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\date{February 1993}
\begin{document}
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\title{Microscopic Shock Profiles: Exact Solution of a Nonequilibrium
System}
\author{B. Derrida\\Service de Physique Th\'eorique,
CE Saclay\\[-2pt]F91191 Gif-sur-Yvette, France\\ and\\
S. A. Janowsky\thanks{Permanent address: Department of Mathematics,
Rutgers University, New Brunswick, NJ 08903.},
J. L. Lebowitz\footnotemark[1]\hspace{2.5pt} and
E. R. Speer\footnotemark[1]\\
Institut Des Hautes Etudes Scientifiques\\[-2pt]
91440 Bures-sur-Yvette, France}
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\begin{abstract}
The full microscopic structure of macroscopic shocks is obtained exactly
in the one dimensional totally asymmetric simple exclusion process from
the complete solution of the uniform stationary nonequilibrium state of
a system containing two types of particles---``first'' and ``second''
class.  The width of the shock as seen from a second class particle
diverges as the inverse square of its strength when the latter goes to
zero.  In that limit the steady state exhibits long range correlations
and weakly bound states of second class particles.
\end{abstract}

Short title:  Microscopic Shock Profiles

PACS: 05.40 05.60 05.70L 64.60

To be submitted to Europhysics Letters

\subsection*{Introduction}
There is much interest, physical and mathematical, in the behavior of
model systems of particles on a lattice evolving microscopically under
conservative stochastic dynamical rules and described macroscopically
by a continuous density field satisfying equations of hydrodynamic
type~\cite{Spohn,LPS}.  Of particular interest are models for which
the macroscopic equations produce shocks.  In such cases it is clear
that the hydrodynamic equations do not tell the full story; in fact,
because of the discontinuities in the density and consequent
infinities in the derivatives which enter the equations, shocks
correspond to nonunique weak solutions of these equations, and to
 find the physical solution the equations have to be supplemented by
additional conditions or be treated as the limit of equations with
nonzero viscosity.  The microscopic level has of course no room for
such extra conditions and its study is therefore essential for a
complete understanding of the behavior of shocks~\cite{Sardinia}.

Unfortunately even molecular dynamics simulations of shocks in real
particle systems can only be carried out very partially with the
currently available supercomputers~\cite{Sardinia}, so model systems
such as the one dimensional asymmetric simple exclusion process are
important.  In this model, particles on the lattice jump independently
at random times with rate 1 to adjacent sites, choosing the site to
their right with probability $p$, and to their left with probability
$1-p$, provided the target site is unoccupied~\cite{Spohn,L}.  The
density profile of this system on macroscopic spatial and temporal
scales $x$ and $t$ is described for $p\ne 1/2$ in the appropriate
hydrodynamic scaling limit by the inviscid Burgers'
equation~\cite{Spohn,LPS},
\begin{equation}\label{Burger}
\frac{\partial u(x,t)}{\partial t} + (2p-1) \frac{\partial}{\partial x}
[u(1-u)] = 0.
\end{equation}
This means that there exists a limiting scaling for which the actual
random microscopic density exactly tracks the solution of the Burgers'
equation.

This equation can produce shocks; for example, there is a solution of
(\ref{Burger}) such that $u(x,t)=u_+$ for $x>x_0(t)$ and
$u(x,t)=u_-$ for $x<x_0(t)$, where $u_+>u_-$ (when $p>1/2$) and the
shock position $x_0(t)$ moves with the constant velocity
$(2p-1)(1-u_+-u_-)$.  We now ask about the corresponding microscopic
behavior: can one see an abrupt change at that scale or is the jump in
densities spread out over very large microscopic distances?

To analyze the shock microscopically one must first locate it---that is,
define precisely its position---on the microscopic level.  This may be
done by introducing into the system a special {\em second class
particle}~\cite{ABL}, which evolves by a modified dynamical rule: it
behaves like a regular (first class) particle when jumping to an empty
site but must give way to any regular particle that tries to jump on it
(by exchanging places with the latter).  For $p>1/2$ this dynamics gives
the second class particle a drift towards regions of positive density
gradient and so keeps it near the shock.  The position of this particle
may then conveniently be taken as the definition of the (microscopic)
shock location.  With this definition it has been established that
shocks are sharp even on the microscopic level~\cite{DKPS,FKS};
specifically, that the microscopic (ensemble averaged) particle density
at site $j$, as viewed from the second class particle, has a time
invariant distribution which approaches the densities $\rho_\pm=u_\pm$
at $\pm\infty$.  In practice, this means that one can see the
discontinuity on the scale of a few intermolecular distances, something
consistent with computer simulations on real particle systems and with
experiments~\cite{Sardinia}.  Computer studies of this shock profile are
described in~\cite{BCFN}, see also~\cite{JL}.

In this note we present the exact microscopic structure of such a shock
 for the case $p=1$.  We obtain our results by first finding the exact
translation invariant stationary state of a related model system,
introduced in \cite{FKS}, which considers a system with uniform
densities $\rho_1$ and $\rho_2$ of first and second class particles
({\bf fcp} and {\bf scp}).  It is in fact easy to show that when this
uniform system is viewed from some chosen scp, all scp's to the right of
the chosen one behave as fcp's while all scp's to the left of it behave
as unoccupied sites or holes.  With this identification the
chosen scp in the uniform system sees exactly the same microscopic shock
profile as a single scp in a system containing only fcp's in which the
density approaches $\rho_+ =\rho_1+\rho_2$ at $+\infty$ and $\rho_-
=\rho_1$ at $-\infty$~\cite{FKS}.

This two species model is of independent interest as an example of a
driven diffusive system~\cite{BS}.  Such systems have particle conserving
transition rates which do not satisfy detailed balance with respect to the
stationary state and are expected to exhibit long range spatial
correlations in two or more dimensions~\cite{BS,GLMS,Grinstein}.  Our one
dimensional model also exhibits such behavior when the density of the
second class particles goes to zero.

 For $\rho_2$ nonzero, the exponential decay of correlations in the
uniform system gives an exponential approach to the asymptotic densities
for the one component shock, with a characteristic length $\xi$ which
diverges as $(\rho_+ - \rho_-)^{-2}$ when $(\rho_+ - \rho_-) = \rho_2
\rightarrow 0$.  In the limit $\rho_2\rightarrow0$ there still exists a
microscopic ``shock'' consisting of an excess density of particles in
front of any given scp and of holes in back of it, which decays to zero
as $|j|^{-1/2}$; in contrast the mean field result found in \cite{BCFN}
is $|j|^{-1}$.  In this limit there are also long range correlations
between the second class particles: a pair of scp's in a system with a
uniform nonzero density of fcp's will form a bound state with the
probability of separation $r$ between them decaying as $r^{-3/2}$.

\subsection*{Exact Solution}
The exact stationary state of our two species totally asymmetric simple
exclusion process is based on earlier work~\cite{DEHP}, see
also~\cite{DDM}, for a one component finite system with open boundaries;
the solution can be extended to the partially asymmetric
case~\cite{DJLS}.  We work first on a ring of $N$ sites containing $N_1$
fcp's, $N_2$ scp's and $N_0\equiv N-N_1-N_2$ holes.  A configuration is
specified by an $N$-tuple $(\tau_1,\tau_2,\ldots,\tau_N)$, where
$\tau_i=0$ if site $i$ is empty, $\tau_i=1$ if there is an fcp at site
$i$, and $\tau_i=2$ if there is an scp at site $i$.  During a time
interval $dt$ each adjacent pair of sites in the system of the type
$1\,0$, $1\,2$ or $2\,0$ has probability $dt$ of being exchanged.  It is
easy to see that any configuration can evolve into any other, so that
the system is ergodic and there is a unique stationary probability
measure.  This measure which assigns a weight $w(\tau)$ to each
configuration $ \tau = (\tau_1,\tau_2,\ldots,\tau_N)$ is determined by
the stationary form of the master equation. It satisfies the following
balance condition
\begin{equation}
\sum_{\{\,i\mid\tau_i\tau_{i+1}=10,\,12,\;{\rm or}\;20\,\}}w(\tau)
        \;\;=\;\;\sum_{\{\,i\mid\tau_i\tau_{i+1}=01,\,21,\;{\rm or}\;02\,\}}
        w(\tau^{i,i+1})\,,\label{bal}
\end{equation}
where $\tau^{ij}$ denotes the configuration obtained from
$\tau=(\tau_1,\ldots,\tau_N)$ by interchanging $\tau_i$ and $\tau_j$.

If either $N_0$, $N_1$ or $N_2$ equals zero then the system reduces to
the one component asymmetric simple exclusion process whose stationary
measure is well known: all configurations are equally likely.  In the
general case the stationary measure will be constructed with the aid of
matrices or operators $D$ and $E$ which satisfy~\cite{DEHP}
\begin{equation}
DE = D+E
\label{DE}
\end{equation}
and the matrix $A = DE-ED$ which, from (\ref{DE}), satisfies the
relations
\begin{equation}\label{DAAE}
DA = AE = A.
\end{equation}
We will show that the probability in the stationary measure for fixed
$N_1, N_2$ of the configuration $\tau$ is then given~\cite{DJLS}, up to an
overall normalizing factor, by the weight $w(\tau)$ defined by
\begin{equation}
w(\tau_1,\ldots,\tau_N)=\Tr[X_1\cdots X_N],\label{fund}
\end{equation}
where $X_i=E$ if $\tau_i=0$, $X_i=D$ if $\tau_i=1$, and $X_i=A$ if
$\tau_i=2$.

To verify that (\ref{fund}) provides the weights of a stationary
measure, we substitute this formula on both sides of (\ref{bal});
it is then convenient to write $X_1\cdots X_N$, possibly after a cyclic
permutation which will not affect the trace, in terms of blocks of
consecutive identical matrices: $X_1\cdots X_N=Y_1^{k_1}\cdots
Y_m^{k_m}$, where each $Y_j$ is $D$, $A$, or $E$, and $Y_j\ne Y_{j+1}$,
$Y_m\ne Y_1$.  Now in the term on the left hand side of (\ref{bal})
indexed by $i$ we use the algebraic relations (\ref{DE}) and
(\ref{DAAE}) to replace the product $X_iX_{i+1}$ by a sum of two
operators or a single operator---that is, $DE$ is replaced by $D+E$
while $DA$ and $AE$ are replaced by $A$.  The net effect of these
replacements is that each $D$ block (respectively $E$ block) gives rise
to a term in which the rightmost (respectively leftmost) factor in the
block has disappeared; notice that in the case of a DE boundary we get a
term for each block.  Thus we find
\begin{equation}
\sum_{\{\,i\mid\tau_i\tau_{i+1}=10,\,12,\;{\rm or}\;20\,\}}
        \Tr[X_1\cdots X_iX_{i+1}\cdots X_N]
= \sum_{\{\,j\,|\, Y_j=D,E\,\}}
        \Tr[Y_1^{k_1}\cdots Y_j^{k_j-1}\cdots Y_m^{k_m}].
\label{left}
\end{equation}
If we make the same replacements for the product $X_{i+1}X_i$ in each
term on the right hand side of (\ref{bal}), then the net effect is that
$D$ blocks lose their leftmost and the $E$ blocks their rightmost
 factors,  and we find that  (\ref{fund}) satisfies (\ref{bal}).

To obtain the matrices $D$, $E$ and $A$, write $D=I+d$ and $E=I+e$,
where $I$ is the identity matrix.  Then (\ref{DE}) gives $de=I$ which
shows that to obtain a nontrivial solution we must take infinite
matrices:
an explicit solution~\cite{DEHP} is that $d$ is the right
and $e$ the left shift on semi-infinite vectors:
\begin{equation}
D_{ij} = \delta_{i,j} + \delta_{i,j-1}, \qquad
E_{ij} = \delta_{i,j} + \delta_{i,j+1} \quad {\rm and} \quad
A_{ij} = \delta_{i,1}\delta_{j,1},\qquad
i,j\ge 1.
\end{equation}
$A$ is then the projector on the first component,
$A=|1\rangle \langle 1|$ with $|1\rangle = [1,0,0,\ldots]$, and has
trace 1.

An important consequence of this matrix formulation and of the fact that
the matrix $A$ is a one dimensional projector is the absence of
correlations between regions separated by second class particles.  In
particular, in the infinite volume limit, there is a factorization about
any site $i$ occupied by a second class particle,
\begin{equation}
\langle f(\tau) g(\tau) | \tau_i=2\rangle =
        \langle f(\tau) | \tau_i=2\rangle
        \langle g(\tau) | \tau_i=2\rangle,
\end{equation}
whenever $f$ (respectively $g$) depends only on the configuration
to the left (respectively right) of $i$.

\subsection*{Shock structure and other properties}
Once the weight of each configuration in the steady state is known, one
can in principle calculate all the correlation functions for arbitrary
$N_1$, $N_2$ and $N$, and then consider various infinite volume limits
$N\rightarrow\infty$.  In practice this limit is actually taken in a
``grand canonical'' ensemble after introducing fugacity-like parameters
which are later converted back to densities.  We consider several
situations: a finite density of both first class and second class
particles ($N_1/N\rightarrow\rho_1$, $N_2/N\rightarrow\rho_2$), a single
second class particle with a nonzero density of first class particles
($N_1/N\rightarrow\rho$, $N_2=1$), and two second class particles with
a nonzero density of first class particles ($N_1/N\rightarrow\rho$,
$N_2=2$).  All these cases can be treated using similar techniques.
Here we will summarize our results in the first two cases, leaving the
detailed derivation for a future longer version of the present
work~\cite{DJLS}, and will sketch the calculation in the third and
easiest case, that of two second class particles.

We first consider the case of finite densities of both types of
particles, in which for each choice of $\rho_1$ and $\rho_2$ the
infinite volume limit yields a translation invariant measure which is
stationary for the two species asymmetric exclusion dynamics.  One
interesting feature of the measure that emerges from explicit
computation~\cite{DJLS} is that, conditioned on the presence of an scp
at some site $i$, the distribution of fcp's to the {\em right} of $i$
and of holes to the {\em left} of $i$ are just product measures with
densities $\rho_1$ and $1-\rho_1-\rho_2$ respectively.

To obtain from this measure the shock profile in the one component
system with asymptotic densities $\rho_+=\rho_1+\rho_2$ and
$\rho_-=\rho_1$, we pick any scp in the two component system and let its
instantaneous position specify the origin of the lattice.  We then
define $\delta_j$ as the probability that site $j$ is occupied by an fcp
 for $j<0$ and that it is occupied by either an fcp or scp for $j>0$.  An
explicit computation gives~\cite{DJLS}
\begin{equation}\label{eta1-1}
\delta_{-1} = \rho_+\rho_-, \qquad
\delta_{1} = \rho_+ + \rho_- - \rho_+\rho_-,
\end{equation}
and for all $j \geq 2$,
\begin{equation}
\delta_{-j} - \delta_{-1} =
        \delta_1 - \delta_j  =
        \sum_{n=1}^{j-1} \sum_{p=0}^{n-1}
        { n \choose p}
        { n-1 \choose p}
\frac{(\rho_+ \rho_-)^{n -p} [(1- \rho_+)(1- \rho_-)]^{p+1}}{p+1}.
                \label{deltaeta}
\end{equation}
That this is indeed the shock profile in the one component totally
asymmetric simple exclusion process, seen in the randomly moving frame
of a single scp, follows from the observation made earlier that each scp
in the two component system behaves as if all the scp's at its left were
holes and all scp's at its right were fcp's.  Notice that these shocks
have the symmetry property $\delta_j +
\delta_{-j} = \rho_+ + \rho_-$.  It is
also easy to check from (\ref{eta1-1}) and (\ref{deltaeta}) that
$\delta_j$ converges exponentially: $ |\delta_j - \rho_{\pm}|
\sim \exp [-|j|/ \xi] $ as $j \rightarrow \pm \infty$ when $\rho_+ >
\rho_-$, where the characteristic length $\xi$ measures
the microscopic width of the shock as seen from the scp.  This length
$\xi$ diverges
as $(\rho_+ - \rho_-)^{-2}$ when $\rho_+ \rightarrow \rho_-$.

We can obtain the limiting shock shape when $\rho_-=\rho_+$ directly
 from (\ref{deltaeta}) or we can equivalently consider the case of a
single scp in a system of fcp's at a uniform nonzero density $\rho$.
Physically, one expects that the scp will be attracted by fluctuations
with a positive density gradient of fcp's, so that the average density $
d_j$ of fcp's at position $j$ (in the frame where the scp is at the
origin) will have a nontrivial profile. We have calculated this profile
exactly \cite{DJLS} from the steady state (\ref{fund}) and found in the
limit of an infinite system size ($N \rightarrow \infty$ with $N_1 / N
\rightarrow \rho$ and $N_2=1$) that $d_1 = 2\rho -\rho^2$, $d_{-1} =
\rho^2$, and for all $j \geq 2$,
\begin{equation}
d_{-j} - d_{-1} = d_1 - d_j = \sum_{n=1}^{j-1} \sum_{p=0}^{n-1} {n
\choose p} {n-1 \choose p} \frac{(1 - \rho)^{2n - 2 p} \rho^{2 p + 2
}}{p+1}.
\end{equation}
 From this exact expression, which is equivalent to that obtained from
(\ref{deltaeta}) for $\delta_j$ when
$\rho_2\searrow0$ and $\rho_1=\rho$, one can show that for large
$j$ the profile is a power law: $ |d_j - \rho| \sim |j|^{- 1 / 2}$.
The exponent $1/2$ disagrees with the exponent $1$ obtained from a mean
 field argument~\cite{BCFN}.

The last case we consider is the case of a pair of scp's in an
infinite system of fcp's at a nonzero density $\rho$. In that case, we
 find that the probability $P_\infty(r)$ that two scp's are a
distance $r $ apart (in an infinite system) is given by
\begin{equation}\label{P-separation}
P_\infty(r) =  \sum_{p=0}^{r-1}
        {r \choose p}   {r-1 \choose p}
        \frac{\rho^{2n -2p-1} (1- \rho)^{2p+1}}{p+1}.
\end{equation}
This shows that the two second class particles form a bound state, of a
rather unusual kind since $ P_\infty(r) \sim r^{-3/2}$ and the average
distance is infinite. (We can say that this is an {\em algebraic} bound
state in contrast with the normally encountered exponentially decaying
bound states.)

The derivation of (\ref{P-separation}) is rather easy when one uses the
grand canonical ensemble. If we introduce a parameter $x$ which plays
the role of a fugacity for the fcp, so that the weight of any configuration
with $N_1$ first class particles is multiplied by $x^{N_1}$, then the
probability (in the grand canonical ensemble) of finding the two second
class particles a distance $r$ apart, on a ring of $N$ sites (for
simplicity we choose here $N$ to be odd and $r$ to be the shorter
distance between the two particles on the ring) is
\begin{equation}
P_N(r) = \langle 1 | C^{r-1} | 1\rangle
        \langle 1 | C^{N-r-1} | 1\rangle
        \left[\, \sum_{s=1}^{(N-1)/2}
        \langle 1 | C^{s-1} | 1\rangle
        \langle 1 | C^{N-s-1} | 1\rangle  \right]^{-1},
                \label{E1}
\end{equation}
where the matrix $C$ is defined by
\begin{equation}
C= x D + E,
\end{equation}
i.e., $C_{ij} = \delta_{i,j+1} +(1+x) \delta_{i,j} + x \delta_{i,j-1} $
 for $i \geq 1 $ and $j \geq 1$. The expression (\ref{E1}) arises
because when one sums over all the occupations of the $N-2$
sites one is adding at each site a matrix $D$ with weight $x$ and a
matrix $E$ with weight $1$, giving at
each site a matrix $C$.  The limit $N \rightarrow \infty$
then follows easily: from the explicit expression of the matrix $C$,
whose elements can be interpreted in terms of random walks, one
can show that for any $n$~\cite{DJLS},
\begin{equation}
\langle 1 |  C^n | 1  \rangle =  \sum_{p=0}^{n}
        {n+1 \choose p} {n \choose p}
        \frac{x^p}{p+1},
                \label{E2}
\end{equation}
which gives for large $n$
\begin{equation}
\langle 1 |  C^n | 1  \rangle  \sim (\sqrt{x} +1)^{2n} \Big/ n^{3/2}.
\label{E3}
\end{equation}
 From this asymptotic behavior, it follows that the density $\rho$ is
related in the large $N$ limit to the fugacity $x$ by $\rho = \sqrt{x}\big/
( \sqrt{x} + 1 )$; then replacing $x$ by $\rho^2/(1- \rho)^2 $ in
(\ref{E1}) and (\ref{E2}) one obtains in the large $N$ limit
(\ref{P-separation}).

\subsubsection*{Acknowledgments}
This work was supported in part by AFOSR Grant 91-0010; SAJ was
supported in part by NSF Mathematical Sciences Postdoctoral Research
 Fellowship DMS 90-07206.  BD thanks the Mathematics Department of
Rutgers University, where this work was initiated, for its
hospitality.  We also thank the ENS in Paris where it was completed.

\renewcommand\baselinestretch{1.08}\large\normalsize
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\end{thebibliography}

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