,\mu_N\)=\(u,\omega_{N,1}^{(k)}(t)\),\nonumber\\
&&
\end{eqnarray}
where we made use of the symmetry of the coefficient  
$W_{N}^{(k)}(Z_N;t)$ with respect to any permutation of the variables  
(the computation is the same we did in Section 4.4 for  
$W_{N,1}^{(1)}(t)$).
 From (\ref{eq: W_N,1sufunztestSEC7}), we can see that  
$W_{N,1}^{(k)}(t)$ and $\omega_{N,1}^{(k)}(t)$ are equal as  
distributions in $\mathcal{S}'\(\R^3\times\R^3\)$ (in particular, we  
can choose test functions belonging to  
$\mathcal{C}^{\infty}_b\(\R^3\times\R^3\)$), then the convergence of  
$W_{N,1}^{(k)}(t)$ is proven. Moreover, for $j\geq 2$, a  
straightforward computation shows that, by fixing an index  
$\overline{j}$, we have
\begin{eqnarray}\label{eq: ERR4}
&&\(u_{\overline{j}},\omega_{N,\overline{j}}^{(k)}(t)\)=
\frac{N(N-1)\dots  
(N-\overline{j}+1)}{N^{\overline{j}}}\(u_{\overline{j}},W_{N,\overline{j}}^{(k)}(t)\)+\frac{C_{j<\overline{j}}}{N},
\end{eqnarray}
where $C_{j<\overline{j}}<\infty$ provided that  
$\(u_j,W_{N,j}^{(k)}(t)\)$ is uniformly bounded for each  
$j<\overline{j}$.
Then, to conclude the proof of Theorem \ref{THEO6.1}, it is enough to  
use a recurrence argument.

\section{Outlooks and Perspectives}
\setcounter{equation}{0}
\def\theequation{6.\arabic{equation}}

In this chapter we give an overview of possible developments of our  
research, both in the perspective of
%improving or
generalizing our result, and from the point of view of applications in  
other (somewhat related) fields.\\

First we observe that the we can apply Theorem \ref{THEO6.1} even by  
considering as one-particle initial datum a suitable mixtures of WKB  
states. Indeed, let us consider the one-particle WKB state described  
by the wave function $\psi_{WKB}\in L^2\(\R^3\)$ given by:
\begin{eqnarray}\label{eq: WKB}
\psi_{WKB}(x\vert v_0)=a(x)e^{i \frac{v_0\cdot x}{\var}}, \ \ \ \ \ \  
\ \ \ a\in\mathcal{S}(\R^3),\ a(x)\in \R\ \ \forall\ \ x,\ \ v_0\ \  
\text{fixed in}\  \R^3,
\end{eqnarray}
where the amplitude $a$ is assumed to verify
\begin{eqnarray}\label{eq: ASSsuA}
\int \ud x\  a^2(x)=1,
\end{eqnarray}
namely, $a^2(x)$ can be interpreted as a one-particle probability  
density in the position phase space $\R^3$.
The Wigner function associated with (\ref{eq: WKB}), given by
\begin{eqnarray}\label{eq: WKB1}
f^\var_{WKB}(x,v\vert v_0)&&=\(2\pi\)^{-3}\int_{\R^3} \ud y\  e^{i\   
v\cdot y}\ \overline{\psi}_{WKB}\(x+\frac{\var y}{2}\vert  
v_0\)\psi_{WKB}\(x-\frac{\var y}{2}\vert v_0\)=\nonumber\\
&&=
\(2\pi\)^{-3}\int_{\R^3} \ud y \ e^{i\  y\cdot v}\
a\(x+\frac{\var y}{2}\)e^{-i \frac{v_0\cdot \(x+\frac{\var  
y}{2}\)}{\var}}\ a\(x-\frac{\var y}{2}\)e^{i \frac{v_0\cdot  
\(x-\frac{\var y}{2}\)}{\var}}=\nonumber\\
&&=\(2\pi\)^{-3}\int_{\R^3} \ud y \ e^{i\  y\cdot \(v- v_0 \)}\
a\(x+\frac{\var y}{2}\)\ a\(x-\frac{\var y}{2}\),
\end{eqnarray}
can be expanded as follows
\begin{eqnarray}\label{eq: WKB2}
f^\var_{WKB}(x,v\vert v_0)=f^{(0)}_{WKB}(x,v\vert v_0)+\var  
f^{(1)}_{WKB}(x,v\vert v_0)+\var^2 f^{(2)}_{WKB}(x,v\vert v_0)+\dots
\end{eqnarray}
where
\begin{eqnarray}
&& f_{WKB}^{(0)}(x,v\vert v_0)=a^2(x)\delta(v-v_0),\label{eq: coeff_0WKB}\\
&& f_{WKB}^{(2n+1)}(x,v\vert v_0)=0\ \ \ \ \ \ \forall\ \  
n=0,1,2,\dots,\label{eq: coeffODDWKB}\\
&& f_{WKB}^{(2n)}(x,v\vert v_0)=-\frac{1}{(2)^{2n}}\sum_{l=0}^{2n}  
\frac{1}{l!}\frac{1}{(2n-l)!}(-1)^l D^l a(x) D^{2n-l }a(x)  
D_v^{2n}\delta (v-v_0),\ \  \ \forall\ n=1,2,\dots \nonumber\\
&& \label{eq: coeffEVENwkb}
\end{eqnarray}
and $D_v^m\delta(v-v_0)$, for any $m>0$, is the distribution acting as
\begin{eqnarray} \label{eq: WKB3}\\
\(u,D_v^m\delta(v-v_0)\)&&=\int
%\ud x
\ud v\ u(v) D_v^m\delta(v-v_0)=\nonumber\\
&&=(-1)^m\int \ud v\ D_v^m u(v)\delta(v-v_0)=(-1)^m D_v^m u(v_0),\nonumber\\
&&
\end{eqnarray}
for any smooth test function $u$.

Then, we consider the Wigner function
\begin{eqnarray}\label{eq: WKB4}
f^\var_{W}(x,v)=\int \ud v_0\ g_W(v_0)f^\var_{WKB}(x,v\vert v_0), \ \  
\ \ \ \ g_W\in\mathcal{S}\(\R^3\)
\end{eqnarray}
associated with the (continuum) mixed state (see paragraph ''Mixed  
states'' in Section 2.1) described by the density matrix (kernel)
\begin{eqnarray}\label{eq: WKB5}
\rho_{W}(x,v)= \int \ud v_0 g_W(v_0)\overline{\psi}_{WKB}(x\vert v_0)  
\psi_{WKB}(y\vert v_0)
\end{eqnarray}
and we assume $g_W$ to be a probability density with respect to the  
velocity variable, $g_W$ not depending on $\var$ (states similar to  
(\ref{eq: WKB4}) have been considered in \cite{Graffi}).
By (\ref{eq: WKB2}), (\ref{eq: coeff_0WKB}), (\ref{eq: coeffODDWKB})  
and (\ref{eq: coeffEVENwkb}) we obtain that (\ref{eq: WKB4}) is  
expanded as:
\begin{eqnarray}\label{eq: WKB6}
f^\var_{W}(x,v)=f^{(0)}_{W}(x,v)+\var f^{(1)}_{W}(x,v)+\var^2  
f^{(2)}_{W}(x,v)+\dots
\end{eqnarray}
where
\begin{eqnarray}
&& f_{W}^{(0)}(x,v)=a^2(x)g_W(v),\label{eq: coeff_0WKBmix}\\
&& f_{W}^{(2n+1)}(x,v)=0\ \ \ \ \ \ \forall\ \  
n=0,1,2,\dots,\label{eq: coeffODDWKBmix}\\
&& f_{W}^{(2n)}(x,v)=-\frac{1}{(2)^{2n}}\sum_{l=0}^{2n}  
\frac{1}{l!}\frac{1}{(2n-l)!}(-1)^l D^l a(x) D^{2n-l }a(x)  
D_v^{2n}g_W(v).\ \  \ \forall\ n=0,1,2,\dots \nonumber\\
&& \label{eq: coeffEVENwkbMIX}
\end{eqnarray}
By virtue of our assumptions on $a$ and $g_W$ we find that  
$f_{W}^{(0)}$ defined by (\ref{eq: coeff_0WKBmix}) is a one-particle  
probability density and $f_{W}^{(0)}\in\mathcal{S}\(\R^3\)$.  
Furthermore, we choose $a$ in such a way that
\begin{eqnarray} \label{eq: A}
&& D^m a(x)= \alpha^{(m)}(x) a(x), \ \forall\ m\geq 1\ \ \ \  
\text{with}\ \ \ \alpha^{(m)}(x)\in C^0\(\R^3\),
\end{eqnarray}
$C^0\(\R^3\)$ being the space of continuous functions. Therefore, by  
(\ref{eq: coeffEVENwkbMIX}) we find
\begin{eqnarray}
&& f_{W}^{(2n)}(x,v)=-\frac{1}{(2)^{2n}}\sum_{l=0}^{2n}  
\frac{1}{l!}\frac{1}{(2n-l)!}(-1)^l D^l a(x) D^{2n-l }a(x)  
D_v^{2n}g_W(v)=\nonumber\\
&&\qquad\qquad\ \  = \beta^{(2n)}(x)  
a^2(x)D_v^{2n}g_W(v)=\beta^{(2n)}(x) D_v^{2n}f_{W}^{(0)}(x,v)\ \  \  
\forall\ n=0,1,2,\dots \nonumber\\
&& \label{eq: coeffEVENwkbMIXagain}
\end{eqnarray}
where $\beta^{(2n)}(x)=-\frac{1}{(2)^{2n}}\sum_{l=0}^{2n}  
\frac{1}{l!}\frac{1}{(2n-l)!}(-1)^l \alpha^{(l)}(x)\alpha^{(2n-l)}(x)$  
for any $n\geq 1$ and in the last equality of (\ref{eq:  
coeffEVENwkbMIXagain}) we used (\ref{eq: coeff_0WKBmix}). By the  
smoothness of $\alpha$ (see (\ref{eq: A})) it follows that  
$\beta^{(2n)}(x)\in C^0\(\R^3\)$ for all $n$, thus (\ref{eq:  
coeffEVENwkbMIXagain}) together with the smoothness of $f_{W}^{(0)}$  
implies that $f_{W}^{(2n)}\in \mathcal{S}\(\R^3\times\R^3\)$ for all  
$n\geq 1$.

Therefore, we have
\begin{eqnarray}
&& f_{W}^{(0)}\in \mathcal{S}\(\R^3\times\R^3\)
\label{eq: coeff_0WKBmixREG}\\
&& f_{W}^{(2n+1)}(x,v)=0\ \ \ \ \ \ \ \ \forall\  
n=0,1,2,\dots,\label{eq: coeffODDWKBmixREG}\\
&& f_{W}^{(2n)}\in \mathcal{S}\(\R^3\times\R^3\)
\ \  \ \forall\ n=1,2,\dots \nonumber\\
&& \label{eq: coeffEVENwkbMIXREG}
\end{eqnarray}
and, by applying Proposition \ref{PROP6.1} as in Section 4.1, we can  
identify  the time-evolved coefficients $f_W^{(k)}(t)$, for each  
$k\geq 0$, as the unique solutions of problems (\ref{eq:  
WigLIOUseqk=0}) and (\ref{eq: WigLIOUseq}) in $L^1
\(\R^3\times\R^3\)$. Clearly, by (\ref{eq: coeffODDWKBmixREG}) we find
\begin{eqnarray}\label{eq: dispariNULLI}
&& f_{W}^{(2n+1)}(x,v;t)\equiv 0,\ \ \  \text{for each}\ n\geq 0 \  
\text{and}\ \forall\ \ t,
\end{eqnarray}
while $f_W^{(2n)}(t)\in\mathcal{S}
\(\R^3\times\R^3\)$ for any $n\geq 0$. In particular, $f_W^{(0)}(t)$  
is the unique solution of the Vlasov equation (\ref{eq: Vlasov}) with  
initial datum $f_W^{(0)}$, thus we are guaranteed that $f_W^{(0)}(t)$  
is a one-particle probability density for all times.

Let us consider the following factorized initial datum for the  
$N$-particle Wigner-Liouville equation (\ref{eq: WigLIOUhartreeNsec3})
\begin{eqnarray}\label{eq: WKBindatN}
&& W_{N,W}^\var(X_N,V_N)=\(f_W^{\var}\)^{\otimes N}(X_N,V_N).
\end{eqnarray}
Then we find
\begin{eqnarray}\label{eq: WKBindatNexp}
&& W_{N,W}^\var(X_N,V_N)=W_{N,W}^{(0)}(X_N,V_N)+\var  
W_{N,W}^{(1)}(X_N,V_N)+\var^2 W_{N,W}^{(2)}(X_N,V_N)+\dots,\nonumber\\
&&
\end{eqnarray}
where, by (\ref{eq: WKB6}),
\begin{eqnarray}
&& W_{N,W}^{(0)}(X_N,V_N)=\(f_W^{(0)}\)^{\otimes N}(X_N,V_N)\label{eq:  
coeff_0WKBmixFACT}\\
&& W_{N,W}^{(n)}(X_N,V_N)=\sum_{\substack{s_1\dots s_N\\0\leq s_j\leq  
n\\ \sum_j s_j=n}}\prod_{j=1}^{N}f_W^{(s_j)}(x_j, v_j)\ \ \   
\text{for}\ n\geq 1,\label{eq: coeffNindat}
\end{eqnarray}
and we note that, as in the case we considered in Chapter 4,  
factorization holds only for the zero order coefficient which,  
furthermore, turns to be an $N$-particle probability density.

By (\ref{eq: coeffNindat}), thanks to (\ref{eq: coeffODDWKBmix}) and  
(\ref{eq: coeffEVENwkbMIXagain}), we find
\begin{eqnarray}
&& W_{N,W}^{(0)}(X_N,V_N)=\(f_W^{(0)}\)^{\otimes N}(X_N,V_N)\label{eq:  
coeff_0WKBmixFACTII}\\
&& W_{N,W}^{(n)}(X_N,V_N)=\sum_{\substack{s_1\dots s_N  
\text{even}\\0\leq s_j\leq n\\ \sum_j  
s_j=n}}\prod_{j=1}^{N}\beta^{(s_j)}(x_j)D_{v_j}^{s_j}W_{N,W}^{(0)}(X_j,V_j)\ \  
\  \text{for}\ n\geq 1.\label{eq: coeffNindatII}
\end{eqnarray}

Defining the operator $\hat{\mathcal{D}}^{r}$, for any $r$ even, as:
\begin{eqnarray}\label{eq: opDstortoGENwkb}
&&\hat{\mathcal{D}}^{0}=\texttt{1},\nonumber\\
&&\hat{\mathcal{D}}^{r}=\sum_{\substack{s_1\dots  
s_N\text{even}:\\0\leq s_j\leq k\\\sum_j  
s_j=r}}\prod_{j=1}^{N}\beta^{(s_j)}(x_j)D_{v_j}^{s_j},\ \ \ r\geq 2,
\end{eqnarray}
we have:
\begin{eqnarray}
&&W_{N,W}^{(2n+1)}(X_N,V_N)=0,\ \ \ \ \  \ \ \ \ \ \ \ \ \  \ \ \ \ \  
\ \ \ \ \text{for}\ n=0,1,2,\dots\label{eq: W_N,W^kODD}\\
&&W_{N,W}^{(2n)}(X_N,V_N)=\hat{\mathcal{D}}^{2n}W_{N,W}^{(0)}(X_N,V_N)\ \ \ \  
\text{for}\ n=0,1,2,\dots.\label{eq: W_N,W^kEVEN}
\end{eqnarray}

Now, let us consider the factorized $j$-particle Wigner function  
$\(f_W^{\var}(t)\)^{\otimes j}$, where $f_W^{\var}(t)$ is the solution  
of the (Hartree) nonlinear Wigner-Liouville equation (\ref{eq:  
WigLIOUhartree}) with initial datum $f_W^\var$ given by (\ref{eq:  
WKB4}). The product $\(f_W^{\var}(t)\)^{\otimes j}$ can be expanded as
\begin{eqnarray}\label{eq: WKBindatINFexp}
&& \(f_W^{\var}(t)\)^{\otimes j}=f_{j,W}^{(0)}(t)+\var  
f_{j,W}^{(1)}(t)+\var^2 f_{j,W}^{(2)}(t)+\dots,\nonumber\\
&&
\end{eqnarray}
where, by the analysis done in Section 4.3 and thanks to (\ref{eq:  
coeffODDWKBmix}), we have
\begin{eqnarray}
&& f_{j,W}^{(0)}(t)=\(f_W^{(0)}(t)\)^{\otimes j}\label{eq: coeffF_jwkb}\\
&& f_{j,W}^{(2n+1)}(t)=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \  \ \  
\ \ \ \ \forall\ n\geq 0\label{eq: coeffF_jwkbODD}\\
&& f_{j,W}^{(2n)}(t)=\sum_{\substack{s_1\dots s_j \text{even}\\0\leq  
s_m\leq 2n\\ \sum_m s_m=2n}}\prod_{m=1}^{j}f_{W}^{(s_m)}(t),\ \ \  
\forall\ n\geq 1,
\end{eqnarray}
$f_W^{(0)}(t)$ solving the Vlasov equation (\ref{eq: Vlasov}) with  
initial datum $f_W^{(0)}$ and $f_{W}^{(s_m)}(t)$, with $1\leq s_m\leq  
2n$, obtained by (\ref{eq: WigLIOUseq}).

By the analysis done in Section 4.2, we find that the $N$-particle  
zero order coefficient $W_{N,W}^{(0)}(t)$ solves
\begin{equation}\label{eq: WigLiouNk=0WKB}
\left\{
\begin{aligned}
&\left(\pa_t+V_N\cdot  
\nabla_{X_N}\right)W_{N,W}^{(0)}(t)=T_{N}^{(0)}W_{N,W}^{(0)}(t),\\
&\left.W_{N,W}^{(0)}(X_N,V_N;t)\right\vert_{t=0}=\(f_W^{(0)}\)^{\otimes  
N}(X_N,V_N)
\end{aligned}
\right.
\end{equation}
the odd coefficients $W_{N,W}^{(2n+1)}(t)$ are determined by
\begin{equation}\label{eq: WigLiouWKBODD}
\left\{
\begin{aligned}
&\left(\pa_t+V_N\cdot  
\nabla_{X_N}\right)W_{N,W}^{(2n+1)}(t)=T_N^{(0)}W_{N,W}^{(2n+1)}(t),\\
&\left.W_{N,W}^{(2n+1)}(X_N,V_N;t)\right\vert_{t=0}=0,\ \ \  \ \ \ \  
k=0,1,2,\dots
\end{aligned}
\right.
\end{equation}
and the even terms $W_{N,W}^{(2n)}(t)$ solve
\begin{equation}\label{eq: WigLiouNWKBeven}
\left\{
\begin{aligned}
&\left(\pa_t+V_N\cdot  
\nabla_{X_N}\right)W_{N,W}^{(2n)}(t)=T_N^{(0)}W_{N,W}^{(2n)}(t)+\Theta_{N,W}^{(2n)}(t),\\
&\left.W_{N,W}^{(2n)}(X_N,V_N;t)\right\vert_{t=0}=\hat{\mathcal{D}}^{2n}W_{N,W}^{(0)}(X_N,V_N),\ \ \  \ \ \ \  
n=1,2,\dots
\end{aligned}
\right.
\end{equation}
where
\begin{equation}\label{eq: Teta_Neven}
\Theta_{N,W}^{(2n)}(t)=\sum_{\substack{0\leq  
l<2n}}T_N^{(2n-l)}W_{N,W}^{(l)}(t).
\end{equation}

We observe that the odd coefficients $W_{N,W}^{(2n+1)}(t)$ solve  
homogeneous Liouville equations with zero initial data, then  
$W_{N,W}^{(2n+1)}(t)\equiv 0$ for all $n\geq 0$. On the contrary, the  
zero order term $W_{N,W}^{(0)}(t)$ solve the Liouville problem  
(\ref{eq: WigLiouNk=0WKB}), thus, denoting by  
$\{W_{j,W}^{(0)}(t)\}_{j=1}^N$ the corresponding $j$-particle  
marginals, by the classical mean-field theory we obtain
\begin{equation}\label{eq: convWKBzero}
W_{j,W}^{(0)}(t)\rightarrow \(f_W^{(0)}(t)\)^{\otimes j},	\ \ \  
\text{as}\ N\to\infty,\ \ \ \ \text{for any fixed}\ \ j
\end{equation}
in the weak topology of probability measures and, in particular, in  
$\mathcal{S}'\(\R^{3j}\times\R^{3j}\)$, where  
$\(f_W^{(0)}(t)\)^{\otimes j}$ is given by (\ref{eq: coeffF_jwkb}).

As regard to the higher order terms, we can apply exactly the same  
strategy presented in Chapter 4, replacing the operator $D^k_G$  
defined in (\ref{eq: operatorD_G}) with $\beta^{(k)}(x)D_v^k$ (see  
(\ref{eq: coeffEVENwkbMIXagain}) ) and the operator $\mathcal{D}^r$  
(see Section 4.6) with $\hat{\mathcal{D}}^r$ (see (\ref{eq:  
opDstortoGENwkb})). In the end we prove that, for $k\geq 1$
\begin{equation}\label{eq: convWKB}
W_{j,W}^{(k)}(t)\rightarrow f_{W,j}^{(k)}(t),	\ \ \ \text{as}\  
N\to\infty,\ \ \ \ \text{for any fixed}\ \ j
\end{equation}
in $\mathcal{S}'\(\R^{3j}\times\R^{3j}\)$. Clearly, (\ref{eq:  
convWKB}) is trivially verified for $k=2n+1,$
because we have already noticed that
\begin{equation}\label{eq: DISPARItrivial}
W_{j,W}^{(2n+1)}(t)\equiv f_{j,W}^{(2n+1)}(t)\equiv 0,\ \ \ \forall	 \  
n\geq 0.
\end{equation}
\vspace{0.3cm}

As we observed in Remark \ref{Remark6.1}, the assumption on the  
initial $N$-particle Wigner function to be a product of mixed states  
prevents the possibility of considering bosonic states.
Indeed,
factorized states could be compatible with the bosonic statistics if  
pure states were considered.\\

It is easy to check that the classical limit is equivalent to the  
limit of heavy particles. In fact, if we set $\var= 1$ in the  
$N$-particle Hamiltonian (\ref{eq: quant2}) but we let the particle  
mass $m$ (previously chosen equal to 1) become large, by imposing the  
condition that the kinetic
energy per particle is independent of $m$, we find exactly the  
mean-field hamiltonian (\ref{eq: quant2}) where $\var$ is replaced by  
the ''effective Planck constant'' $\var_{m}=1/\sqrt{m}$ going to zero  
as $m\to\infty$. Then a possible application of our result would be  
that of studying the approximations of dynamics of this type, in which  
one looks at a particular scaling which corresponds to the  
semiclassical one, even if interpreted in a different sense.












\newpage
\section*{Appendix A}
\setcounter{equation}{0}
\def\theequation{A.\arabic{equation}}
{\bf Proof of Proposition \ref{DERIVATE1}}\\

To avoid inessential notational complications, we deal with the  
one-dimensional case. \\
By the Newton equations, we have:
\begin{eqnarray}\label{eq:est7v}
&&\frac{\pa x_{i}(t)}{\pa v_r}=\delta_{ir}t+\int_{0}^{t}\ud s  
(t-s)\frac{1}{N}\sum_{ j\neq i}^N  
\pa_{x}F\left(x_{i}(s)-x_{j}(s)\right)\left(\frac{\pa x_{i}(s)}{\pa  
v_{r}}-\frac{\pa x_{j}(s)}{\pa v_{r}}\right),\\
&&\nonumber\\
&&\frac{\pa v_{i}(t)}{\pa v_{r}}=\delta_{ir} + \int_{0}^{t} \ud  
s\frac{1}{N}\sum_{j\neq i}^{N}\pa_{x}  
F\left(x_{i}(s)-x_{j}(s)\right)\left(\frac{\pa x_{i}(s)}{\pa  
v_r}-\frac{\pa x_{j}(s)}{\pa v_{r}}\right),\label{eq:est7vBIS}
\end{eqnarray}
where:
\begin{eqnarray}\label{eq:est1}
F=-\nabla_{x}\phi,
\end{eqnarray}
is the force associated with the potential $\phi$.

Let us analyze in detail the derivative of $x_{i}(t)$. From  
(\ref{eq:est7v}), we get:
\begin{eqnarray}\label{eq:estAppA}
&&
\max_{\substack{i,r\\t\leq T}}\left\vert \frac{\pa x_{i}(t)}{\pa  
v_r}\right\vert\leq C.
\end{eqnarray}
Inserting this estimate again in (\ref{eq:est7v}), we realize that we  
can obtain a better bound for $\frac{\pa v_{i}(t)}{\pa v_{r}}$ in the  
case $r\neq i$ (see \cite{Graffi}), namely:
\begin{eqnarray}\label{eq:estAppA1}
\left\vert \frac{\pa x_{i}(t)}{\pa v_r}\right\vert &&\leq C  
\int_{0}^{t}\ud s (t-s) \left\vert\frac{\pa x_{i}(s)}{\pa  
v_{r}}\right\vert+\\
&&+C \int_{0}^{t}\ud s (t-s)\frac{1}{N}\left\vert \frac{\pa  
x_{r}(s)}{\pa v_{r}}\right\vert +\nonumber\\
&&+C \int_{0}^{t}\ud s (t-s)\frac{1}{N}\sum_{ \substack{j\neq i\\j\neq  
r}}^N \pa_{x}F\left(x_{i}(s)-x_{j}(s)\right)\left\vert \frac{\pa  
x_{j}(s)}{\pa v_{r}}\right\vert.\nonumber\\
&&
\end{eqnarray}
Hence, by virtue of the Gronwall lemma, we find:
\begin{eqnarray}\label{eq:estAppA2}
&&
\max_{\substack{i\neq r\\ t\leq T}}\left\vert \frac{\pa x_{i}(t)}{\pa  
v_r}\right\vert\leq \frac{C}{N}.
\end{eqnarray}
By (\ref{eq:est7vBIS}), we find that the same estimate holds for the  
derivative of $v_i(t)$ with respect to $v_r$.
Analogous estimates hold for the derivatives with respect to the  
initial positions (see also \cite{Graffi}).\\
Therefore the claim of Proposition \ref{DERIVATE1} is proven for  
derivatives of order one.

Now, let us consider a sequence $I:=(j_1,\dots, j_k)$ of possibly  
repeated indices. We show that:
\begin{eqnarray}\label{eq:estAppA3}
%\text{sup}_{t >0}
\frac{1}{N}\sum_{i=1}^N
\left\vert \frac{\pa^k x_{i}(t)}{\pa v_{j_1}\dots \pa  
v_{j_k}}\right\vert \leq \frac{C}{N^{d_k}},
\end{eqnarray}
where $d_k$ is the number of different indices in the sequence  
$j_1,\dots, j_k$.
We know that (\ref{eq:estAppA3}) is verified for $k=1$ (it follows  
directly by (\ref{eq:estAppA}) and (\ref{eq:estAppA2})), thus we prove  
(\ref{eq:estAppA3}) by induction on $k$.
Denoting by:
\begin{eqnarray}\label{eq:estAppA4}
D(I):= \frac{\pa^k }{\pa v_{j_1}\dots \pa v_{j_k}},
\end{eqnarray}
estimate (\ref{eq:estAppA3}) can be rewritten as:
\begin{eqnarray}\label{eq:estAppA3bis}
  \frac{1}{N}\sum_{i=1}^{N}\left\vert D(I)x_i(t)\right\vert \leq  
\frac{C}{N^{d_k}}.
\end{eqnarray}
By (\ref{eq:est7v}) we derive the following estimate for $ D(I)x_i(t)$:
\begin{eqnarray}\label{eq:estAppA5prima}
&&
%\frac{1}{N}\sum_{i=1}^{N}
\left\vert D(I) x_{i}(t)\right\vert \leq  \int_{0}^{t} \ud s (t-s)
%\frac{1}{N^2}
\frac{C}{N}
%\sum_{i=1}^{N}
\sum_{ j\neq i}^N
%\pa_{x}F\left(x_{i}(s)-x_{j}(s)\right)
\left\vert D(I)\left( x_{i}(s)-x_{j}(s)\right)\right\vert +M_i(t),\nonumber\\
&&
\end{eqnarray}
where the term $M_i(t)$ can be computed from (\ref{eq:est7v})  
according to the Leibniz rule. Let $\mathcal{P}_n:=\{I_1,\dots, I_n\}$  
be a partition of the set $I$ of cardinality $n$, with $2\leq n\leq  
k$, then we have:
\begin{eqnarray}\label{eq:estAppA6prima}
&& M_i(t) \leq\int_{0}^{t} \ud s (t-s)
%\frac{1}{N^2}
\frac{1}{N}
\sum_{ j\neq i}^N  
\sum_{n=2}^{k}\sum_{\mathcal{P}_n}C(\mathcal{P}_n)\left\vert\prod_{H\in  
\mathcal{P}_n} \[D(H)  
\(x_{i}(s)-x_{j}(s)\right)\]\right\vert\leq\nonumber\\
&&\leq\int_{0}^{t} \ud s  
(t-s)\sum_{n=2}^{k}\sum_{\mathcal{P}_n}C(\mathcal{P}_n)\frac{1}{N}
\sum_{ j=1}^N \left\vert\prod_{H\in \mathcal{P}_n} \[D(H)  
\(x_{i}(s)-x_{j}(s)\right)\]\right\vert,
\end{eqnarray}
where $D(H):= \prod_{h\in H}\frac{\pa}{\pa v_{h}}$ and  
$C(\mathcal{P}_n)$ are coefficients depending on the partition  
$\mathcal{P}_n$ and on suitable derivatives of $F$.
By
%(\ref{eq:estAppA6prima}) and
(\ref{eq:estAppA5prima}), it follows that:
\begin{eqnarray}\label{eq:estAppA5}
&&
\frac{1}{N}\sum_{i=1}^{N}
\left\vert D(I) x_{i}(t)\right\vert \leq \int_{0}^{t} \ud s (t-s)
\frac{C}{N}
\sum_{i=1}^{N} \left\vert D(I)x_{i}(s)\right\vert
+M(t),
\end{eqnarray}
where $M(t)=\frac{1}{N}\sum_{i=1}^{N}M_i(t)$ and, by  
(\ref{eq:estAppA6prima}), we have:
\begin{eqnarray}\label{eq:estAppA6}
&& M(t) \leq\int_{0}^{t} \ud s  
(t-s)\sum_{n=2}^{k}\sum_{\mathcal{P}_n}C(\mathcal{P}_n)\frac{1}{N^2}
%\frac{1}{N}
\sum_{i=1}^{N}\sum_{ j=1}^N \left\vert\prod_{H\in \mathcal{P}_n}  
\[D(H) \(x_{i}(s)-x_{j}(s)\right)\]\right\vert,\nonumber\\
&&
\end{eqnarray}
We observe that:
\begin{eqnarray}\label{eq:estAppA6bis}
&&\frac{1}{N^2}
%\frac{1}{N}
\sum_{i,j=1}^{N}
%\sum_{ j=1}^N
\left\vert\prod_{H\in \mathcal{P}_n}\left[D(H)  
\(x_{i}(s)-x_{j}(s)\right)\]\right\vert\leq\frac{1}{N}
\sum_{i=1}^{N}\prod_{H\in\mathcal{P}_n}\left\vert D(H)x_{i}(s)\right\vert+
%(-1)^n
\frac{1}{N}\sum_{j=1}^{N}\prod_{H\in\mathcal{P}_n}\left\vert  
D(H)x_{j}(s)\right\vert+\nonumber\\
&&+\sum_{\mathcal{Q}\subset \mathcal{P}_n}C(\mathcal{Q})\(\frac{1}{N}
\sum_{i=1}^{N}\prod_{Q\in\mathcal{Q}}\left\vert  
D(Q)x_{i}(s)\right\vert\)\(\frac{1}{N}
\sum_{j=1}^{N} \prod_{J\in\mathcal{P}_n\setminus\mathcal{Q}  
}\left\vert D(J)x_{j}(s)\right\vert\),\nonumber\\
&&
%&&+\sum_{J,H:J\sqcup H=I }C(J,H)D(J)x_{i}(s)D(H)x_{i}(s).
\end{eqnarray}
where $\mathcal{Q}$ is any subpartition of $\mathcal{P}_n$ and  
$C(\mathcal{Q})$ are coefficients depending on $\mathcal{Q}$.\\
We assume that the estimate (\ref{eq:estAppA3bis}) holds for any  
$m\leq k-1$, namely:
\begin{eqnarray}\label{eq:estAppA7}
%%\text{sup}_{t >0}
%\max_{\substack{i,\ M\subset I:\\\vert M\vert=m}}
  \frac{1}{N}\sum_{i=1}^{N}\left\vert D(M) x_{i}(t)\right\vert \leq  
\frac{C}{N^{d_m}},\ \ \text{for any } M\subset I\ \text{s.t.\ } \vert  
M\vert=m\leq k-1,
\end{eqnarray}
where $d_m$ is the number of different indices in the sequence $M$.\\
%which are different from $i$ (counted without their multiplicity).\\
Indeed, if we consider a partition $\mathcal{P}_n$ of cardinality  
$n\geq 2$, we are guaranteed
%by (\ref{eq:estAppA7})
that $\vert M\vert\leq k-1$ for each $M\in\mathcal{P}_n$.  Then, by  
noting that:
\begin{eqnarray}\label{eq:estAppAnuova}
  \frac{1}{N}\sum_{i=1}^{N}\prod_{H\in\mathcal{H}}\left\vert D(H)  
x_{i}(t)\right\vert \leq\prod_{H\in\mathcal{H}}  
\frac{1}{N}\sum_{i=1}^{N}\left\vert D(H) x_{i}(t)\right\vert,\  
\forall\ \text{subpartition} \ \mathcal{H}\subseteq \mathcal{P}_n,
\end{eqnarray}
we can apply the inductive hypotheses (\ref{eq:estAppA7}) to estimate  
the derivatives of $x_i(s)$ and $x_j(s)$ appearing in  
(\ref{eq:estAppA6bis}). Thus, we obtain:
%in estimating (\ref{eq:estApp6bis}) and we get:
\begin{eqnarray}\label{eq:estAppA8}
  \frac{1}{N}\sum_{i=1}^{N}\prod_{H\in \mathcal{P}_n}\left\vert  
D(H)x_{i}(s)\right\vert&&\leq \prod_{H\in \mathcal{P}_n}  
\frac{1}{N}\sum_{i=1}^{N}\left\vert  
D(H)x_{i}(s)\right\vert\leq\nonumber\\
&& \leq\prod_{H\in \mathcal{P}_n}\frac{C}{N^{d_h}}=\frac{C}{N^{\sum  
d_h}}\leq \frac{C}{N^{d_k}} ,
\end{eqnarray}
where $d_h$ is the number of different indices in the sequence $H$ and  
we used that $\sum_{H\in \mathcal{P}_n}d_h\geq d_k$.\\
% because $H\subset I$.\\
In a similar way, we find
\begin{eqnarray}\label{eq:estAppA9}
\frac{1}{N}\sum_{i=1}^{N}\prod_{Q\in \mathcal{Q}}\left\vert  
D(Q)x_{i}(s)\right\vert\leq \prod_{Q\in \mathcal{Q}}\frac{C}{N^{d_q}},
%\leq \frac{C}{N^{d}},
\end{eqnarray}
where $d_q$ is the number of different indices in the sequence  
$Q$.\\Moreover, we have:
\begin{eqnarray}\label{eq:estAppA10}
&&\frac{1}{N}\sum_{j=1}^{N}\prod_{H\in \mathcal{P}_n}\left\vert  
D(H)x_{j}(s)\right\vert\leq \prod_{H\in  
\mathcal{P}_n}\frac{C}{N^{d_h}}=\frac{C}{N^{\sum d_h}}\leq  
\frac{C}{N^{d_k}},
\end{eqnarray}
and
\begin{eqnarray}\label{eq:estAppA11}
&&\frac{1}{N}\sum_{j=1}^{N}\prod_{J\in  
\mathcal{P}_n\setminus\mathcal{Q}}\left\vert D(J)x_{j}(s)\right\vert
\leq \prod_{J\in \mathcal{P}_n\setminus\mathcal{Q}} \frac{C}{N^{d_j}},
\end{eqnarray}
where $d_j$ is the number of different indices in the sequence $J$.
Then, putting together (\ref{eq:estAppA9}) and (\ref{eq:estAppA11}), we find:
\begin{eqnarray}\label{eq:estAppA12}
&&\sum_{\mathcal{Q}\subset  
\mathcal{P}_n}C(\mathcal{Q})\(\frac{1}{N}\sum_{i=1}^{N}\prod_{Q\in\mathcal{Q}}\left\vert D(Q)x_{i}(s)\right\vert\)\(\frac{1}{N}\sum_{j=1}^{N}\prod_{J\in\mathcal{P}_n\setminus\mathcal{Q} }\left\vert  
D(J)x_{j}(s)\right\vert\)\leq\nonumber\\
&&\sum_{\mathcal{Q}\subset  
\mathcal{P}_n}C(\mathcal{Q})\prod_{Q\in\mathcal{Q}}\prod_{J\in\mathcal{P}_n\setminus\mathcal{Q}  
}\frac{C}{N^{d_{q}+d_j}}\leq\nonumber\\
&&\leq \sum_{\mathcal{Q}\subset  
\mathcal{P}_n}C(\mathcal{Q})\prod_{Q\in\mathcal{Q}}\prod_{J\in\mathcal{P}_n\setminus\mathcal{Q} }\frac{C}{N^{d_k}}\leq  
\frac{C}{N^{d_k}}.
%%
%%\leq \delta_j(I)\frac{C}{N^{d-1}}+(1-\delta_j(I))\frac{C}{N^{d}}.
\end{eqnarray}
In the end, we have just proven that each term in  
(\ref{eq:estAppA6bis}) is bounded by $\frac{C}{N^{d_k}}$. Therefore,  
by using this estimate in (\ref{eq:estAppA6}), we find:
\begin{eqnarray}\label{eq:estAppA13}
&& M(t) \leq\frac{C}{N^{d_k}}.
\end{eqnarray}
By (\ref{eq:estAppA13}) and (\ref{eq:estAppA5}), it follows that:
\begin{eqnarray}\label{eq:estAppA14}
&&\frac{1}{N}\sum_{i=1}^{N}\left\vert D(I) x_{i}(t)\right\vert \leq  
\int_{0}^{t} \ud s (t-s)
%\frac{1}{N}
\frac{C}{N}\sum_{i=1}^{N}\left\vert D(I)x_{i}(s)\right\vert
+\frac{C}{N^{d_k}}.\nonumber\\
&&
\end{eqnarray}
Therefore, by using the Gronwall lemma, we find:
\begin{eqnarray}\label{eq:estAppA15}
&&\frac{1}{N}\sum_{i=1}^{N}\left\vert D(I) x_{i}(t)\right\vert  
\leq\frac{C}{N^{d_k}}.
\end{eqnarray}

As regard to the derivatives of $v_i(t)$ with respect to some initial  
velocities $v_{j_1},\dots,v_{j_k}$, an analogous estimate holds and  
the proof  works in the same way.
% by computing the derivative $D(I)$ of $v_i(t)$ and by doing  
estimates through an induction procedure.
Furthermore, this strategy leads to the same estimate for the  
derivatives of the function $\frac{1}{N}\sum_{i=1}^{N}z_i(t)$ with  
respect to some initial positions $x_{j_1},\dots,x_{j_k}$.

Now, thanks to the estimate we have just proven for the derivatives of  
the function $\frac{1}{N}\sum_{i=1}^{N}z_i(t)$, we are able to prove  
the claim of Proposition \ref{DERIVATE1}. In fact, we have:
\begin{eqnarray}\label{eq: specificoIV}
&&\frac{1}{N}\sum_{i=1}^{N}\left\vert D(I)z_i(t)  
\right\vert=\frac{1}{N}\sum_{\substack{i=1\\i\in D}}^{N}\left\vert  
D(I) z_{i}(t)\right\vert+\frac{1}{N}\sum_{\substack{i=1\\i\notin  
D}}^{N}\left\vert D(I)z_i(t) \right\vert\leq \frac{C}{N^{d_k}},
%$\nonumber\\
\end{eqnarray}
where $D\subset I$ contains the different indices appearing in the  
sequence $I$. Thus, according to our previous notation, $\vert D\vert  
=d_k$ and we denote the elements of $D$ by  
$\tilde{j}_1,\dots,\tilde{j}_{d_k}$. Then by (\ref{eq: specificoIV})  
we find:
\begin{eqnarray}\label{eq: correzME1}
\frac{1}{N}\sum_{i=1}^{N}\left\vert D(I)z_i(t) \right\vert&&=\frac{1}{N}
%\sum_{\substack{i\in \(\tilde{j}_1,\dots,\tilde{j}_{d_k}}\)
\left\vert D(I)z_{\tilde{j}_1}(t) \right\vert+\dots +  
\frac{1}{N}\left\vert D(I)z_{\tilde{j}_{d_k}}(t) \right\vert+\nonumber\\
&&+\frac{1}{N}\sum_{\substack{i=1\\i\notin D}}^{N}\left\vert  
D(I)z_i(t) \right\vert\leq \frac{C}{N^{d_k}},
\end{eqnarray}
which implies
\begin{eqnarray}\label{eq: correzME2}
&&\left\vert D(I)z_i(t) \right\vert\leq   
C\(\frac{\sum_{\ell=1}^{d_k}\delta_{i \tilde{j}_\ell}}{N^{d_k  
-1}}+\frac{1}{N^{d_k}}\),
\end{eqnarray}
or
\begin{eqnarray}\label{eq: correzME3}
&&\left\vert D(I)z_i(t) \right\vert\leq  \frac{C}{N^{d_k^{(i)}}},
\end{eqnarray}
where $d_k^{(i)}$ is the number of different indices in the sequence  
$I$ which are also different from $i$.
%So that, the claim of Proposition 5.1 is proven for any $k$.
\\ \spaz \hspace{1cm} \hfill $\square$ \newline
\section*{Appendix B}
\setcounter{equation}{0}
\def\theequation{B.\arabic{equation}}
{\bf Proof of Proposition \ref{PROP6.1}}\\

Let $U_h(t,s)$ be the two parameters semigroup solution of the linear problem:
\begin{equation}\label{eq: AppB1}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)U_h(t,s)\gamma_0=\left(\nabla  
\phi\ast h\right)\ast \nabla_v U_h(t,s)\gamma_0,\\
&U_h(s,s)\gamma_0=\gamma_0.
\end{aligned}
\right.
\end{equation}
The solution of (\ref{eq: AppB1}) is obtained by carrying the initial  
datum $\gamma_0$ along the characteristic flow
\begin{equation}\label{eq: AppB2}
\left\{
\begin{aligned}
&\dot{x}=v,\\
&\dot{v}=-\nabla \phi \ast h.
\end{aligned}
\right.
\end{equation}

Next, we consider the problem
\begin{equation}\label{eq: AppB3}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)\tilde{\gamma}=L(h)\tilde{\gamma},\\
&\tilde{\gamma}\vert_{t=0}=\gamma_0.
\end{aligned}
\right.
\end{equation}
which can be reformulated in integral form:
\begin{equation}\label{eq: AppB4}
\tilde{\gamma}(t)=U_h(t,0)\gamma_0+\int_0^t\ud s \  
U_h(t,s)\left[\left(\nabla \phi\ast \tilde{\gamma}(s)\right)\cdot  
\nabla_v h(s)\right].
\end{equation}
The above formula can be iterated to yield the formal solution
\begin{eqnarray}\label{eq: expAppB5}
\tilde{\gamma}(x,v;t)=&& U_h(t,0)\gamma_0(x,v)+\sum_{n\geq  
1}\int_0^{t}\ud t_1\int_{0}^{t_1}\ud t_2\dots \int_{0}^{t_{n-1}}\ud  
t_n\int \ud x_1\int \ud v_1\dots \int \ud x_n\int \ud v_n\nonumber\\
&&U_h(t,t_1)\left[\nabla_v h(x,v;t_1)\cdot  
\nabla_x\phi(x-x_1)\right]\nonumber\\
&&U_h(t_1,t_2)\left[\nabla_{v_1} h(x_1,v_1;t_2)\cdot  
\nabla_{x_1}\phi(x_1-x_2)\right]\nonumber\\
&&\dots\nonumber\\
&&U_h(t_{n-1},t_n)\left[\nabla_{v_{n-1}} h(x_{n-1},v_{n-1};t_n)\cdot  
\nabla_{x_{n-1}}\phi(x_{n-1}-x_n)\right]\nonumber\\
&&U_h(t_{n},0)\gamma_0(x_n, v_n).
\end{eqnarray}
We remark that $U_h(t_k,t_{k+1})$ acts on the variables $x_k, v_k$  
with the convention that $(x_0,v_0)=(x,v)$ and, furthermore, $U_h$ is  
multiplicative and preserves the $L^p(\R^3\times\R^3)$ norms  
$(p=1,2,\dots,\infty)$.\\
Under the assumptions of Proposition \ref{PROP6.1}, the above series  
is bounded in $L^1(\R^3\times\R^3)$ by:
\begin{eqnarray}\label{eq: expUNIFBOUNDAppB6}
&&\sum_{n\geq 0}\frac{t^n}{n!}\left(\text{sup}_{\tau \in  
[0,t]}\left\|\nabla_v h(\tau)\right\|_{L^1(\R^3\times\R^3)}\right)^n  
\left\|\nabla_x \phi\right\|_{L^{\infty}(\R^3)}^n  
\left\|\gamma_0\right\|_{L^{1}(\R^3\times\R^3)},\nonumber\\
&&
\end{eqnarray}
which is converging for each $t$. Now, we denote by  
$\Sigma_h(t,s):L^1(\R^3\times\R^3)\rightarrow L^1(\R^3\times\R^3)$,  
the two parameters semigroup given by the series (\ref{eq: expAppB5}).  
Then, the solution $\gamma$ to the problem (\ref{eq: Prop6.1}) is  
given by:
\begin{equation}\label{eq: AppB7}
\gamma(t)=\Sigma_h(t,0)\gamma_0+\int_0^t\ud s \ \Sigma_h(t,s)\Theta(s),
\end{equation}
and, thanks to the assumption we made on $\Theta$ and to the fact that  
the above series (\ref{eq: expAppB5}) is converging for any $t$, we  
are guaranteed that $\gamma\in  
\mathcal{C}^0\(L^1(\R^3\times\R^3),\R^+\)$.

The $\mathcal{C}^k$ regularity of  
$\tilde{\gamma}(t)=\Sigma_h(t,0)\gamma_0$ follows by (\ref{eq:  
expAppB5}) and
%the $\mathcal{C}^k$ regularity and
the fact that $U_h(t,t_1)$ propagates the $\mathcal{C}^k$ regularity.
\newline \spaz \hspace{1cm} \hfill $\square$ \newline

{\bf Proof of Proposition \ref{PROP6.2}}\\

The proof consists of two steps.\\\\
\emph{Step 1)}:

Let $\gamma_N$ be as in Proposition \ref{PROP6.2}. Then, we show that  
$\gamma_N$ solves the problem:
\begin{equation}\label{eq: AppB8}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot \nabla_x\right)\gamma_N=L(h)\gamma_N+\Theta_{N}',\\
&\gamma_N\vert_{t=0}=\gamma_{N,0},
\end{aligned}
\right.
\end{equation}
with
\begin{equation}\label{eq: AppB9}
\Theta_{N}'=\Theta_N +R_N,
\end{equation}
and $R_N$ is such that:
\begin{equation}\label{eq: AppB10}
R_N\rightarrow 0 ,\ \ \ \mathcal{C}_{b}^{\infty}-\text{weakly}.
\end{equation}
In proving (\ref{eq: AppB10}), the assumption {\bf ii)} on $\gamma_N$  
is crucial.\\\\
\emph{Step 2)}:

By virtue of Step 1), the hypotheses we made on $\nabla_v h$ and  
Proposition \ref{PROP6.1}, we find that:
\begin{equation}\label{eq: AppB11}
\gamma_N(t)=\Sigma_{h}(t,0)\gamma_{N,0}+\int_0^t\ud s \  
\Sigma_{h}(t,s)\Theta_{N}'(s).
\end{equation}
Then, reminding that:\\
$\circ$ $h(t)\in \mathcal{C}_b^{\infty}(\R^3\times\R^3)$ for any $t$,\\
$\circ$ the flow $\Sigma_{h}$ propagates the $\mathcal{C}^k$ regularity,\\
$\circ$ $R_N\rightarrow 0 ,\ \ \ \mathcal{C}_{b}^{\infty}-\text{weakly}$,\\
and by virtue of
the assumptions on $\gamma_{N,0}$ and $\Theta_N$,
% and to what has been proven in Step 1) as regard to $R_N$,
we can easily show that:
\begin{equation}\label{eq: AppB12}
\gamma_N\rightarrow\gamma,\ \text{as}\ \ N\to\infty,\ \  
\mathcal{C}_{b}^{\infty}-\text{weakly},
\end{equation}
where
\begin{equation}\label{eq: AppB13}
\gamma(t)=\Sigma_{h}(t,0)\gamma_{0}+\int_0^t\ud s \ \Sigma_{h}(t,s)\Theta(s).
\end{equation}
Therefore, we recognize that $\gamma$ solves the problem (\ref{eq: Prop6.1})
and, by virtue of Proposition \ref{PROP6.1}, it is uniquely determined  
by (\ref{eq: AppB13}) and hence it is in  
$\mathcal{C}^0\(L^1(\R^3\times\R^3), \R^+\)$.\\\\
\emph{Proof of Step 1)}:\\

We have:
\begin{equation}\label{eq: Prop6.2APP}
\left\{
\begin{aligned}
&\left(\pa_t+v\cdot  
\nabla_x\right)\gamma_N=L(h)\gamma_N+\Theta_N+L(h_N - h)\gamma_N\\
&\left. \gamma_N(x,v;t)\right\vert_{t=0}=\gamma_{N,0}(x,v),
\end{aligned}
\right.
\end{equation}
where
\begin{equation}\label{eq: AppB15}
R_N=R_N(x,v;t):=L(h_N - h)\gamma_N.
\end{equation}
We want to show that $R_N\to 0$,\  \ $\mathcal{C}_b^{\infty}$-weakly.  
According to the definition of the operator $L$, we have:
\begin{equation}\label{eq: AppB16}
R_N=\(\nabla_x\phi\ast (h_N -h)\)\nabla_v\gamma_N+\(\nabla_x\phi\ast  
\gamma_N\)\nabla_v(h_N -h),
\end{equation}
thus, we have to show that
\begin{equation}\label{eq: AppB17}
\(u,\(\nabla_x\phi\ast (h_N -h)\)\nabla_v\gamma_N\)\rightarrow 0,\  
\text{as}\ \ N\to\infty,\ \ \forall\  u\in  
\mathcal{C}_b^{\infty}(\R^3\times\R^3),
\end{equation}
and
\begin{equation}\label{eq: AppB18}
\(u,\(\nabla_x\phi\ast \gamma_N\)\nabla_v(h_N -h)\)\rightarrow 0,\  
\text{as}\ \ N\to\infty,\ \ \forall\  u\in  
\mathcal{C}_b^{\infty}(\R^3\times\R^3).
\end{equation}
We show only (\ref{eq: AppB18}) in detail because (\ref{eq: AppB17})  
will follow the same line.
We have:
\begin{eqnarray}\label{eq: AppB19}
&&\(u,\(\nabla_x\phi\ast \gamma_N\)\nabla_v(h_N -h)\)=\int \ud x \ud  
v\int \ud y\ud w\ u(x,v)\nabla_x\phi(x-y)\gamma_N(y,w;t)\cdot\nonumber\\
&&\qquad\qquad\qquad\qquad  
\qquad\qquad\qquad\qquad\qquad\nabla_v(h_N(x,v;t) -h(x,v;t))=\nonumber\\
&&\qquad\qquad\qquad\qquad=-\int \ud x \ud v\int \ud y\ud w\ \nabla_v  
u(x,v)\nabla_x\phi(x-y)\gamma_N(y,w;t)\cdot\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (h_N(x,v;t)  
-h(x,v;t))=\nonumber\\
&&\qquad\qquad\qquad\qquad=\int \ud x \ud v\int \ud y\ud w\ \nabla_v  
u(x,v)\(\nabla_x\phi\ast\gamma_N\)(x,v;t)(h-h_N)(x,v;t).\nonumber\\
&&
\end{eqnarray}

Setting
\begin{eqnarray}\label{eq: AppB20}
\zeta_N(x,v):=\nabla_v u(x,v)\int \ud y\ud w\nabla_x\phi(x-y)\gamma_N(y,w;t),
\end{eqnarray}
we can write (\ref{eq: AppB19}) as:
\begin{eqnarray}\label{eq: AppB21}
&&\(u,\(\nabla_x\phi\ast \gamma_N\)\nabla_v(h_N -h)\)=\int \ud x\ud v\  
\zeta_N(x,v)(h(x,v;t) -h_N(x,v;t))=\nonumber\\
&&\qquad\qquad\qquad\quad=\int \ud x\ud v\int \ud x'\ud  
v'\(\zeta_N(x,v)-\zeta_N(x',v')\)P_N(x,v;x',v';t),\nonumber\\
&&
\end{eqnarray}
where $P_N$ is a coupling of $h$ and $h_N$, namely a probability  
density in $\R^6\times \R^6$ with marginals given by $h$ and $h_N$.  
Now we observe that:
\begin{eqnarray}\label{eq: AppB22}
\nabla_{x,v}\zeta_N(x,v):=\int \ud y\ud w\nabla_{x,v}\left[\nabla_v  
u(x,v)\nabla_x\phi(x-y)\right]\gamma_N(y,w;t),\nonumber\\
&&
\end{eqnarray}
and, thanks to the assumption {\bf ii)} we made on $\gamma_N$, we know  
that there exists a constant $C=C(u,\phi)>0$ such that:
\begin{eqnarray}\label{eq: AppB23}
\sup_{x,v}\left\vert \nabla_{x,v}\zeta_N(x,v)\right\vert  
=\left\|\nabla\zeta_N\right\|_{L^{\infty}(\R^3\times\R^3)}<C<+\infty.\nonumber\\
&&
\end{eqnarray}
Therefore, coming back to (\ref{eq: AppB21}), we find:
\begin{eqnarray}\label{eq: AppB24}
\left\vert \(u,\(\nabla_x\phi\ast \gamma_N\)\nabla_v(h_N  
-h)\)\right\vert&&\leq \int \ud z\int \ud z'\left\vert  
\zeta_N(z)-\zeta_N(z')\right\vert P_N(z;z';t)\nonumber\\
&&\leq \int \ud z\int \ud z'C\left\vert z-z'\right\vert  
P_N(z;z';t).\nonumber\\
&&
\end{eqnarray}
where we used the standard notation  $z=(x,v)$ and $z'=(x',v')$. Then,  
taking in (\ref{eq: AppB24}) the infimum over all couplings between  
$h$ and $h_N$, we obtain that:
\begin{eqnarray}\label{eq: AppB25}
\left\vert \(u,\(\nabla_x\phi\ast \gamma_N\)\nabla_v(h_N  
-h)\)\right\vert\leq C \mathcal{W}(h_N,h),
\end{eqnarray}
where, as in Section 1.2, $\mathcal{W}$ denotes the Wasserstein distance.
%(see \ref{DOB}).
But we know that the right hand side of (\ref{eq: AppB25}) goes to  
zero because of the assumption {\bf i)}, then we have just proven that:
\begin{eqnarray}\label{eq: AppB26}
\left\vert \(u,\(\nabla_x\phi\ast \gamma_N\)\nabla_v(h_N  
-h)\)\right\vert\rightarrow 0,\ \ \forall\  
u\in\mathcal{C}_b^{\infty}(\R^3\times\R^3).\nonumber\\
&&
\end{eqnarray}

Analogously, we can prove that
\begin{eqnarray}\label{eq: AppB27}
\left\vert \(u,\(\nabla_x\phi\ast (h_N  
-h)\)\nabla_v\gamma_N\)\right\vert\rightarrow 0,\ \ \forall\  
u\in\mathcal{C}_b^{\infty}(\R^3\times\R^3).\nonumber\\
&&
\end{eqnarray}
Therefore we have just proven that $R_N$ goes to zero in the  
$\mathcal{C}_b^{\infty}$-weak sense and the proof of Step 1) is done.
\newpage
\emph{Proof of Step 2)}:\\

Thanks to Step 1) and to the assumption on $\nabla_v h$, we know that  
$\gamma_N(t)$ can be written as in (\ref{eq: AppB11}). Then, for any  
function $u$ in $\mathcal{C}_b^{\infty}(\R^3\times\R^3)$, we have that:
\begin{equation}\label{eq: AppB28}
\(u,\gamma_N(t)\)=\(u,\Sigma_{h}(t,0)\gamma_{N,0}\)+\int_0^t\ud s \  
\(u,\Sigma_{h}(t,s)\Theta_{N}'(s)\),
\end{equation}
namely
\begin{equation}\label{eq: AppB29}
\(u,\gamma_N(t)\)=\(\left(\Sigma_{h}(t,0)\right)^*  
u,\gamma_{N,0}\)+\int_0^t\ud s \  
\(\(\Sigma_{h}(t,s)\)^*u,\Theta_{N}'(s)\),
\end{equation}
where $\Sigma_{h}^*$ is the adjoint of $\Sigma_{h}$. We remind that  
the two-parameters semigroup $\Sigma_{h}(t,s)$ propagates the  
$\mathcal{C}^k$ regularity, provided that $\nabla_v h\in  
\mathcal{C}^k\(\R^3\times\R^3\)$. In particular, if $\Sigma_h$ acts on  
a function $u$ which is in $\mathcal{C}_b^{\infty}(\R^3\times\R^3)$  
and the function $h(t)$ is supposed to be in  
$\mathcal{C}_b^{\infty}(\R^3\times\R^3)$ for any $t$, as it is in the  
assumptions of Proposition \ref{PROP6.2}, we are clearly guaranteed  
that $\nabla_v h(t)$ is in $\mathcal{C}_b^{\infty}(\R^3\times\R^3)$  
for any $t$, and then, $u(t):=\Sigma_h(t,0)u(x,v)$ is also in  
$\mathcal{C}_b^{\infty}(\R^3\times\R^3)$ for any $t$. Obviously, the  
same holds for $\Sigma_{h}^*$. Thus, the functions  
$\left(\Sigma_{h}(t,0)\right)^* u$ and $\(\Sigma_{h}(t,s)\)^*u$  
appearing in (\ref{eq: AppB29}) are in  
$\mathcal{C}_b^{\infty}(\R^3\times\R^3)$ for any $t$. Therefore,  
thanks to the assumptions we made on $\gamma_{N,0}$ and $\Theta_N$,  
and of what we know about $R_N$, we find that:
\begin{eqnarray}\label{eq: AppB29BIS}
&&\(\left(\Sigma_{h}(t,0)\right)^* u,\gamma_{N,0}\)+\int_0^t\ud s \  
\(\(\Sigma_{h}(t,s)\)^*u,\Theta_{N}'(s)\)\nonumber\\
&&\nonumber\\
&&\qquad\qquad\qquad \downarrow\ \ \ \ \ \ N\to\infty\nonumber\\
&&\nonumber\\
&&\(\left(\Sigma_{h}(t,0)\right)^* u,\gamma_{0}\)+\int_0^t\ud s \  
\(\(\Sigma_{h}(t,s)\)^*u,\Theta(s)\)=\nonumber\\
&&=\( u,\Sigma_{h}(t,0)\gamma_{0}\)+\int_0^t\ud s \  
\(u,\Sigma_{h}(t,s)\Theta(s)\).
\end{eqnarray}
Finally, by Proposition \ref{PROP6.1}, we know that the expression  
(\ref{eq: AppB29BIS}) identifies properly the unique solution of the  
problem (\ref{eq: Prop6.1}) in  
$\mathcal{C}^0\(L^1\(\R^3\times\R^3\),\R^+\)$ and Proposition  
\ref{PROP6.2} is proven.
\newline \spaz \hspace{1cm} \hfill $\square$ \newline

%\addcontentsline{toc}{section}{Appendix C}
\section*{Appendix C}
\setcounter{equation}{0}
\def\theequation{C.\arabic{equation}}
%%{\bf Estimate of the error term $\mathbf{E_N^1}$}\\\\
{\bf Lemma C.1:} \emph{For each time $\tau>0$, let us define the  
operator $\hat{T}_N^{(n)}(\tau)$ as follows:}
$$
\hat{T}_N^{(n)}(\tau):=S_N(-\tau)\hat{T}_N^{(n)}S_N(\tau).
$$

\emph{Then, for each $m\geq 0$ and for each  
$u\in\mathcal{C}_b^{\infty}(\R^3\times\R^3)$, there exists a constant  
$C>0$, not }

\emph{depending on $N$, such that:}
\begin{eqnarray}
&&\text{{\bf i)}}\left\vert \left(u,\hat{T}_N^{(r_m)}(t_m)\dots  
\hat{T}_{N}^{(r_1)}(t_1)\mu_N(t)\right)\right\vert<C.\ \ \label{eq:  
streak2O1N}
\end{eqnarray}

\emph{Moreover, we have:}
%there exists a positive constant $C$ such that:
\begin{eqnarray}
&&\text{{\bf ii)}}\left\vert \left(u,T_N^{(r_m)}(t_m)\dots  
T_{N}^{(r_1)}(t_1)\mu_N(t)\right)\right\vert\leq \left\vert  
\left(u,\hat{T}_N^{(r_m)}(t_m)\dots  
\hat{T}_{N}^{(r_1)}(t_1)\mu_N(t)\right)\right\vert+O\(\frac{1}{N}\).\nonumber\\
&&\label{eq: streak2O1}
\end{eqnarray}
\\
{\bf Proof:}

We observe that:

\begin{eqnarray}\label{eq: rewrite}
&&\left(u,\hat{T}_N^{(r_m)}(t_m)\dots  
\hat{T}_{N}^{(r_1)}(t_1)\mu_N(t)\right)=\hat{T}_N^{(r_m)}(t_m)\hat{T}_N^{(r_{m-1})}(t_{m-1})\dots  
\hat{T}_N^{(r_{1})}(t_1)\mathcal{U}(Z_N(t)),\nonumber\\
&&
\end{eqnarray}
where:
\begin{eqnarray}\label{eq: funzU}
\mathcal{U}(Z_N(t)):=\(u,\mu_N(t)\)=\frac{1}{N}\sum_{\ell=1}^{N}u(z_\ell(t)).
\end{eqnarray}

We assume $m>0$ being the case $m=0$ obvious.
\\
By using the notations:
\begin{eqnarray}\label{eq: notazSTRINGA}
&& S({\bf \underline{r}}_m,{\bf  
\underline{t}}_m):=T_N^{(r_m)}(t_m)\dots T_{N}^{(r_1)}(t_1)
\end{eqnarray}
and
\begin{eqnarray}\label{eq: notazSTRINGAdiag}
&&\hat{S}({\bf \underline{r}}_m,{\bf  
\underline{t}}_m):=\hat{T}_N^{(r_m)}(t_m)\dots \hat{T}_{N}^{(r_1)}(t_1),
\end{eqnarray}
we have (see the first term in the right hand side of (\ref{eq:  
opTtemposuG})):
\begin{eqnarray}\label{eq: azSTRINGAdiag}
\hat{S}({\bf \underline{r}}_m,{\bf  
\underline{t}}_m)\mathcal{U}(Z_N(t))&&=\frac{C}{N^m}\sum_{j_1\dots  
j_m}\sum_{l_1\dots l_m}D_x^{r_m+1}\phi(x_{j_m}(t_m)-x_{l_m}(t_m))\cdot  
D_{v_{j_m}}^{r_m+1}(t_m)\nonumber\\
&&D_x^{r_{m-1}+1}\phi(x_{j_{m-1}}(t_{m-1})-x_{l_{m-1}}(t_{m-1}))\cdot  
D_{v_{j_{m-1}}}^{r_{m-1}+1}(t_{m-1})\nonumber\\
&&\dots\nonumber\\
&&D_x^{r_1+1}\phi(x_{j_1}(t_1)-x_{l_1}(t_1))\cdot  
D_{v_{j_1}}^{r_1+1}(t_1)U(Z_N(t)),
\end{eqnarray}
$C$ depending on ${\bf \underline{r}}_m$. By setting:
\begin{eqnarray}
&&\Phi_{j_n}(Z_N(t_n)):=\frac{1}{N}\sum_{l_n=1}^{N}D_x^{r_n+1}\phi(x_{j_n}(t_n)-x_{l_n}(t_n))\label{eq:  
PHI}\\
&& \forall\ \ n=1,2,\dots, m\nonumber
\end{eqnarray}
(\ref{eq: azSTRINGAdiag}) can be rewritten as
\begin{eqnarray}\label{eq: azSTRINGAdiagII}
\hat{S}({\bf \underline{r}}_m,{\bf  
\underline{t}}_m)\mathcal{U}(Z_N(t))&&=C\sum_{j_1\dots  
j_m}\Phi_{j_m}(Z_N(t_m))\cdot D_{v_{j_m}}^{r_m+1}(t_m)\nonumber\\
&&\Phi_{j_{m-1}}(Z_N(t_{m-1}))\cdot  
D_{v_{j_{m-1}}}^{r_{m-1}+1}(t_{m-1})\nonumber\\
&&\dots\nonumber\\
&&\Phi_{j_{1}}(Z_N(t_1))\cdot D_{v_{j_1}}^{r_1+1}(t_1)U(Z_N(t)).
\end{eqnarray}

We observe that, thanks to the smoothness of the potential $\phi$,  
$\Phi_{j_n}$ (for each $n$) is a uniformly bounded function of the  
configuration $Z_N$, together with its derivatives.\\
Performing the derivatives in (\ref{eq: azSTRINGAdiagII}), we realize  
that $\hat{S}({\bf \underline{r}}_m,{\bf  
\underline{t}}_m)\mathcal{U}(Z_N(t))$ is a linear combination of terms  
of the following type:
\begin{eqnarray}\label{eq: azSTRINGAdiagIII}
&&\sum_{j_1\dots j_m}\Phi_{j_m}(Z_N(t_m))\cdot  
D_{v_{j_m}}^{a_{m,1}}(t_m)\dots  
D_{v_{j_2}}^{a_{2,1}}(t_2)D_{v_{j_1}}^{a_{1,1}}(t_1)U(Z_N(t))\nonumber\\
&& \qquad \quad D_{v_{j_m}}^{a_{m,2}}(t_m)\dots  
D_{v_{j_2}}^{a_{2,2}}(t_2)\Phi_{j_1}(Z_N(t_1))\nonumber\\
&&\qquad \quad\dots\nonumber\\
&&\qquad  \quad  
D_{v_{j_m}}^{a_{m,m-1}}(t_{m})D_{v_{j_{m-1}}}^{a_{m-1,m-1}}(t_{m-1})\Phi_{j_{m-2}}(Z_N(t_{m-2}))\nonumber\\
&&\qquad \quad D_{v_{j_m}}^{a_{m,m}}(t_m)\Phi_{j_{m-1}}(Z_N(t_{m-1})),
\end{eqnarray}
with the constraint
\begin{equation}\label{eq: constraint}
\left\{
\begin{aligned}
& a_{1,1}=r_1 +1\\
& a_{2,1}+a_{2,2}=r_2+1\\
&\dots\\
& a_{m,1}+a_{m,2}+\dots +a_{m,m}=r_m+1.
\end{aligned}
\right.
\end{equation}

For a fixed sequence $a_{\ell,s}$, we have to compensate the  
divergence arising from the sum $\sum_{j_1\dots j_m}$, which is  
$O\(N^m\)$, by the decay of the derivatives as given by Proposition  
\ref{DERIVATE1} and Corollary \ref{PROP5.2}. Indeed we have:
\begin{eqnarray}\label{eq: correzMARIO1}
&&\left\vert D_{v_{j_m}}^{a_{m,1}}(t_m)\dots  
D_{v_{j_2}}^{a_{2,1}}(t_2)D_{v_{j_1}}^{a_{1,1}}(t_1)\mathcal{U}(Z_N(t))\right\vert\leq  
\frac{C}{N^d},
\end{eqnarray}
where $d$ is the number of different indices in the sequence  
$j_1,j_2,\dots,j_m$ for which $a_{m,1},\dots, a_{2,1},a_{1,1}$ are  
strictly positive. Note that the fact that the derivatives are not  
computed at time $t=0$ but at different times $t_1,t_2,\dots,t_m$,  
does not change the estimate in an essential way.\\
An analogous estimate holds when we replace $\mathcal{U}$ by some  
$\Phi_{j_s}$, namely
\begin{eqnarray}\label{eq: correzMARIO2}
&&\left\vert  
D_{v_{j_m}}^{a_{m,k}}(t_m)D_{v_{j_{m-1}}}^{a_{m-1,k}}(t_{m-1})\dots  
D_{v_{j_k}}^{a_{k,k}}(t_k)\Phi_{j_{k-1}}(Z_N(t_{k-1}))\right\vert\leq  
\frac{C}{N^{d_{k-1}}},
\end{eqnarray}
where $d_{k-1}$ is the number of different indices in the sequence  
$j_k,\dots,j_m$ which are also different from $j_{k-1}$ and from which  
$a_{m,k}, \dots,a_{k,k}$ are strictly positive.

As regard to the term in the sum $\sum_{j_1\dots j_m}$ in which all  
the indices are different (which is the only one of size $O(N^m)$),  
the constraints (\ref{eq: constraint}) together with estimates  
(\ref{eq: correzMARIO1}) and (\ref{eq: correzMARIO2}) ensure that the  
product of derivatives on the right hand side of (\ref{eq:  
azSTRINGAdiagIII}) is bounded by $1/N^m$. Thus this term is of order  
one.
Now for each $s=1,\dots, m-1$ consider the $\frac{m!}{s!(m-s)!}$ terms  
in the sum $\sum_{j_1\dots j_m}$ in which $s$ indices are equal. The  
sum is bounded by $N^{m-s}$. On the other hand, the constraints  
(\ref{eq: constraint}) together with (\ref{eq: correzMARIO1}) and  
(\ref{eq: correzMARIO2}) ensure that the product of derivatives on the  
right hand side of (\ref{eq: azSTRINGAdiagIII}) is bounded by  
$1/N^{m-s}$. Thus even these terms are of size one and {\bf i)} is  
proven.

To prove {\bf ii)} we observe that:
\begin{eqnarray}\label{eq: correzMARIO3}
&& S({\bf \underline{r}}_m,{\bf \underline{t}}_m)\mathcal{U}(Z_N(t)) -  
\hat{S}({\bf \underline{r}}_m,{\bf \underline{t}}_m)\mathcal{U}(Z_N(t))
\end{eqnarray}
can be expanded as in (\ref{eq: azSTRINGAdiag}) and (\ref{eq:  
azSTRINGAdiagIII}). However now we have an extra derivative, arising  
from the definition of $R_{N}^{(n)}$ (see (\ref{eq: OFFdiagOPt})),  
which yields an additional $1/N$. We omit the details of the proof  
which follows the same line of {\bf i)}.
\spaz \hspace{1cm} \hfill $\square$ \newline

In the same way we can also prove the following\\
\\
{\bf Lemma C.2}: \emph{For each $m\geq 0$, $k>0$ and  
$u\in\mathcal{C}_b^{\infty}(\R^3\times\R^3)$, there exists a constant  
$C>0$, not depending on $N$, such that:}
\begin{eqnarray}
&&\left\vert \mathcal{D}^{2k}S({\bf \underline{r}}_m,{\bf \underline{t}}_m)
\mathcal{U}(Z_N(t))
\right\vert<C.\ \ \label{eq: Dtag1}
\end{eqnarray}
\emph{where $\mathcal{U}(Z_N(t))$ is defined as in (\ref{eq: funzU})}.
\\\\
{\bf Proof:}

First we look at the case $m>0$. Reminding the structure of the  
operator $\mathcal{D}^{2k}$ (see (\ref{eq: opDstortoGEN})), we are led  
to consider the term  
$D_{G,j}^{2s_j}\hat{S}(\underline{\mathbf{r}}_m,\underline{\mathbf{t}}_m)\mathcal{U}(Z_N(t))$. We remind that $D_{G,j}^{2s_j}$ is a derivation operator with respect to the variable $z_j$ that acts as specified by (\ref{eq: operatorD_G}). By the expansion (\ref{eq: azSTRINGAdiagIII}) we readily arrive to the  
bound:
\begin{eqnarray}\label{eq: correzMARIO4}
&&\left\vert D_{G,j}^{2s_j}\hat{S}({\bf \underline{r}}_m,{\bf  
\underline{t}}_m)
\mathcal{U}(Z_N(t))
\right\vert\leq \frac{C}{N}.
\end{eqnarray}
Indeed by applying $D_{G,j}^{2s_j}$ to (\ref{eq: azSTRINGAdiagIII})  
either $j\notin (j_1\dots j_m)$ so that we gain $1/N$ by the extra  
derivative, or $j\in (j_1\dots j_m)$ so that we reduce the sum  
$\sum_{j_1\dots j_m}$ by a factor $1/N$. More generally, by the same  
argument we find:
\begin{eqnarray}\label{eq: correzMARIO5}
&&\left\vert \prod_{j\in I}D_{G,j}^{2s_j}\hat{S}({\bf  
\underline{r}}_m,{\bf \underline{t}}_m)
\mathcal{U}(Z_N(t))
\right\vert\leq \frac{C}{N^n},
\end{eqnarray}
where $n=\vert I\vert$.\\
Finally by writing the action of the operator $\mathcal{D}^{2k}$ as in  
(\ref{eq: convINDATstruttura}), we obtain
\begin{eqnarray}\label{eq: correzMARIO6}
&&\left\vert \mathcal{D}^{2k} \hat{S}({\bf \underline{r}}_m,{\bf  
\underline{t}}_m)
\mathcal{U}(Z_N(t))
\right\vert\leq  
\sum_{n=1}^{N}\frac{N!}{n!(N-n)!}\sum_{\substack{s_1\dots s_n\\1\leq  
s_j\leq k\\ \sum_j s_j=k}}\frac{C}{N^n}\leq  
B^k\sum_{n=1}^{N}\frac{N!}{n!(N-n)!}\frac{C^n}{N^n}\leq \nonumber\\
&&\leq B^k\(1+\frac{C}{N}\)^N\leq C,
\end{eqnarray}
$B, C$ being positive constants not depending on $N$. Again  
$\mathcal{D}^{2k} \hat{S}({\bf \underline{r}}_m,{\bf  
\underline{t}}_m)\mathcal{U}(Z_N(t))$ is the leading term of  
$\mathcal{D}^{2k} S({\bf \underline{r}}_m,{\bf  
\underline{t}}_m)\mathcal{U}(Z_N(t))$ for the same reasons we  
discussed in Lemma C.1.

If $m=0$, the estimates (\ref{eq: correzMARIO4}) and (\ref{eq:  
correzMARIO5}) follow directly by Proposition 5.2. Thus, even in this  
case, the proof is concluded by (\ref{eq: correzMARIO6}).
\newline \spaz \hspace{1cm} \hfill $\square$ \newline

The fact that the error term $E_N^1$ (see (\ref{eq: eqPEReta})) and  
hence $E_N^2$ (see (\ref{eq: eqPERnuDEF})) are  
$\mathcal{C}^{\infty}_{b}$-weakly vanishing when $N\to\infty$ is an  
immediate consequence of the following\\\\
{\bf Lemma C.3}: \emph{
Let ${\bf \underline{r}}_{J}$ and ${\bf \underline{t}}_{J}$ be defined  
as in Section 7, for any $J\subset I_n$ with $I_n=\{1,2,\dots,n\}$.   
For any
%$k> 0$,
$r
%,n$
\geq 0$
we have:}
\begin{eqnarray}\label{eq: correzMARIO7}
&& \mathcal{D}^{2r} S({\bf \underline{r}}_n,{\bf  
\underline{t}}_n)\mu_N(z'_1\vert Z_N(t))\mu_N(z'_2\vert  
Z_N(t))=\nonumber\\
&&=\sum_{0\leq \ell \leq r}\sum_{0\leq m\leq  
n}\sum_{\substack{I\subset I_n\\ \vert  
I\vert=m}}\(\mathcal{D}^{2\ell}S({\bf \underline{r}}_{I},{\bf  
\underline{t}}_{I})
\mu_N(z'_1\vert Z_N(t)) \)\(\mathcal{D}^{2(r-\ell)}S({\bf  
\underline{r}}_{I_n\setminus I},{\bf \underline{t}}_{I_n\setminus  
I})\mu_N(z'_2\vert Z_N(t))\)+e_{r,N}\nonumber\\
&&
\end{eqnarray}
\emph{where}
\begin{eqnarray}\label{eq: correzMARIO8}
&& e_{r,N}\rightarrow 0\ \ as\ N\to\infty\ \ \ \mathcal{C}_b^{\infty}-weakly.
\end{eqnarray}
\newpage
{\bf Proof:}

It is enough to prove (\ref{eq: correzMARIO7}) and (\ref{eq:  
correzMARIO8}) replacing each streak $S$ with the corresponding  
$\hat{S}$, being the difference $S-\hat{S}$ negligible in the limit.

We start by assuming $r=0$.
%$n>0$.
In that case, testing the left hand side of (\ref{eq: correzMARIO7})  
against a product of two test functions $u_1,u_2$, we are led to  
consider:
\begin{eqnarray}\label{eq: correzMARIO9}
&&\hat{S}({\bf \underline{r}}_n,{\bf  
\underline{t}}_n)\mathcal{U}_1(Z_N(t))\mathcal{U}_2(Z_N(t))
\end{eqnarray}
for which we can apply the expansion (\ref{eq: azSTRINGAdiag}).

Proceeding as in the proof of Lemma C.1 (see (\ref{eq:  
azSTRINGAdiagIII})), we have to consider:
\begin{eqnarray}\label{eq: correzMARIO10}
&& D_{v_{j_m}}^{a_{m,1}}(t_m)\dots  
D_{v_{j_2}}^{a_{2,1}}(t_2)D_{v_{j_1}}^{a_{1,1}}(t_1)\mathcal{U}_1(Z_N(t))\mathcal{U}_2(Z_N(t)),
\end{eqnarray}
where $a_{1,1}=r_1+1>0$. Now any contribution of the form
\begin{eqnarray}\label{eq: correzMARIO11}
&&  
D_{v_{j_1}}^{\alpha}(t_1)\mathcal{U}_1(Z_N(t))D_{v_{j_1}}^{\beta}(t_1)\mathcal{U}_2(Z_N(t)),
\end{eqnarray}
with $\alpha>0$, $\beta>0$, $\alpha+\beta=a_{1,1}$ is  
$O\(\frac{1}{N^2}\)$, therefore it is negligible in the limit. The  
same argument applies to $D_{v_{j_k}}^{a_{k,1}}(t_k)$ whenever  
$a_{k,1}>0$ . This means that each derivative appearing in $\hat{S}$  
either applies to $\mu_N(z'_1\vert Z_N(t))$ or to $\mu_N(z'_2\vert  
Z_N(t))$ up to an error $e_{0,N}$ vanishing in the limit. This is  
exactly what (\ref{eq: correzMARIO7}) and (\ref{eq: correzMARIO8}) say  
for $r=0$.

For $r>0$
%and $n>0$
we have to apply $\mathcal{D}^{2r}$ to (\ref{eq: correzMARIO7})  
(replacing $S$ by $\hat{S}$) with $r=0$. Clearly $\mathcal{D}^{2r}  
e_{0,N}$ vanishes in the limit. Moreover:
\begin{eqnarray}\label{eq: correzMARIO12}
&& D_{G,j}^{2s_j}\[\hat{S}({\bf \underline{r}}_I,{\bf  
\underline{t}}_I)\mathcal{U}_1(Z_N(t))\hat{S}({\bf  
\underline{r}}_{I_n\setminus I},{\bf \underline{t}}_{I_n\setminus  
I})\mathcal{U}_2(Z_N(t))\]=\nonumber\\
&&= \(D_{G,j}^{2s_j}\hat{S}({\bf \underline{r}}_I,{\bf  
\underline{t}}_I)\mathcal{U}_1(Z_N(t))\)\hat{S}({\bf  
\underline{r}}_{I_n\setminus I},{\bf \underline{t}}_{I_n\setminus  
I})\mathcal{U}_2(Z_N(t))+\nonumber\\
&&+\hat{S}({\bf \underline{r}}_I,{\bf  
\underline{t}}_I)\mathcal{U}_1(Z_N(t))\(D_{G,j}^{2s_j}\hat{S}({\bf  
\underline{r}}_{I_n\setminus I},{\bf \underline{t}}_{I_n\setminus  
I})\mathcal{U}_2(Z_N(t))\)+O\(\frac{1}{N^2}\)
\end{eqnarray}
By simple algebraic manipulation we finally arrive to (\ref{eq:  
correzMARIO7}) and (\ref{eq: correzMARIO8}).


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