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\author{Francis J. Alexander\thanks{Supported in part by a Rutgers University
Excellence Fellowship.  Current address: Center for Nonlinear Studies,
	MS-B258, Los Alamos National Lab, Los Alamos, NM 87545},
	Zheming Cheng\thanks{Current address: Program Development Corp.,
300~Hamilton Ave., Suite~409, White Plains, NY 10601},\\
	Steven A. Janowsky\thanks{Supported in part by National
Science Foundation Mathematical Sciences Postdoctoral Research 	Fellowship DMS
90-07206}\hspace{.25em} and
	Joel L. Lebowitz\\[5pt]
	Departments of Mathematics and Physics\\
	Rutgers University\\New Brunswick, NJ 08903}
\title{Shock Fluctuations in the Two-Dimensional Asymmetric Simple
	Exclusion Process\thanks{Supported in part by National Science
	Foundation grant DMR 89-18903.}}
\date{February 26, 1992}
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\begin{abstract}

We study via computer simulations (using various serial and parallel updating
techniques) the time evolution of shocks, particularly the shock width
$\sigma(t)$, in several versions of the two-dimensional asymmetric simple
exclusion process (ASEP).  The basic dynamics of this process consists of
particles jumping independently to empty neighboring lattice sites with rates
$p_{\rm up} = p_{\rm down} = p_\perp$, $p_{\rm left} < p_{\rm right}$.  If the
system is initially divided into two regions with densities $\rho_{\rm left} <
\rho_{\rm right}$, the boundary between the two regions corresponds to a shock
front.  Macroscopically the shock remains sharp and moves with a constant
velocity $v_{\rm shock} = (p_{\rm right} - p_{\rm left})(1 - \rho_{\rm left} -
\rho_{\rm right})$.  We find that microscopic fluctuations cause $\sigma$ to
grow as $t^\beta$, $\beta\approx 1/4$. This is consistent with theoretical
expectations.

We also study the nonequilibrium stationary states of the ASEP on a periodic
lattice, where we break translation invariance by reducing the jump rates
across the bonds between two neighboring columns of the system by a factor
$r$.  We find that for fixed overall density $\rho_{\rm avg}$ and reduction
factor $r$ sufficiently small (depending on $\rho_{\rm avg}$ and the jump
rates) the system segregates into two regions with densities $\rho_1$ and
$\rho_2=1-\rho_1$, where these densities do not depend on the overall density
$\rho_{\rm avg}$.  The boundary between the two regions is again
macroscopically sharp.  We examine the shock width and the variance in the
shock position in the stationary state, paying particular attention to the
scaling of these quantities with system size.  This scaling behavior shows
many of the same features as the time-dependent scaling discussed above,
providing an alternate determination of the result $\beta\approx 1/4$.\\[3pt]

\noindent
KEY WORDS: Stochastic particle systems; shock waves; surface growth
\end{abstract}
\clearpage
\section{Introduction}
Macroscopic equations that describe fluid flow result from an averaging over
the rapidly fluctuating microscopic motion of a large number of
molecules~\cite{LPS}.  This procedure yields deterministic (typically
nonlinear) partial differential equations for the conserved quantities
(momentum, energy, mass density)---which vary slowly on the microscopic scale.
The familiar hydrodynamic equations of motion (Navier-Stokes and Euler) which
describe the dynamics of fluids are of this form.

These equations describe quite well what happens to fluids on a large scale
when the flow is smooth.  Problems arise when gradients in the hydrodynamic
variables become very large and the assumptions made in the derivation of the
hydrodynamic equations break down---for example, where discontinuities
(shocks) in macroscopic variables (such as the density) appear. We would like
to gain a better understanding of these situations both at the macroscopic and
microscopic levels.

\subsection{The Burgers Equation}
One of the simplest examples of a nonlinear macroscopic equation with a single
conserved quantity is the Burgers equation,
\begin{equation}\label{Burger1}
\frac{\partial {\bf u}}{\partial t} = -{\bf u}\cdot\! \nabla{\bf u} +
\nu\Delta {\bf u},
\end{equation}
originally proposed to study turbulence with ${\bf u}$ representing a
velocity field~\cite{Burgers,Tatsumi}; in our considerations, ${\bf u}$
will represent a scalar density field.  What makes the Burgers equation
interesting is that initially smooth density profiles can evolve after a
finite time into traveling wave fronts.  The transitions between low and
high density regions in these fronts occur in very narrow spatial
regions---regions with width proportional to $\sqrt{\nu}$.  In the limit
$\nu\rightarrow 0$ the profile becomes discontinuous and we say that
shocks form.  If $\nu$ is finite but small ({\em i.e.}\ microscopic) we
may still refer to the narrow transition region as a shock.  These
shocks as given by the Burgers equation move with a deterministic
velocity.  We are interested in what happens to shocks when viewed on
the microscopic level.

The non-viscous Burgers equation, with $\nu=0$, can be derived
rigorously from a number of computationally efficient particle
models~\cite{LPS,Spohn}.  The field ${\bf u}$ represents the space and time
rescaled particle configurations, {\em i.e.}\ the hydrodynamic limit
with Euler scaling in which time and space are scaled by a fixed ratio.
(Certain special cases with other scalings can result in limits with
$\nu> 0$.)  Here we use the asymmetric simple exclusion process.  On a
lattice, particles hop to unoccupied neighboring sites with a drift
(asymmetry) along one of the lattice directions. The hard-core exclusion
(only one particle per site) is the source of the nonlinearity and
provides for some interesting phenomena.  Even on the microscopic scale
shocks form. At this level, instead of traveling with some definite
deterministic velocity, there is a a fluctuating velocity, due to the
initial conditions and the dynamics, that must be superposed upon the
macroscopic one.  Therefore, the location of the shock will deviate from
the location given by the Burgers equation by some fluctuating
quantity~\cite{DKPS}.

\subsection{The ASEP}
The ASEP is a continuous time stochastic process in which particles
occupy sites of the lattice ${\Bbb Z}^d$ and move according to simple
rules.  Configurations in this process are denoted by $\eta \in
\{ 0,1 \}^{{\Bbb Z}^d}$, where the individual site occupation variable $\eta
(\vec{\bf r}) = 1$ if $\vec{\bf r}$ is occupied, and $0$ if unoccupied.
An exclusion rule prevents more than one particle from simultaneously
inhabiting the same site.  Independently and randomly, each particle
waits for an exponentially distributed time with mean 1 and attempts to
jump to a neighboring site.  If the target site is unoccupied, then the
jump succeeds; if not, then it fails.  An asymmetry enhances jump
attempts in one direction and induces a net particle current. In the
one-dimensional model, a particle attempts to jump to the right with
rate $p_{\rm right}$ and to the left with rate $p_{\rm left}$, $p_{\rm
right}>p_{\rm left}$, $p_{\rm right}+p_{\rm left} = 1$. In higher
dimensions the asymmetry along the $x$ axis persists, but the jump rates
along both directions of the perpendicular axes are equal (symmetric)
and given by $p_\perp$ such that $p_{\rm left}+p_{\rm right}+2(d-1)
p_\perp =1$, recalling that $d$ is the dimension.  Shocks will now
correspond to $(d-1)$-dimensional fronts which will fluctuate in space
and time.

\subsection{Shock Growth $=$ Surface Growth; KPZ approach}
We may also interpret the shock evolution in the $d$-dimensional ASEP as a
model of $(d-1)$-dimensional surface growth where holes are driven to the left
and stick to the ``hole substrate,'' with the surface of this substrate
traveling to the right.  This interpretation is particularly useful given the
recent interest in surface deposition models and the kinetic roughening of
surfaces~\cite{KS}.  The model we describe here is complicated for analysis as
well as for simulation since we are interested in the statistics of the shock
(= surface) and its dependence on a priori {\em unknown} properties of the
ASEP.  In other surface problems one usually assumes that the particles (or
holes) hitting the surface are uncorrelated or have {\em known} correlations.
One can then construct a stochastic partial differential equation which models
the surface dynamics.

Typically the approach to such problems in surface roughening models is to
introduce a Langevin type equation which governs the local interface position
$h(\vec{\bf r},t)$, the usual choice being the Kardar-Parisi-Zhang (KPZ)
equation~\cite{KPZ}:
\begin{equation}\label{KPZ1}
\frac{\partial h}{\partial t} = v_0 + \nu\Delta h + \frac{\lambda}{2}
	(\nabla h)^2 + \zeta(\vec{\bf r},t).
\end{equation} 
The random noise term $\zeta(\vec{\bf r},t)$ is such that
\begin{equation}
\langle \zeta (\vec{\bf r},t) \rangle = 0,
\end{equation}
but otherwise is chosen according to the specific nature of the model being
studied~\cite{Medina}.

The KPZ equation provides a phenomenologically based description of a growing
surface. Each term represents a different aspect of the growth process: the
constant $v_0$ is the growth rate for a completely flat interface.  The
Laplacian term accounts for surface restructuring as particles diffuse on the
surface and move to fill gaps, while the nonlinear gradient term describes an
inclination dependent growth rate. The noise term represents growth due to
fluctuations at the microscopic level; without this term the KPZ equation
(\ref{KPZ1}) can be related to the Burgers equation (\ref{Burger1}) via the
transformation ${\bf u} = -\nabla h$.  Higher order terms are not included in
the KPZ equation because they are irrelevant in the renormalization group
sense; without loss of generality we change variables $h\rightarrow h-v_0t$
and take $v_0=0$.

\subsubsection{Linearized KPZ}
We believe that the one-dimensional KPZ equation, with the nonlinear
term absent ($\lambda=0$), describes the interface behavior of the
two-dimensional ASEP. (This reduces to a noisy, linear diffusion
equation.)  We argue this on several grounds.  In the first place, the
nonlinear term arises from the dependence of surface growth on the local
orientation.  In the ASEP, however, the shock velocity is independent of
surface orientation and so averaging over orientations results in
cancellation of the nonlinear term~\cite{Spohn}.

Secondly, in the case of weak asymmetry, where $p_\perp$ is fixed, $p_{\rm
right} - p_{\rm left} = \epsilon$, space is rescaled by $\epsilon$ and time is
rescaled by $\epsilon^2$, it is known rigorously that the equation describing
the interface fluctuations in the limit $\epsilon\rightarrow0$ is indeed just
the linear KPZ equation~\cite{Ravi}
\begin{equation}\label{eq:ravishan}
\frac{\partial h(r,t)}{\partial t} = \nu\Delta h(r,t) + \zeta(r,t),
\end{equation}
with $\zeta(r,t)$ Gaussian white noise: $\langle \zeta (r,t) \rangle = 0$ and
\begin{equation}
\langle \zeta(r,t) \zeta (r',t') \rangle = K\delta(r-r')\delta(t-t'),
\end{equation}
where $K$ depends on the asymptotic densities on either side of the
interface.  One can solve (\ref{eq:ravishan}) exactly by using Fourier
transforms; the result for the shock width starting with an initially
flat interface in a strip of width $W$ is~\cite{Family}
\begin{equation}\label{scalingform}
\sigma(t) \sim \left\{
\begin{array}{l@{\quad}l}
t^\beta, &t\ll W^{\alpha/\beta},\\[3pt]
W^\alpha, &t\gg W^{\alpha/\beta};
\end{array}\right.
\end{equation}
where
\begin{equation}
\sigma^2(t) \equiv \frac1W\int dr\,\left(h(r,t)-\bar{h}(t)\right)^2,
\qquad
\bar{h}(t)  \equiv \frac1W\int dr\, h(r,t),
\end{equation}
$W$ is the width of the system (length of the interface) and the scaling
exponents are
\begin{equation}\label{alphabeta}
\alpha = 1/2, \qquad \beta = 1/4.
\end{equation}
We shall see later that these exponents are indeed consistent with the results
of our simulations---although the approach to scaling behavior can take a
very long time.

\subsubsection{Logarithmic Correction}
In fact there is a defect in the above analysis; the two-dimensional ASEP is
known to exhibit superdiffusive behavior~\cite{SvB} and thus one should not
expect a linear diffusion equation to accurately model it.  Analysis similar
to that of \cite{vB} indicates that the correct behavior is~\cite{vB:private}
\begin{equation}
\sigma(t) \sim t^{1/4} (\log t)^{1/3}\quad{\rm for}\quad
	t^{1/4} (\log t)^{1/3} \ll W^{1/2},
\end{equation}
while for $t^{1/4} (\log t)^{1/3} \gg W^{1/2}$ the result remains $W^{1/2}$.
This correction is sufficiently small that it will be unobservable in any
numerical simulation likely to be done before the next century, and
(\ref{eq:ravishan}) remains a reasonable approximation.
\section{Models}
We studied two basic classes of systems undergoing ASEP dynamics:  The
time-dependent behavior of an effectively infinite system, and the stationary
states of a finite periodic model.

\subsection{Two-Dimensional ASEP---Time-Dependent Model}
The specific system we considered for our time dependent studies consisted of
a $W\times L$ lattice with periodic boundary conditions in the vertical
(perpendicular to the field) direction.  Parallel to the field no effort was
made to apply suitable boundary conditions as simulations were always ended
prior to the arrival of information about the boundaries.  In most cases we
chose the jumps to be totally asymmetric in the parallel direction, {\em
i.e.}\ we took $p_{\rm left} = 0$.

\subsubsection*{Simulation of a Continuous Time Process}
Since each lattice site has an independent exponentially distributed
waiting time for attempting a jump, the probability of two sites
attempting to jump at the same time is zero.  Thus we can simulate the
ASEP dynamics in discrete time by choosing at each time step one site
where we attempt to schedule a jump.  All sites in $[1,W]\times [1,L-1]$
are chosen with equal probability.  If the chosen site is occupied, we
pick a direction $\in \{\rm up,\ down,\ right,\ left\}$ with
probabilities $p_\perp$, $p_\perp$, $p_{\rm right}$ and $p_{\rm left}$,
respectively.  If the nearest neighbor site in the chosen direction is
unoccupied, the particle jumps to that site; if it is occupied, the
particle does not move for the given time step.

\subsection{Stationary Model}
Here we consider ASEP dynamics on an $W\times L$ torus.  We break the
translation invariance of this periodic system by inserting a blockage into
the system between columns $L$ and 1, which reduces the probability of a
particle traveling between those two columns---a set of ``slow bonds'' which
act as a traffic jam for the particles.  In the language of driven diffusive
systems, the introduction of ``slow bonds'' is similar to altering the driving
field at this one column~\cite{AL}.  This blockage is analogous to a
restriction in a pipe through which fluid flows; the corresponding model also
provides an example of the dramatic global effects caused by a local
perturbation in conservative systems which do not satisfy detailed
balance~\cite{GLMS}.  It also provides an alternate method for observing the
same behavior as in the time-dependent model.  The one dimensional version of
this system was examined in~\cite{JL}.

More specifically, we reduce the jump rates between columns $L$ and 1 by a
factor $r$, $0\le r\le1$.  For $r=1$ the model is translation invariant.  For
$r=0$ the model is fully blocked; the stationary state has density one behind
the blockage and density zero in front of it; there is no current flowing
through the system.  For $0<r<1$ the model has nontrivial behavior, with the
stationary state satisfying the requirement that the current through any
column of bonds must be independent of the column location.

In this model with a blockage we are not particularly interested in the time
evolution of the shock but instead study the properties of the shock in the
stationary state, and how those properties depend on system size.

\subsubsection{Simulation of ASEP dynamics with blockage}
The simulation of the ASEP dynamics in the periodic system with a blockage is
basically the same as that in the ordinary model.  All sites in $[1,W]\times
[1,L]$ are chosen with equal probability.  If the chosen site is occupied, we
pick a direction $\in \{\rm up,\ down,\ right,\ left \}$ with probabilities
$p_\perp$, $p_\perp$, $p_{\rm right}$ and $p_{\rm left}$, respectively.  We
now must distinguish between the blockage and non-blockage columns.  If the
chosen site is in column $1$ or $L$ and the chosen direction is left or right,
respectively, then the particle is attempting to jump the blockage and the
attempt is completed (assuming the destination site is vacant) with
probability $r$.  Other jumps take place in the same fashion as above but are
completed (assuming the destination site is vacant) with probability $1$.

Of course we must allow an adequate time for this system to evolve so that we
can be confident that the properties we observe are indicative of the
stationary state.  The necessary time is determined simply by observing the
time evolution of shock width, position, etc., and waiting until they reach
asymptotic values.  For all system sizes except for the very largest a
substantial ``safety'' factor was also included.

\subsubsection{Parallel and semi-parallel models}
In order to examine very large system sizes, we devised various modifications
of the ASEP that would permit effective utilization of vector and massively
parallel supercomputers.  Unfortunately attempting to perform updates on all
sites at once is impossible as this leads to two or more particles vying for
the same site.  Sublattice updating is a traditional technique; however, it
may introduce spurious correlations into the model.  We did have limited
success with the following variations on sublattice updating:
\begin{itemize}
\item 
We were able to consider 32 different realizations of the same size system at
once by treating each bit of a 32 bit word as an individual system.  Although
the choices for which sites to update were the same for all the systems, the
choice of directions in each system were independent.  We were thus able to
improve the statistics over that which would have otherwise been available;
the 32 (correlated) systems appeared to provide data equivalent to that which
would have been given by approximately 10 truly independent systems.  Most of
our data sampling (for both time-dependent and stationary models) made use of
this procedure.
\item
We considered a partial sublattice updating where 32 sites in each column
(spaced $W/32$ apart) were updated simultaneously.  {\em Qualitatively} the
model remained the same, but significant correlations were introduced at
vertical distances of $W/32$, $W/16$, etc.  None of our quantitative data
makes use of this procedure.
\item
In an attempt to reduce the correlations introduced by a fixed sublattice
updating, we decided to examine the possibility of using a sublattice which at
each step had a random origin and spacing.  First fix the system width $W$ as
$W = a w_1 w_2 w_3 w_4$, where all the factors $\{w_i\}$ are relatively prime.
Thus we have four possible sublattice spacings available.  Updating proceeds
by first choosing a sublattice spacing (stride) $w_i$ with probability
$w_i\big/\sum w_i$, then choosing a site, and finally updating that site and
all (vertical) translates of that site by multiples of $w_i$.  For $W =
9\times10\times11\times13 = 12870$ ($a=1$) this allowed us to update between
990 and 1430 sites simultaneously using vectorized instructions on a Cray YMP.
This large speedup allowed us to examine systems that were much larger than we
could have otherwise considered.
\end{itemize}
\section{Time-Dependent Results}
\subsection{Shock identity}
When studying systems with shocks, we can either wait for shocks to
develop from smooth initial data, or else we can impose them on the
initial data by hand.  We opted for the latter approach to facilitate
computer simulation.  In either case we must be able to identify the
location of the shock.  We begin by giving a particular operational
definition of what is meant by a shock at the microscopic level in one
dimension.

Consider the integer lattice ${\Bbb Z}$. On each site $x \geq L/2$ we
independently place a particle with probability $\rho_0$.  All sites $x
< L/2$ are empty.  Asymmetry in the jump rates will drive particles to
the right.  At time $t=0$ we label the leftmost particle in the system
the ``first'' particle.  As the system evolves, there is no reordering
of particles since jumps are to nearest neighbor sites only (no
crossing).  Therefore, this ``first particle'' label is permanently
attached to the same particle.  If we view the process from the first
particle, then to the left the density is always zero while to the
right, the density of particles approaches $\rho_0$ at an exponentially
(in space) fast rate, where the decay length is $1/\log(p_{\rm
right}/p_{\rm left})$~\cite{Ferrari}.  Therefore, we have an abrupt
change in particle density (over a few sites)---a shock.  Accordingly,
we define the shock to be located at the site containing the first
particle~\cite{Wick}, and we know that even on the microscopic scale the
shock remains sharp.  Note that this picture breaks down when $p_{\rm
right} = p_{\rm left}$; in this case the behavior is purely diffusive
and the average distance between the first and second particles grows
like $t^{1/2}$.

In two dimensions this picture needs modification.  The asymmetry drives
particles to the right while allowing for diffusion in the vertical direction.
What results is a right-moving, rough interface across which there is an
abrupt change in average particle density.  We also refer to this fluctuating
interface as a shock.  Now, however, the first particle in each row is no
longer a permanent label as a result of perpendicular diffusion.  But the
basic principle is still valid, and the most natural and computationally
efficient way to characterize the shock (when the density is zero to the left)
is to look at the distribution of first particles in different rows (different
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\caption{Example of a particle configuration and the resulting shock
position $h(y,t)$.}\label{shockdef}
\end{center}\end{figure}
$y$ values)---see Figure~\ref{shockdef}.  The instantaneous width is then
defined by fluctuations in these locations $h(y,t)$ about their average:
\begin{equation}
\sigma (t) \equiv \left(\frac1W \sum_{y=1}^W (h(y,t)-{\bar h}(t))^2
	\right)^{1/2},
\end{equation}
where ${\bar h}(t)$ is the average location of the first particles,
\begin{equation}
{\bar h}(t) = \frac1W \sum_{y=1}^W h(y,t).
\end{equation}

\subsection{Results}
With the exception of some limited runs to check the gross behavior with
varied parameters, all of our time-dependent simulations were carried
out with an initial particle density of $\rho_0 = .5$ and with jump
rates $p_{\rm right}=.75$, $p_{\rm left}=0$ and $p_\perp =.125$.  The
field drove particles to the right and holes to the left.  Since
particles eventually moved away from the left wall (and were not
replenished) and holes moved away from the right wall (and were not
replenished), we were left with particle substrate on the right of the
system moving to the left, and the hole substrate on the left, moving to
the right.  Particle-hole symmetry (at $\rho_0 = .5$) implies they
should have the same behavior.  With this approach we were able to study
the statistics of two nearly independent shocks at once.

In Figure~\ref{width}
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\caption{Shock width {\em vs}.\ time for $W=256(\diamond)$ and $W=1024(+)$.}
\label{width}
\end{center}\end{figure}
we compare the width of the shock interface for system widths $W=256$ and
$W=1024$; in each case the data is averaged over 75 independent runs.  The
available run time was not long enough for the shock width to saturate,
although the effect of the finite system width is apparent as we see the
$W=256$ and $W=1024$ curves separate for $t>400$.  We expect that for $W$
large enough there will an ``asymptotic'' time regime for the growth of
$\sigma(t)$ before it saturates, with behavior given by equations
(\ref{scalingform}--\ref{alphabeta}).

The results in Figure~\ref{width} indicate that the growth of fluctuations has
not reached the asymptotic (in time) regime. If we think of the exponent
$\beta$ characterizing this growth as a time-dependent quantity, then $\beta
\to .17$ for the longest times that we were able to observe (in the $W=1024$
system), and is apparently still increasing.  It may be that the long time
behavior will still be consistent with the predictions of the linear KPZ
equation given in (\ref{scalingform}--\ref{alphabeta}), but that the approach
to this asymptotic behavior is extremely slow.

There appear to be two reasons for this slow convergence.  First, the shock
width at $t=0$ does not strictly vanish.  There are fluctuations inherent to
the initializing process of setting down particles randomly with probability
$\rho_0$.  This accounts for a ``zero point fluctuation'' of a few lattice
spacings.  We can eliminate this effect by starting with an artificially
perfectly flat interface.  The true asymptotic behavior will not be affected
by this.

This, however, is not the main problem---the main difficulty is that there are
natural, short wavelength fluctuations which result from the dynamics and form
at an early time.  Letting $\langle\cdot\rangle$ represent an average over the
random dynamics, {\em i.e.}\ a sampling average, we see that there is a
natural separation distance $\langle |(h(y,t) - h(y+1,t))| \rangle$ between
the shock locations in neighboring rows, which builds up rapidly and then
saturates.  If the perpendicular jump rate is small, then this length can be
quite large.  In this case, $p_\perp \ll p_{\rm left}+ p_{\rm right}$,
neighboring rows are essentially noninteracting for long periods of time.  The
first particles in each row then execute a random walk relative to each other
and can become quite distant.  As their distance increases, it becomes more
probable to have a transition from one row to another which will reduce the
distance between the first particles in the two rows.  These perpendicular
transitions prevent the first particles in neighboring rows from moving
arbitrarily far from each other (confinement), and thus $\langle |h(y,t) -
h(y+1,t)| \rangle$ approaches an asymptotic value.  For the jump rates we
used, we found that this natural extension was on the order of a few lattice
spacings, just enough to mask the overall shock broadening which was typically
less than 8 lattice spacings.

This latter problem could not be eliminated with a simple adjustment of the
initial conditions or jump rates. Therefore, we looked at other manifestations
of shock broadening which in the asymptotic time regime should be equivalent
to the shock width defined above.  Defining
\begin{equation}
G(m,t) = \langle h(y,t) h(y+m,t) \rangle - \langle h(y,t)\rangle^2,
\end{equation}
we see that $G(0,t) = \sigma^2(t)$.  For $m=1$ or $2$, $G(m,t)$ represents the nearest
(row) neighbor, $m=1$, and next nearest neighbor, $m=2$, truncated first
particle correlation functions.  These functions have none of the $t=0$
fluctuations mentioned above and will be affected to a much lesser extent by
the short-wavelength fluctuations.  In fact, $G(1)$ differs from $G(0)$ by
exactly the nearest-neighbor fluctuations:
\begin{equation}
G(0,t) - G(1,t) = \frac12 \left\langle (h(y,t) - h(y+1,t))^2 \right\rangle.
\end{equation}

In Figure~\ref{g1k} we show these functions and include the width ($m=0$) for
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\caption{Time dependence of $G(m,t)$ for $W=1024$.
$m=0 (\Diamond)$, $m=1(+)$, $m=2(\Box )$.  Line has slope 1/4.}\label{g1k}
\end{center}\end{figure}
comparison. Note that for $m= 1$ and $m=2$ that the ``effective growth
exponent,'' characterized by the slope of the tangent to this curve, is
actually greater than $.25$ and is decreasing.
For the width $m=0$ it is increasing.  

As well as studying $G(m,t)$ for fixed $m$ and varying $t$, we considered the
complementary case of fixed $t$ and varying $m$.  The results appear in
Figure~\ref{cortim1k}.
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\caption{Correlations along the shock at times 36 ($\diamond $),
133(+), and 662($\Box $).}\label{cortim1k}
\end{center}\end{figure}
The self-correlation $(m=0)$ is the square of the shock width.

As $G(m,t)$ decays exponentially in $m$ (for $m$ not too large), we can
define a correlation length along the shock, $\xi_{\parallel}(t)$, determined
by
\begin{equation}
G(m,t) \sim \exp [-m\xi_{\parallel}(t)]
\end{equation}
for small $m$.  In Figure~\ref{corlen1k}
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\caption{Growth of correlation length. $W=1024$.  Line has slope
1/2.}\label{corlen1k}
\end{center}\end{figure}
we show time time evolution of the correlation length. The asymptotic behavior
corresponds to a diffusive growth: $\xi_{\parallel} \sim t^{1/2}$; this is in
agreement with the evolution determined by equation~(\ref{eq:ravishan}).  Note
that the initial behavior is much more rapid as the short wavelength
fluctuations develop.

\subsubsection{Linearized surface equation}
The short wavelength fluctuations in the ASEP are of the same order of
magnitude as the shock width and therefore mask its growth.  We checked to see
if this behavior persisted in a discretized version of the linear (1d) KPZ
equation, where we know the asymptotics exactly.  Consider the discretization
of (\ref{eq:ravishan}):
\begin{equation} \label{eq:linapp}
h(y,t+1)-h(y,t) = D [ h(y-1,t) -2 h(y,t) + h(y+1,t) ] +  \gamma \zeta (y,t).
\end{equation}
The noise term $\zeta$ is Gaussian  with covariance
\begin{equation}
\langle \zeta(y,t) \zeta(y',t')\rangle = \delta(y-y')\delta(t-t').
\label{eq:linnoise}
\end{equation}

We solved for time dependence of the shock width and $m=1$ correlations by
simulating the process governed by (\ref{eq:linapp}) and (\ref{eq:linnoise}).
The size of the system (corresponding to the length of the interface which is
the dimension of the ASEP perpendicular to the field, {\em i.e.}\ $W$) was
10000 sites, and we averaged over 50 independent samples.  The diffusion
constant is $D=0.5$, and the noise amplitude is $\gamma =0.5$.  These
parameters were chosen to compare with the results of the ASEP simulations.
The initial conditions on $h(y,t)$ were equivalent to what would result from
the initialization process outlined above with a density $\rho_0 = 0.5$.

What we found was qualitatively very similar to the simulations of the ASEP,
and is presented in Figure~\ref{phenom}.
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\end{picture}
\caption{$G(m,t)$ for a discretized linear KPZ equation,
$m=0 (\diamond )$ and $m=1(+)$.  Solid line has slope 1/4.}\label{phenom}
\end{center}\end{figure}
Namely, we observed an initial regime of fast growth in the nearest neighbor
correlation function $G(1,t)$ and an eventual tapering off of the effective
time-dependent exponent.  Meanwhile, for $G(0,t)$ (shock width squared) we
found that the effective growth rate {\em increased} with time consistent with
an exponent $\beta = 1/4$.

\subsubsection{Compatibility with Existing Theory}
We have presented computer simulations of the two dimensional asymmetric
simple exclusion model.  The growth of shock width in the ASEP is consistent
with the current theory, but the natural width inherent to the dynamics make
it difficult strengthen this claim.  Since similar behavior is also observed
in simulations of the discretized interface equation, we are, however,
supportive of the linear theory.
\section{Stationary Results}
When we study the stationary state of our periodic model with a blockage, we
have an advantage over the time-dependent case in that we need not worry
about initial conditions---only the total particle number is relevant.
Thus we need not worry about the best way to produce a shock, {\em e.g.}\
whether we should use a checkerboard pattern or simple product measure.
However, this lessens the freedom we have in determining the type of shock
that results.  Since a shock in the stationary state must have no net drift,
the densities on either side must be symmetric with respect to density 1/2
(the shock velocity, which must be zero in the stationary state, is given by
\(v_{\rm shock} = 1 - \rho_{\rm left} - \rho_{\rm right}\)).  Thus the simple
technique of identifying the shock by the first particle in each row, valid
for $\rho_{\rm left} = 0$, will not be effective.

\subsection{Shock identity}
The difficulty of identifying the shock also appeared in the
one-dimensional case~\cite{JL}, where a so-called second class
particle~\cite{BCFG} was used to track the shock.  The second class
particle is an extra particle added to the system, which is treated as a
hole in exchanges with particles and as a particle in exchanges with
holes; this does not change the dynamics of the original particles.
When the second class particle is in a high density region (of ordinary
particles), it is forced to the left by the particles jumping (to the
right) and landing on it; when it is in a low density region it moves to
the right.

In two dimensions one can also add second class particles to the system, but
as they can diffuse from row to row we no longer have a single second class
particle associated with the shock position in a given row.  One possibility is
to add a large number of second class particles to the system and determine
their distribution, unfortunately a computationally intensive procedure.

The second class particle is actually a much more powerful tool than we need
for determining the shock position, so we discard it in favor of something
simpler.  The motion of the second class particle is basically that of a
biased random walk with a drift towards the shock position; instead of second
class particles we introduce shadow particles whose dynamics is {\em exactly}
that of a biased random walk with a drift towards the shock position.  These
shadow particles do not affect the motion of the ordinary particles but move
in a ``potential'' determined by the ASEP configuration.  Specifically, after
each sweep of Monte-Carlo updates, each shadow particle (of which there is one
per row) moves according to the following rule:
\begin{equation}
h(y,t+1) = \left\{
\begin{array}{ll}
h(y,t)&\mbox{with probability $1/4$},\\[3pt]
h(y,t) - s(h(y,t),y; t) &\mbox{with probability $3/4$},
\end{array}\right.
\end{equation}
where $s(x,y; t) = 1$ if there is a particle at site $(x,y)$ at time $t$, and
$s(x,y; t) = -1$ if there is no particle there.  Thus the shadow particles
move to the left in regions of high density and to the right in regions of low
density, driving them towards the shock where these regions meet.  The
probabilities 1/4 and 3/4 were chosen simply for convenience; other similar
rules for the shadow particle evolution were tried but did not yield
significantly different behavior.

Note that we continue to label the location of the surface by $h(y,t)$, even
though our definition of this location has changed from the time-dependent
case, where we made use of the first particle position, since both definitions
represent the same physical idea.
Ideally, one would like to allow the ``shadow'' random walk to evolve for a
long time for each given ASEP configuration; under such conditions it is clear
that the shadow particles will accurately identify the shock position,
provided a shock does in fact exist.  The ratio of one shadow update per ASEP
Monte Carlo step is a compromise between the need to accurately identify the
shock postion and the desire to reduce the computational load of tracking it.

\subsubsection*{Quantities studied}
We were interested in studying both the shock profile as well as its
fluctuations.  To this end we computed the following quantities from the
sampled shock positions $h(y,t)$, where $\langle\cdot\rangle$ represents a
sampling ({\em i.e.}\ time) average:
\begin{itemize}
\item the average shock position
\(\displaystyle \langle\bar{h}\rangle = \left\langle \frac1W \sum_{y=1}^W
	h(y,t) \right\rangle
	= \frac1W \sum_{y=1}^W \langle h(y,t) \rangle
\),
\item the variance in the shock position
\(\displaystyle \langle\delta^2\rangle = \left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle
\),
\item the average shock width
\(\displaystyle \langle\sigma\rangle = \left\langle\left(\frac1W \sum_{y=1}^W
	(h(y,t)-{\bar h}(t))^2 \right)^{1/2}\right\rangle
\),
\item the rms shock width
\(\displaystyle \langle\sigma^2\rangle^{1/2} = \left\langle\left(\frac1W
	\sum_{y=1}^W (h(y,t)-{\bar h}(t))^2 \right)\right\rangle^{1/2}
\),
\item the truncated height-height correlation function
\(\displaystyle \langle G(m)\rangle = \left\langle h(0,t)h(m,t) - \bar{h}^2(t)
	\right\rangle =
\left\langle \frac1W \sum_{y=1}^W \left(h(y,t) h(y+m,t) - \bar{h}^2(t)\right)
\right\rangle\).
\end{itemize}

\subsection{Results}
We attempted to determine the behavior of the shock width as we varied the
size of our system.  We found that if either the system size $L$ or the system
width $W$ was taken very large, the shock width approached an asymptotic value.
Thus we were able to reduce our two parameter system to a single parameter, by
considering the regimes $L\gg W$ and $L\ll W$.

For $L\gg W$, the finite width of the system prevents the shock from growing
indefinitely.  This behavior should correspond to growth saturation in the
time-dependent case.  We present our data for a system with $\rho_{\rm avg} =
0.5$, $p_{\rm right} = 0.75$, $p_\perp = .125$ and $r=0.25$ in
Figure~\ref{width-vs-width}. 
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\caption{Shock width {\em vs.}\ system width $W$ for $L\gg W$.  Solid line
has slope 0.5.}
\label{width-vs-width}
\end{figure}
The error bars (three symbols are plotted for each measurement: the actual
value and that value shifted up or down by the error bound) represent
statistical error based on the approximate number of independent samples
selected from the steady state in each system.  To fully saturate the shock
width required that we consider a system length of $3200$ for a system width
of $180$; typically we had $L\gtrsim 16W$.

We only plot the data for the rms shock width; the average shock width behaved
similarly.  Along with $\langle\sigma^2\rangle^{1/2}$ we plot the nearest
neighbor correlation $\langle G(1)\rangle^{1/2}$ which should have the same
asymptotic behavior as the shock width.  The difference between the two is
due to the short-wavelength fluctuations, just as in the time-dependent case.
It is clear that we are just beginning to access the asymptotic behavior; our
results are consistent with $\alpha=0.5$ but cannot be considered conclusive.

For $L\ll W$, the finite width of the system is irrelevant to the width of the
shock.  In this case the system length $L$ determines the shock width.  We
present our data for a system with $\rho_{\rm avg} = 0.5$, $p_{\rm right} =
0.75$, $p_\perp = .125$ and $r=0.125$ in Figure~\ref{width-vs-length}.
\begin{figure}
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\end{center}
\caption{Shock width {\em vs.}\ system length $L$ for $L\ll W$.  Solid line
has slope 0.25.}
\label{width-vs-length}
\end{figure}
To reach the asymptotic shock width for the longest system, $L=360$, required
a system width of $720$; typically we had $W\gtrsim 2L$.

If one associates length with time, we should expect to see the same exponent
$\beta$ as we saw in the time-dependent case, {\em i.e.}
\begin{equation}
\langle\sigma^2\rangle^{1/2} \sim L^\beta \sim t^\beta \sim \sigma(t).
\end{equation}
Indeed, our results are consistent with $\beta=0.25$.

In an attempt to examine systems larger than $720\times 360$, we used the
random-sublattice updating technique described earlier.  With a system width
of $12870$ we were able to examine lengths up to $800$ and be quite sure that
the finite width of the system was not affecting the shock width.
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\caption{Shock width {\em vs.}\ system length $L$ for the semi-parallel update
model.  Solid line has slope 0.25.}
\label{width-vs-length-cray}
\end{figure}
We present the data from our semi-parallel model in
Figure~\ref{width-vs-length-cray}.  The parameters are the same as for
Figure~\ref{width-vs-length}, and the data is in fact very similar.

We should be somewhat careful in interpreting the results from our
semi-parallel model, because there are differences from the serial model.  In
Figure~\ref{hhcor} we plot the height-height correlation function $G$ for the
ASEP with length $L=64$ and width $W=256$, along with the semi-parallel model
with the same length.
\begin{figure}
\begin{center}
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\caption{Height-height correlation function.}
\label{hhcor}
\end{figure}
Although both systems have an initial exponential decay, the serial model has a
significantly larger ``dip'' of negatively correlated surface heights than
that which we see in the semi-parallel model.

The basic structure of the shock seems to be fairly impervious to the details
of the model.  The scaling behavior of its intrinsic width, for example, does
not depend upon the overall density $\rho_{\rm avg}$.  On the other hand, the
location $\bar{h}$ and the fluctuations of that location {\em do} depend upon
$\rho_{\rm avg}$, specifically, whether or not $\rho_{\rm avg}=0.5$.  This is
the same behavior observed in the one-dimensional model, where the shock
fluctuations were reduced when there was particle-hole symmetry~\cite{JL}.

For a system where $L\gg W$, we would expect the system to be effectively
one-dimensional, and we recover the one-dimensional results~\cite{JL}.  For
$\rho_{\rm avg} \ne 0.5$, we expect $\left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle^{1/2}$ to scale like $L^{1/2}$.  In
Figure~\ref{fluc5625} we plot
\begin{figure}
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\caption{Rescaled shock fluctuation: $W^{1/2}\left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle^{1/2}$ {\em vs.}\ system length $L$.
Average density different from 1/2.  The dotted line has slope 1/2.}
\label{fluc5625}
\end{figure}
the shock fluctuation, rescaled by the square root of the system width,
against the system length.  The system parameters are $\rho_{\rm avg} =
0.5625$, $p_{\rm right}=0.75$, $p_\perp=.125$ and $r=0.25$.  It is
clear from this data that
\begin{equation}
\langle\delta^2\rangle = \left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle \sim \frac{L}{W}.
\end{equation}
The collapse of the data shows that this scaling is valid even for quite
narrow systems ($W=4$).  Thus the $\rho_{\rm avg} \ne 0.5$, two-dimensional
system behaves as a collection of $W$ independent one-dimensional systems, at
least as far as the overall fluctuations of the shock position is concerned.

The situation is different when $\rho_{\rm avg} = 0.5$.  The only difference
between the data in Figure~\ref{fluc50} and Figure~\ref{fluc5625} is that in
Figure~\ref{fluc50} we have $\rho_{\rm avg} = 0.5$.
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\caption{Rescaled shock fluctuation: $W^{1/2}\left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle^{1/2}$ {\em vs.}\ system length $L$.
Average density is 1/2.  Dotted lines have slope 1/4 and 1/3.}
\label{fluc50}
\end{figure}
Here the rescaled data for different system widths superimpose on each other
only for $L$ small (compared with $W$).  Examining the data in
Figure~\ref{fluc50} carefully, we see that the fluctuations for each width
cross over from $L^{1/4}$ behavior to $L^{1/3}$ behavior, with the crossover
point increasing with $W$.  The fact that the data superimposes for small $L$
indicates that in this regime fluctuations scale as $L^{1/4}/W^{1/2}$, while
the $L^{1/3}$ behavior scales with a power of $W$ different from $1/2$.  To
get a better handle on this phenomenon, we plot the rescaled fluctuations
against the system width
in Figure~\ref{flucvw-res}.  For $L\gg W$ the slope approaches $-1/6$, which
indicates that the fluctuations scale as $L^{1/3}W^{-1/2 -1/6} =
L^{1/3}/W^{2/3}$.  While we do not have a complete understanding of this
exponent, its makes sense from general scaling arguments as the crossover from
$L^{1/4}$ to $L^{1/3}$ occurs when
\begin{equation}
L^{1/4}/W^{1/2} \sim L^{1/3}/W^{2/3},
\end{equation}
or when $L\sim W^2$---{\em i.e.}\ we observe one-dimensional behavior when
particles have a chance to diffuse across the entire width of the system.
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\end{picture}
\end{center}
\caption{Rescaled shock fluctuation: $W^{1/2}\left\langle(\bar{h}(t) -
\langle\bar{h}\rangle)^2\right\rangle^{1/2}$ {\em vs.}\ system width $W$.
Average density is 1/2.}
\label{flucvw-res}
\end{figure}

We can also think of the system at $L\gg W$ as a pseudo-one-dimensional system
where the noise is reduced from that in the one-dimensional ASEP because of
the average over the width.  If we insert a noise intensity parameter $I$ into
the (one-dimensional) KPZ equation (\ref{KPZ1}), {\em i.e.}\ the fluctuating
Burgers equation~\cite{Medina,vB}, yielding
\begin{equation}
\frac{\partial h}{\partial t} = \nu\frac{\partial^2 h}{\partial x^2} +
\frac{\lambda}{2} \left(\frac{\partial h}{\partial x}\right)^2
	+ I\zeta(x,t),
\end{equation} 
then effectively $I\propto W^{-1/2}$.  A scaling analysis of Amar and
Family~\cite{AF} indicates that fluctuations in $h$ are proportional to
$I^{4/3}$, so that we obtain fluctuations proportional to $W^{-2/3}$.  This
factor multiplies $L^{1/3}$ which is the behavior of the fluctuations in the
one-dimensional system~\cite{JL,GP,Dit,CLS,vB}.  This is confirmed by
simulations we have performed on the one-dimensional ASEP~\cite{inprogress}.
For hole density $\rho_h$ (=$1-\rho$) we find that the fluctuations in the
shock position scale as $\rho_h^{2/3} t^{1/3}$, which is consistent with
\cite{AF} since $I\sim \rho_h^2$.

This crossover point also makes sense from the point of view of understanding
the $L^{1/4}/W^{1/2}$ behavior of the fluctuations in the shock position, and
connects this with the $L^{1/4}$ behavior of the shock width.  For $L\ll W^2$
treat each row of the system as an almost independent one-dimensional ASEP.
By ``almost'' we mean we allow coupling only through the total density in each
row; otherwise the rows are treated independently.

If a particular row has a density of $\rho_{\rm row}$, its (local) shock
position will have variance proportional to $|\rho_{\rm row} -
1/2|L$~\cite{JL}.  The diffusive coupling between rows will produce
fluctuations in $\rho_{\rm row} - 1/2$ that are $O(L^{-1/2})$.  Thus the
typical deviation of the local shock from the center of the system is
$\left(O(L^{-1/2})L\right)^{1/2} = O(L^{1/4})$.  This is the correct
contribution to the shock width.  The overall shock position is the average of
the local shock position; we are treating each of the $W$ local positions as
an independent random variable, so the standard deviation of the average is
$W^{-1/2}O(L^{1/4})$.  Of course this neglects the $L^{1/3}$ term which must
eventually be larger.

Our results for the stationary model are summarized in Table~\ref{table}.
\begin{table}
\caption{Scaling behavior for the variance in shock position
$\langle\delta^2\rangle$ and shock width $\langle\sigma^2\rangle^{1/2}$.}
\label{table}
\[
\renewcommand{\arraystretch}{1.4}
\begin{array}{l||c|c|c|c|}
\cline{2-5}
&\multicolumn{2}{c|}{\rho_{\rm avg} = 1/2}
&\multicolumn{2}{c|}{\rho_{\rm avg} \ne 1/2}\\
\cline{2-5}
& L\gg W & L \ll W & L\gg W & L \ll W\\
\hline
\langle\delta^2\rangle & \left(L/W^2\right)^{2/3} &
	\left(L/W^2\right)^{1/2}&\multicolumn{2}{c|}{L/W}\\
\hline
\langle\sigma^2\rangle^{1/2} & W^{1/2} & L^{1/4} & W^{1/2} & L^{1/4}\\
\hline
\end{array}
\]
\end{table}
\section{Discussion}
\subsection{Relationship Between Stationary and Time-Dependent Models}
Studying the time-dependent behavior of the ASEP is significantly
different from studying the stationary states of the ASEP with a
blockage.  While there is no rigorous argument that the same exponents
should describe both the time-dependent and size-dependent scaling of
the shock width, it is not suprising to expect that they are the same,
considering that the underlying physics is identical in both models.
Namely, the transit time for traversal of the distance from block to
shock is of order $L$, so that if the time dependence is $t^\beta$,
the length dependence should be $L^\beta$.  The experimental
correspondence is unmistakable---Figure~\ref{g1k}, where we plot the
correlations $G$ {\em vs.}\ time $t$, and
Figure~\ref{width-vs-length}, where we plot the correlations {\em
vs.}\ system length $L$, are virtually copies of each other.

Of course, our only interpretations of this phenomenon are heuristic.
Although it is clear that fluctuations due to the transit time will
produce $L^\beta$ behavior, we can not exclude the existence of
stronger noise sources which would overwhelm this behavior---which in
fact we do observe when particle-hole symmetry is broken.

\subsection{Difficulties}
The precise determination of scaling exponents, particularly small ones, is
often not a straightforward task. This is especially true if one is limited in
analyzing the range over which the scaling holds.  In our case the constraints
on simulation run-time, due to lattice size and computer memory
considerations, proved crucial.

There were several options available to us which would have alleviated these
problems, but all involved tampering with the ASEP dynamics, usually by
destroying (or introducing spurious) correlations---small perturbations in
models such as the ASEP that have a conservation law but do not satisfy
detailed balance can have dramatic global effects~\cite{GLMS}.  It was
precisely the ASEP, and not some approximate variant, that we wished to study.
As a result, we were forced to simulate a truly two-dimensional problem and
not just some restricted (effectively one dimensional) domain containing the
shock surface.

\subsection{Parallel {\em vs.}\ Serial Models}
The future of large-scale scientific computing will rely on massively parallel
computers and thus systems with parallel dynamics.  As opposed to certain
disciplines where the utilization of parallelism has been difficult, there are
no inherent problems with the use of parallelism in physics---certainly ``real
world'' dynamics are parallel.  However, although they may not accurately
represent the real universe, serial models are generally more amenable to
analytical study~\cite{Spohn}.

We attempted to utilize parallel models that differed only very slightly from
the original serial models.  Even so, we observed behavior that was
significantly different in certain respects.  Thus the parallel dynamics must
be checked carefully for consistency with the serial dynamics.

\subsection{Other Models}
\subsubsection{Non-infinite temperature}
The ASEP dynamics can be viewed as the infinite temperature limit of a driven
diffusive system where the jump rates of the particles depend on the local
environment.  Models with temperature are clearly more realistic models than
those without it, and the immediate question is whether or not the behavior we
have observed is limited to the infinite temperature case.

In one dimension we have found no significant difference in the behavior
between infinite and finite temperature models.  The addition of temperature
makes the determination of the dependence of the current on the density more
difficult, but once this has been done, the shock fluctuations scale as
predicted by the fluctuating Burgers equation (with the appropriate $J$ {\em
vs.}\ $\rho$ behavior)~\cite{inprogress}.

In two dimensions many driven diffusive systems exhibit a phase
transition~\cite{KLS,GMD};
the low temperature behavior is qualitatively different from the ASEP.
At higher temperatures, the behavior is at least qualitatively similar to the
ASEP, in that shocks form and we can have segregation perpendicular to the
field as in our stationary model~\cite{AL}, but we have no detailed data on
the behavior of the interface.

\subsubsection{Non-lattice models ({\em e.g.}\ molecular dynamics)}
In addition to studying more realistic lattice models, it would be interesting
to examine the behavior of models in continuous space.  Specifically, we would
like to study via nonequilibrium molecular dynamics simulations~\cite{EM} the
statistics of shocks that form when a compressible fluid is forced to flow
through a pinched tube.
\subsection*{Acknowledgments}
We thank Henk van Beijeren and Herbert Spohn for their contributions to this
work.  This work was conducted using the computational resources of the
Pittsburgh Supercomputing Center.
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\end{document}
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