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This paper uses and extends the results given in mp-arc/99-373
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diffusion, Burnett, Lorentz gas, decay of correlations, cumulants, 
combinatorics
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\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{conjecture}[theorem]{Conjecture}

\begin{document}
\title{The Burnett expansion of the periodic Lorentz gas}
\author{
C. P. Dettmann\thanks{Center for Studies in Physics and Biology,
Rockefeller University, New York, NY 10021.}}

\maketitle




\begin{abstract}
The macroscopic limit of some deterministic scattering processes can be
described by the diffusion equation $\partial_t \rho=D\nabla^2\rho$,
where the diffusion coefficient $D$ is given by a sum over two-time
correlation functions, the discrete time version of the Green-Kubo integral
over the velocity autocorrelation function.  The approximation can be
improved by
allowing higher spatial derivatives; in this case the coefficients are
called Burnett coefficients, and are given by sums over
multiple time correlation functions.  The periodic Lorentz gas has exponential
decay of two-time correlation functions, implying the existence of the
diffusion coefficient.
Recent work has established a stretched exponential decay of multiple
correlation functions and the existence of the fourth order Burnett coefficient
for the periodic Lorentz gas.  Here we give expressions for the higher
coefficients, show that these expressions converge, and give a plausible
argument based on similar models that the expansion composed of the Burnett
coefficients has a finite radius of convergence.
\end{abstract}

\section{Introduction}
The {\em Lorentz gas} is a model used in statistical mechanics, consisting of a
point particle moving at constant velocity except for specular collisions
with smooth (specifically $C^3$) convex fixed scatterers in $d\geq 2$
dimensions.  The original model~%\cite{L}
has randomly placed scatterers
in infinite space, and is known to have power law decay of correlations,
so that the Burnett coefficients (defined below as sums of such correlations)
are not generally expected to exist~\cite{vB}. Here we consider a periodic
arrangement of scatterers which is equivalent to a dispersing billiard on
a torus, so that two-time correlations (in discrete time)
decay exponentially~\cite{Y,C99}.
This, together with the {\em finite horizon} condition, that is, that
the time between collisions is bounded, implies the existence of the
diffusion coefficient ($D^{(2)}$ below).
A recent paper~\cite{CD} gives a stretched exponential decay
of multiple correlations~\cite{CD}, and uses this to show
(again with finite horizon) that the fourth order Burnett
coefficient ($D^{(4)}$ below) exists.  Here we extend this result to
all the Burnett coefficients.  A common example for $d=2$ with a finite horizon
is given by circular scatterers on a hexagonal lattice; for $d>2$ the
finite horizon condition requires either nonspherical scatterers, or more
than one scatterer per unit cell. The Lorentz gas and its various extensions
are discussed extensively in Ref.~\cite{S}.

From the point of view of statistical mechanics, the microscopic (Liouville
or Frobenius-Perron) evolution of the probability density $\rho$ of the
particle in phase space $({\bf r},{\bf v})$ can be described using
macroscopic or ``hydrodynamic'' equations in real space ${\bf r}$ at
distance and time scales large with respect
to the size of the scatterers, the periodic cell, and the
time between collisions.
The diffusion equation in a homogeneous but anisotropic medium corresponding
to the periodic Lorentz gas is
\begin{equation}
\partial_t \rho=D^{(2)}_{\alpha\beta}\partial_\alpha\partial_\beta\rho
\end{equation}
where the Greek indices indicate spatial dimensions, and repeated Greek
indices are summed.  One systematic
improvement of the hydrodynamic approximation, suggested by Burnett~\cite{B},
is the addition of higher spatial derivatives corresponding to smaller
distance scales,
\begin{equation}\label{e:gdiff}
\partial_t \rho=D^{(2)}_{\alpha\beta}\partial_\alpha\partial_\beta\rho
+D^{(4)}_{\alpha\beta\gamma\delta}\partial_\alpha\partial_\beta\partial_\gamma
\partial_\delta\rho+\ldots
\end{equation}
and the higher coefficients are called (linear) Burnett coefficients.
Nonlinear terms such as powers of $\partial_\alpha\rho$ are excluded 
here because $\rho$ is a projection onto real space of a phase space
density satisfying a linear evolution equation.  The presence of only
even terms here and in Eq.~(\ref{e:disp}) below is expected by symmetry
arguments; we do not assume this until it is justified at the end of
Sec.~\ref{s:exp}. 

The solutions of the generalized diffusion equation are of the form
\begin{equation}\label{e:sol}
\rho({\bf r},t)=\exp(s({\bf k})t+\i{\bf k}\cdot{\bf r})
\end{equation}
with a dispersion relation
\begin{equation}\label{e:disp}
s({\bf k})=-D_{\alpha\beta}^{(2)}k_\alpha k_\beta
+D^{(4)}_{\alpha\beta\gamma\delta}k_\alpha k_\beta k_\gamma k_\delta+\ldots
\end{equation}
describing the rate of exponential decay of the hydrodynamic mode corresponding
to wave number ${\bf k}$.  Note that the dispersion relation~(\ref{e:disp})
may be a more convenient description than the generalized diffusion
equation~(\ref{e:gdiff}) when the Burnett coefficients do not exist, as
$s({\bf k})$ may still exist, but be nonanalytic.

The dispersion relation is obtained from the microscopic dynamics as
discussed in chapter 7 of Ref.~\cite{G}, from which we need only Eq.~(7.91):
\begin{equation}\label{e:gen}
1=\lim_{n\rightarrow\infty}\la\prod_{i=-n}^{n-1}\exp
\left[-s({\bf k})T(\map^i x)-\i{\bf k}\cdot{\bf a}(\map^i x)\right]\ra
\end{equation}
from which $s({\bf k})$ can be obtained by an expansion in powers of $\bf k$.
Here the billiard map $\map$ is defined on the collision space $M$, that
is, $x=({\bf r},{\bf v})\in M$ when ${\bf r}$ is on the boundary of one of
the scatterers and ${\bf v}$ is in an outward direction from the scatterer.
$T(x)$ is the time between the collision at $x$ and the next; it is a piecewise
H\"{o}lder continuous function~\cite{BSC,C94} (the finite horizon condition
ensures that both $T$ and ${\bf a}$ are bounded).  ${\bf a}(x)$
is the lattice translation vector associated with this free flight; it is
a linear combination of the lattice basis vectors ${\bf e}^{(\alpha)}$
with integer coefficients, and is a piecewise constant function.  The
average $\la\cdot\ra$ denotes integration over $M$ with
respect to the invariant equilibrium measure. 

We are now in a position to state the main result of this paper:

\begin{theorem}\label{th:main}
The coefficients $D^{(m)}$ defined by a formal expansion of
Eqs.~(\ref{e:disp},\ref{e:gen}) in powers of ${\bf k}$ are given
by absolutely convergent series.
\end{theorem}

The proof is in two parts.  Sec.~\ref{s:exp} shows how to expand the
above expression to obtain the Burnett coefficients in terms of sums
of correlation functions; we need only time reversibility and phase
space volume conservation, so these expressions apply to more general
periodic systems than the Lorentz gas.  Sec.~\ref{s:com} uses the results of
Ref.~\cite{CD} together with some combinatorics to show that these series
converge absolutely for the periodic Lorentz gas.  The final section,
Sec.~\ref{s:asym} contains arguments in support of the following conjecture:

\begin{conjecture}\label{c}
The series defined by Eq.(\ref{e:disp}) has a finite radius of convergence.
\end{conjecture}

The author is grateful for helpful discussions with N. I. Chernov, E. G. D.
Cohen, J. R. Dorfman and P. Gaspard, and for the support of the Engineering
Research Program of the Office of Basic Energy Sciences at the US Department
of Energy, contract \#DE-FG02-88-ER13847.


\section{Expressions for the Burnett coefficients}\label{s:exp} 
The coefficient $D^{(4)}$ is obtained in Ref.~\cite{G} by differentiating
Eq.~(\ref{e:gen}) with respect to ${\bf k}$ four times; the calculation
is described as ``long but straightforward''.  Here we are not interested
in the explicit expressions obtained but rather their structure.  We remark
however, that the
equations given here contain enough details to obtain the full
expression for $D^{(6)}$ etc. efficiently if required.

We begin by noting that Theorem~\ref{th:CD} below requires functions $f(x)$
with {\em zero mean}, that is, $\la f \ra=0$.  We note that
$\la {\bf a} \ra=0$ due to time reversibility and phase space
volume conservation.  We define
a new function $\Delta T(x)=T(x)-\la T \ra$ so that $\la \Delta T \ra=0$.
This gives the fluctuations in the collision time about its average.
Writing $T=\la T\ra+\Delta T$ in Eq.~(\ref{e:gen}), removing the $\la T \ra$
term from the average, and taking the logarithm, we find
\begin{equation}\label{e:logexp}
s({\bf k})=\lim_{n\rightarrow\infty}\frac{1}{2n\la T \ra}
\ln\la\exp[F({\bf k})]\ra
\end{equation}
where
\begin{equation}\label{e:Fdef}
F({\bf k})=\sum_{i=-n}^{n-1} f({\bf k})\circ\map^i=\sum_{i=-n}^{n-1}
[s({\bf k})\Delta T\circ\map^i+\i{\bf k}\cdot{\bf a}\circ\map^i]
\end{equation}
is a power series in ${\bf k}$ such that all the coefficients have zero mean. 

Now we put the expression in the form of a cumulant expansion, which
allows cancellation between similar terms of opposite sign, thus permitting
the proof of convergence in Sec.~\ref{s:com}.
Expanding the exponential and the logarithm, and noting that $\la F \ra=0$
we find
\begin{equation}\label{e:Q}
s({\bf k})=\lim_{n\rightarrow\infty}\frac{1}{2n\la T \ra}\sum_{N=2}^{\infty}
Q_N({\bf k})
\end{equation}
where the {\em cumulants} $Q_N({\bf k})$ are defined by
\begin{equation}\label{e:Qdef}
Q_N({\bf k})=\sum_{\{n_j\}:\sum_jjn_j=N}
(-1)^{\nu-1}\frac{(\nu-1)!\prod_j\la F^j \ra^{n_j}}{\prod_j (n_j!j!^{n_j})}
\end{equation}
with $j$ and $n_j$ integers satisfying $j\geq2$ and $n_j\geq0$, and
$\nu=\sum_jn_j$ is the total number of correlations in the product.
For example
\begin{eqnarray}
Q_2({\bf k})&=&\la F^2 \ra/2\\ 
Q_3({\bf k})&=&\la F^3 \ra/6\\
Q_4({\bf k})&=&(\la F^4\ra-3\la F^2 \ra^2)/24\\
Q_5({\bf k})&=&(\la F^5 \ra-10\la F^3 \ra \la F^2 \ra)/120\\
Q_6({\bf k})&=&
(\la F^6 \ra-15\la F^4 \ra \la F^2 \ra-10\la F^3 \ra^2+30\la F^2 \ra^3)/720
\end{eqnarray}

Using the definition of $F$, we can expand each cumulant as a power series
in ${\bf k}$,
\begin{equation}\label{e:qdef}
Q_N({\bf k})=\sum_{m=N}^{\infty}q_{N,m;\alpha_1\ldots\alpha_m}
k_{\alpha_1}\ldots k_{\alpha_m}
\end{equation}
Note that to fixed order ${\bf k}^m$, only $Q_N$ with $N\leq m$ appear, thus
the rearrangement of the expansion from~(\ref{e:logexp}) to~(\ref{e:Q})
involves only a finite number of terms, so questions of absolute convergence
do not appear.  This is the reason for the interpretation of these expressions
as formal power series in ${\bf k}$ as indicated in the statement of
Thm.~\ref{th:main}.  In Sec.~\ref{s:asym} we ask whether such series converge
for finite values of ${\bf k}$; it is apparent from Sec.~\ref{s:com}
that convergence of the individual cumulants depends to a large extent
on cancellations, so it is likely that Eq.~(\ref{e:Q}) rather than
Eq.~(\ref{e:logexp}) gives more consistent results at finite ${\bf k}$.

Writing Eq.~(\ref{e:Q}) as a formal power series in ${\bf k}$ leads to
an infinite set of (finite) equations for the unknowns $D^{(m)}$:
\begin{equation}\label{e:Dsol}
\i^mD^{(m)}_{\alpha_1\ldots\alpha_m}=
\lim_{n\rightarrow\infty}\frac{1}{2n\la T\ra}
\sum_{N=2}^{m}q_{N,m;\alpha_1\ldots\alpha_m}
\end{equation}
Note that the $q_{N,m}$ contain $D^{(j)}$ with $j\leq m/2$ through the
$\Delta T$ term in~(\ref{e:Fdef}); thus we have an explicit equation for
$D^{(m)}$, order by order, if the $n\rightarrow\infty$ limit in~(\ref{e:Q})
converges.

Time reversal invariance and phase space volume conservation lead
to equal and opposite contributions from correlations with odd powers of
${\bf a}$.  Since each ${\bf a}$ corresponds to a single power of ${\bf k}$
and each $\Delta T$ corresponds to powers $m\geq2$ at which $D^{(m)}$ is
nonzero, we find that the terms $\la a_\alpha a_\beta a_\gamma \ra$ and
$\la \Delta T a_\alpha \ra$ contributing to $D^{(3)}$ both vanish.
The vanishing of $D^{(m)}$ for all odd $m$ then follows by induction.
Note that odd cumulants do not vanish, for example $Q_3$ contains the
term $\la \Delta T a_\alpha a_\beta \ra$ which does not vanish, and
contributes to $D^{(4)}$.

We remark that we have not yet used any information about the decay of
correlations of the Lorentz gas, so the above expressions are valid for the
Burnett coefficients of more general diffusive processes, as long as the
$n\rightarrow\infty$ limit converges.  For systems with a decay of
correlations that is a (not too small) negative power of the time, we expect
the first few Burnett coefficients to exist, see for example Ref.~\cite{vB}.


\section{Convergence of the series}\label{s:com}
The averages $\la F^{n_j} \ra$ appearing in the cumulants contain summations
over $m$ variables with range $-n$ to $n-1$, and could grow as fast as
$O(n^{n_j})$ in general.  Thus each term, which is a product of such averages
could grow as $O(n^N)$ in general.  For the limit in Eq.~(\ref{e:Q}) to exist,
we require that the RHS grows only as $O(n)$.  Although the growth of each
product of correlations cannot be controlled this well, the following
theorem is sufficient to imply the existence of the Burnett coefficients:

\begin{theorem}\label{th:conv}
$q_{N,m}$ is defined in Eqs.~(\ref{e:Fdef}, \ref{e:Qdef}, \ref{e:qdef})
for integers $N$ and $m$ satisfying $2\leq N\leq m$. 
$q_{N,m;\alpha_1\ldots\alpha_m}=O(n)$ for all such $N$ and $m$ in the
periodic Lorentz gas.  
\end{theorem}

To prove Thm.~\ref{th:conv}, we make use of the properties (``stretched
exponential decay'') of multiple correlations in the periodic Lorentz gas:

\begin{theorem}\label{th:CD}
(Theorem 2 of Ref.~\cite{CD}) Let $i_1\leq\cdots\leq i_k$ and
$1\leq t\leq k-1$.  Then
\begin{equation}
|\la \f{1}\cdots\f{k}\ra-\la \f{1}\cdots\f{t}\ra\la\f{t+1}\cdots\f{k}\ra|
\leq C_k\cdot|i_k-i_1|^2\lambda^{|i_{t+1}-i_t|^{1/2}}
\end{equation}
where $C_k>0$ depends on the functions $f_1,\ldots,f_k$, and $\lambda<1$
is independent of $k$ and $f_1,\ldots,f_k$.
\end{theorem}

Here (and subsequently) $f_j$ are piecewise H\"older continuous functions
such that $\la f_j \ra=0$ for all $j$.  We put $f_j^i=f_j\circ\map^i$.
In our case the $f_j$ are $\Delta T$ or $a_\alpha$.  As noted in
Ref.~\cite{CD}, we expect based on Refs.~\cite{Y,C99} that it should be
possible to prove a stronger bound $\lambda^{|i_{t+1}-i_t|}$, but the
above bound is sufficient for our purposes here.

The $q_{N,m}$ as defined in the previous section are finite sums of terms of
the form (see Eqs.~(\ref{e:Fdef}, \ref{e:Qdef}, \ref{e:qdef}))
\begin{equation}
\sum_{\{n_j\}:\sum_jjn_j=N}(-1)^{\nu-1}\frac{(\nu-1)!}{\prod_j(n_j!j!^{n_j})}
\sum_{i_1\ldots i_N=-n}^{n-1}\la \f{1}\ldots\f{j} \ra
\la \f{j+1}\ldots \ra\ldots\la\ldots \f{N}\ra
\end{equation}
multiplied by constants such as the lower order Burnett coefficients.

In order to use Thm.~\ref{th:CD} we need to put the times $i_p$ in
numerical order.  It does not matter in which order the $f_j$ are
multiplied within a correlation, or which order correlations of
equal numbers of $f_j$ are multiplied; thus both factorials in
the denominator disappear, leading to
\begin{eqnarray}\nonumber
\sum_{\{n_j\}:\sum_jjn_j=N}&&(-1)^{\nu-1}(\nu-1)!\left[
\sum_{i_1\leq i_2\ldots i_N}\frac{1}{S[i]}\la \f{1}\ldots\f{j} \ra
\la \f{j+1}\ldots \ra\ldots\la\ldots \f{N}\ra\right.\\
&&\left.+\mbox{ permutations}\right]\label{e:sum}
\end{eqnarray}
Here $S[i]$ is a combinatoric factor which takes account of the
symmetry when some of the $i_p$ are equal; the exact form is
irrelevant because it does not depend on the $n_j$ and so appears
as a common prefactor.  Equal times do not play a direct role in the
application of Thm.~\ref{th:CD}, since we use the {\em largest} difference
between the times $i_{t+1}-i_t$.

As an example, we give the expression for $N=6$:
\begin{eqnarray}\nonumber
\sum_{i_1\leq i_2\leq i_3\leq i_4\leq i_5\leq i_6}&&\frac{1}{S[i]}\left\{
\la\f{1}\f{2}\f{3}\f{4}\f{5}\f{6}\ra
-\left[\la\f{1}\f{2}\ra\la\f{3}\f{4}\f{5}\f{6}\ra
+\mbox{14 permutations}\right]\right.\\
&&-\left[\la\f{1}\f{2}\f{3}\ra\la\f{4}\f{5}\f{6}\ra
+\mbox{9 permutations}\right]\\\nonumber
&&\left.+2\left[\la\f{1}\f{2}\ra\la\f{3}\f{4}\ra\la\f{5}\f{6}\ra
+\mbox{14 permutations}\right]\right\}
\end{eqnarray}

Now we apply Thm.~\ref{th:CD} to the largest gap, $i_{t+1}-i_t$.  Notice
in the $N=6$ example, whatever the value of $t$, the theorem combines all the
above correlations to leave a term bounded by $\lambda^{|i_{t+1}-i_t|^{1/2}}$
multiplied by powers of the time differences.  Explicitly, for $t=1$, all
terms cancel individually because $\la f_j \ra=0$.  For $t=2$ the $\la f^6 \ra$
term cancels with one of the $\la f^2 \ra \la f^4 \ra$ terms, six other
$\la f^2 \ra \la f^4 \ra$ terms cancel with three of the $\la f^2 \ra^3$
terms and the remaining terms all split leaving a $\la f \ra$ term.
For $t=3$ the $\la f^6 \ra$ term cancels with one of the $\la f^3 \ra^2$
terms, and all of the others split leaving a $\la f \ra$ term.  $t=4$ is
analogous to $t=2$ and $t=5$ is analogous to $t=1$.

In general we must show that the coefficient $(-1)^{\nu-1}(\nu-1)!$ leads to
complete cancellation for all values of $N$.  Consider a general term
(ignoring the $S[i]$ which is the same for each term) which is
unaffected by a split at time $t$:
\begin{equation}\label{e:ex}
\la \ra \la \ra\ldots\la \ra | \la \ra \la \ra\ldots \la \ra
\end{equation}
where all times $i_p$ to the left of the bar ``$|$'' are less than or equal to
$t$ and all times to the right of the bar are greater than $t$.  There
are $A$ correlations to the left and $B$ correlations to the right, so
$A+B=\nu$.

This term will cancel with any term which is split to the same form, that
is, a term consisting of correlations that are either the same as the
above, or combine one correlation from the left and one from the right.
For example, when $N=8$, a split at $t=4$ combines
$-6\la\f{1}\f{2}\ra\la\f{3}\f{4}\ra|\la\f{5}\f{6}\ra\la\f{7}\f{8}\ra$
with $2\la\f{1}\f{2}\f{5}\f{6}\ra\la\f{3}\f{4}\ra\la\f{7}\f{8}\ra$,
$2\la\f{1}\f{2}\f{7}\f{8}\ra\la\f{3}\f{4}\ra\la\f{5}\f{6}\ra$,
$2\la\f{1}\f{2}\ra\la\f{3}\f{4}\f{5}\f{6}\ra\la\f{7}\f{8}\ra$,
$2\la\f{1}\f{2}\ra\la\f{3}\f{4}\f{7}\f{8}\ra\la\f{5}\f{6}\ra$,
$-\la\f{1}\f{2}\f{5}\f{6}\ra\la\f{3}\f{4}\f{7}\f{8}\ra$ and
$-\la\f{1}\f{2}\f{7}\f{8}\ra\la\f{3}\f{4}\f{5}\f{6}\ra$.  These all
cancel because $-6+2+2+2+2-1-1=0$.

The term given in Eq.~(\ref{e:ex}) has coefficient $(-1)^{\nu-1}(\nu-1)!$.
There are $AB$ terms with coefficient $(-1)^{\nu-2}(\nu-2)!$ obtained by
combining a single correlation on the left and the right.  There are
$A(A-1)B(B-1)/2!$ terms with coefficient $(-1)^{\nu-3}(\nu-3)!$ obtained
by combining two correlations on the left and the right, and so on.  The
total coefficient is thus given by

\begin{equation}\label{e:hyper}
\sum_{p=0}^{min(A,B)} (-1)^{A+B-p-1}(A+B-p-1)!\frac{A!B!}{(A-p)!(B-p)!p!}
\end{equation}

To show that the coefficients cancel, we therefore need the following lemma:

\begin{lemma}\label{l}
The sum given by Eq.~(\ref{e:hyper}) is zero for all positive integers
$A$ and $B$.
\end{lemma}

{\em Proof:}
The sum is symmetric in $A$ and $B$ so suppose that $A\geq B$.  Then
the summand is the product of a constant $(-1)^{A+B-1}A!$, an alternating
binomial of degree $B$, that is, $(-1)^{-p}B!/((B-p)!p!)$ and a polynomial
in $p$ of degree $B-1$, that is, $(A+B-p-1)!/(A-p)!$.  Repeatedly summing
by parts reduces the degree of both the alternating binomial and the
polynomial until the result is zero, thus proving the lemma.

Given that all the correlations in the cumulants cancel leaving terms
bounded by $\lambda^{|i_{t+1}-i_t|^{1/2}}$ times a polynomial in the
time differences, the only terms making an appreciable contribution to
the sum~(\ref{e:sum}) are those with no sizable gap, that is, with
all the times $i_p$ close to each other.  There are $O(n)$ of such terms,
so Thm.~\ref{th:conv} is proved.  Eq.~(\ref{e:Dsol}) gives the Burnett
coefficients in terms of a finite number of the $q_{N,m}$ and so
Thm.~\ref{th:main} is also proved.

\section{Convergence of the Burnett expansion}\label{s:asym}
Having established the existence of the Burnett coefficients $D^{(m)}$,
we make a few remarks about the question of whether the expansion
Eq.~(\ref{e:disp}) converges for finite ${\bf k}$, Conj.~\ref{c}.
It is generally
very difficult to estimate convergence numerically, because the many
cancellations appearing in the cumulant expansion make
an accurate computation beyond $m=4$ very difficult, see for example
Ref.~\cite{DC}.

There are two known results that make a finite radius of convergence
for the Lorentz gas plausible.  The first is that in the Boltzmann
limit of a hard sphere gas, that is, a gas with many moving particles at
low density and with recollisions ignored, the expansion in ${\bf k}$
(in this context called the linearized Chapman-Enskog
expansion) converges, Ref.~\cite{McL}.  Of course, the hard sphere
collisions are similar to that of the Lorentz gas, but recollisions
cannot be ignored in general.

The second result is exact, but for a highly simplified (piecewise linear)
system.  We consider the map
\begin{equation}
\map(x)=\frac{3}{2}-2x+3[x]
\end{equation}
which is equivalent to a random walk where the particle moves with equal
probability
from one interval $I_n=(n-1/2,n+1/2]$ to the left, $I_{n-1}$ or to the right,
$I_{n+1}$.  The dispersion relation $s(k)$ follows directly from the
solution, Eq.~(\ref{e:sol}),
\begin{equation}
\rho(n,t)=\exp(st+ikn) 
\end{equation}
After one iteration,
\begin{eqnarray}
\rho(n,1)&=&\frac{1}{2}[\exp(\i{}k(n-1))+\exp(\i{}k(n+1))]\\
&=&\cos{k}\exp(\i{}kn)
\end{eqnarray}
leading to
\begin{equation}
s(k)=\ln\cos{k}
\end{equation}
which has a power series around $k=0$ with a radius of convergence equal
to $\pi/2$.

\begin{thebibliography}{99}
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