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% (appeared in Phys Rev A 45 618--625 (1992))
% `Finite Size Effects and Shock Fluctuations in the Asymmetric Simple
%	Exclusion Process' by Janowsky/Lebowitz
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\author{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\\
	Departments of Mathematics and Physics\\
	Rutgers University\\New Brunswick, NJ 08903}
\title{Finite Size Effects and Shock Fluctuations in the Asymmetric Simple
	Exclusion Process\thanks{Supported in part by National Science
	Foundation grant DMR 89-18903}}
\date{January, 1992}
\begin{document}\maketitle
\begin{abstract}
We consider a system of particles on a lattice of $L$ sites, set on a circle,
evolving according to the asymmetric simple exclusion process, {\em i.e.}\
particles jump independently to empty neighboring sites on the right (left)
with rate $p$ (rate $1-p$), $1/2<p\leq 1$.  We study the nonequilibrium
stationary states of the system when the translation invariance is broken by
the insertion of a blockage between (say) sites $L$ and $1$; this reduces the
rates at which particles jump across the bond by a factor $r$, $0<r<1$.  For
fixed overall density $\rho_{\rm avg}$ and $r \lessapprox (1-|2\rho_{\rm
avg}-1|)/ (1+|2\rho_{\rm avg}-1|)$ this causes the system to segregate into
two regions with densities $\rho_1$ and $\rho_2=1-\rho_1$, where the densities
depend only on $r$ and $p$, with the two regions separated by a well-defined
sharp interface.  This corresponds to the shock front described
macroscopically in a uniform system by the Burgers equation.  We find
that fluctuations of the shock position about its average value grow like
$L^{1/2}$ or $L^{1/3}$, depending upon whether particle-hole symmetry exists.
This corresponds to the growth in time of $t^{1/2}$ and $t^{1/3}$ of the
displacement of a shock front from the position predicted by the solution of
the Burgers equation in a system without a blockage and provides a new method
for studying such fluctuations.\\[3pt]

PACS numbers: 05.40.+j~~~02.50.+s~~~05.70.Ln~~~47.40.Nm
\end{abstract}
\section{Introduction}
The one-dimensional asymmetric simple exclusion process (ASEP) is a
continuous-time sto\-chastic process in which particles jump independently and
randomly at unit rate with probability $p$ to a vacant neighboring site on the
right and with probability $1-p$ to one on the left, $1/2 < p \le
1$~\cite{Lig}.  It is one of the simplest of the driven diffusive lattice-gas
models in that the only interaction between particles is the hard-core
exclusion which prevents any site from being occupied by more than one
particle.  This makes it one of the simplest models for studying the behavior
of systems whose dynamics do not satisfy detailed balance and for the
derivation of Euler-like hydrodynamical equations from microscopic dynamics,
{\em e.g.}\ the Burgers equation~\cite{LPS,Dit,GP,CLS,vB}.  This equation
reflects the property of physical fluids in that shock fronts, where the
macroscopic density is discontinuous, can form between regions of different
density, even if the initial distribution is smooth.  Since the macroscopic
description breaks down at the shock position we must study the structure of
the shock directly at the microscopic level.  This has been the focus of much
recent research~\cite{LPS,Dit,GP,CLS,vB}.

When a shock forms, or if one is present in the initial state, it will move
with an average velocity determined by the densities on either side of the
shock via the Burgers equation.  One of the interesting phenomena found in
studying the simple exclusion process is the dependence of the fluctuations
about that average on initial conditions---while there are fluctuations due to
the stochastic nature of the evolution, these are usually swamped by
fluctuations caused by microscopic variations in the initial
state~\cite{Dit,GP}.  For example, fluctuations of a shock's position grow
like $t^{1/3}$ if one compares independent stochastic evolutions with
identical initial conditions, but like $t^{1/2}$ (relative to the
deterministic hydrodynamic evolution given by the Burgers equation) if one
takes into account the random fluctuations in the initial state~\cite{CLS,vB}.

In contrast to the time dependent behavior, the stationary states for the
system with translation invariance are not very interesting; the stationary
states on the circle give equal weight to all permissible configurations with
a given number of particles, and their infinite volume limits are uniform
product measures with average density $\rho$, $0\le\rho\le1$.  We can generate
more interesting behavior, however, by breaking the translation invariance and
partially blocking one bond so that particles which occupy sites on either end
of the bond will have different transition probabilities than particles at
other sites.  This is analogous to a restriction in a pipe containing a fluid
flow.  This yields, in addition to its own interest as an example of the
generic long range coherence present in stationary nonequilibrium systems with
particle conserving dynamics~\cite{GLMS} (related to self-organized
criticality), a new, more robust method for studying the time dependent case.

By studying only the stationary state we eliminate most of the dependence on
the initial conditions (only total particle number remains relevant), but the
basic nature of the translation invariant model, which produces macroscopic
shocks with microscopic structure, remains.  The blockage generally introduces
a shock into the stationary state, and by studying its fluctuations we can
recover in a numerically more accessible way the exponents describing the
growth in time of the fluctuations of an initially sharp shock in the infinite
system.  Specifically, for a system of size $L$ the generic shock fluctuation
scales as $L^{1/2}$, but if the average density is $0.5$, placing the shock
exactly opposite the blockage, we obtain fluctuations in the shock position
that scale like $L^{1/3}$.  This can be understood in terms of a cancellation
that takes place when there is particle-hole symmetry combined with the
time-dependent scaling of the infinite system; we also show that by altering
the blocking mechanism we can eliminate the need for a cancellation and
construct a model that always has $L^{1/3}$ behavior.

\section{The Model}
Our system consists of particles on a lattice with unit spacing set on a
circle of circumference $L$, moving (randomly) in one direction, with a limit
of one particle per site, {\em i.e.}\ we consider the totally asymmetric
exclusion process, $p=1$.  This limitation seems to have little impact on the
nature of our results, as determined by simulations for both types of systems.

We break the translation invariance of this periodic system by inserting a
blockage into the system between sites $L$ and 1, which reduces the
probability of a particle traveling between those two sites---a ``slow bond''
which acts as a traffic jam for the particles.  In the language of driven
diffusive systems, this is similar to altering the driving field at this one
bond~\cite{AL}.

More formally, our process is defined by the generator $\cal L$ giving the
rate of change of any function of the configuration
$\eta = \{\eta(1), \eta(2), \ldots, \eta(L)\}$, where
$\eta(k) = 0\ {\rm or}\ 1$ is the occupation number at site $k$:
\begin{equation}
{\cal L} f(\eta) =
	\sum_{i=1}^{L-1}[f(\eta_{i\rightarrow i+1}) - f(\eta)] +
	r[f(\eta_{L\rightarrow 1}) - f(\eta)],
\end{equation}
where 
\begin{equation}
\eta_{i\rightarrow j}(k) = \left\{
\begin{array}{l@{\quad}l}
1,& k=j {\rm ~and~} \eta(i) = 1,	\\
0,& k=i {\rm ~and~} \eta(j) = 0,	\\
\eta(k),& {\rm otherwise};
\end{array}
\right.\end{equation}
the rate $r$, $0\le r\le1$, determines the degree of blocking.  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 and there is no current flowing through the system.  For $0<r<1$ the
model has nontrivial behavior, resulting from the requirement that the current
$J_k$ through any bond $k\rightarrow k+1$, {\em i.e.}\ the expected number of
particles jumping from $k$ to $k+1$ in unit time, must be independent of $k$.

As already mentioned, one motivation for this study comes from the fact that
this is perhaps the simplest model which shows the dramatic global effects
caused by a local perturbation in the dynamics in a system with conservative
dynamics that does not satisfy a detailed balance condition~\cite{GLMS}.
For the homogeneous case $r=1$, the stationary measure $\mu_{\rho_{\rm avg}}$
for a given system size $L$ and number of particles $N=\rho_{\rm avg}L$ is one
in which all $L\choose N$ permissible configurations have equal weight.  While
this is true for all values of $p\in[0,1]$, it is only for the symmetric case
($p=1/2$) that the dynamics satisfy detailed balance with respect to
$\mu_{\rho_{\rm avg}}$.  A local blockage then has only a local effect.

We recover detailed balance even for $p\neq 1/2$ if we consider $r=0$, {\em
i.e.}\ full blockage, since then the current in the stationary state must be
zero.  The stationary state is then given by
\begin{equation}
P(\eta) = \frac{1}{Z} \prod_k \left(\frac{p}{1-p}\right)^k \eta(k),
\end{equation}
where $Z$ is a normalization factor such that the sum of $P(\eta)$ over all
allowed configurations is one.  When $P=1$ the stationary state has
$\eta(k)=0$ for $k=1,\ldots,L-N$ and $\eta(k)=1$ for $k=L-N+1,\ldots,L$.

\subsection*{Equivalent Growth Model}
The ASEP can also be considered as a simple model for surface
growth~\cite{MRSB}.  The surface height is related to the particle occupancy
as follows: the presence of a particle in the ASEP is equivalent to a decrease
in the height of the surface (by 1), the absence of a particle to an increase
(by 1); thus the relative surface height is essentially the integrated
particle number:
\begin{equation}\label{surfaceheight}
h(x) - h(0) = \sum_{k=1}^x (1-2\eta(k)).
\end{equation}
If the total number of particles is $L/2$ the surface is periodic; otherwise
we use helical boundary conditions with $h(L) - h(0) = L(1-2\rho_{\rm avg})$.

Since $\eta(k)$ must be zero or one, $(1-2\eta(k))$ is $\pm 1$ and the height
difference between neighboring sites must have absolute value 1.  A particle
followed by a hole (a decrease followed by an increase) forms a local minimum
in the surface, and a hole followed by a particle (increase/decrease) forms a
local maximum.  Of course the dynamics is the same: local minima become local
maxima with rate $p$ (deposition) as particles move to vacant sites on the
right, and the reverse process (evaporation) occurs with rate $1-p$ as
particles move to the left.  
Our introduction of a blockage is equivalent to
altering the deposition/evaporation rates at a single site~\cite{WT}, {\em
i.e.}\ we consider deposition that is uniform everywhere except at one site
where the rate is reduced; this can have a global effect in the growth of the
surface.

\section{Results}

We simulated our model for a large number of different system sizes, blocking
rates and average densities.  To carry out a simulation, we choose at each
time step one site, say site $i$, at random.  If $i\ne L$ and site $i$ is
occupied and site $i+1$ is not, then the particle at site $i$ jumps to site
$i+1$, otherwise nothing happens.  If we do choose site $L$, then the jump to
site $1$ takes place with probability $r$, provided that site $L$ is filled
and site 1 is empty.

Qualitatively there is little difference in the behavior of the system as we
vary the parameters, provided that $r \ne 0, 1$ (which would correspond to the
two trivial cases mentioned earlier) and (for reasons discussed following
eq.~(\ref{rhoapprox})) that the average density is not too far from 1/2.  A
typical (time-averaged) density profile is presented in figure~\ref{prof35}.
\begin{figure}[htb]
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\end{center}
\caption{Density profile: 1600 sites, blockage transmission=.35, average
density=0.5}\label{prof35}
\end{figure}
As one would expect, particles pile up behind the blockage and are relatively
scarce in front of it.  Away from the blockage the system appears similar to
the infinite model where two uniform phases meet to form a shock~\cite{ABL},
while there is a deviation from uniformity near the blockage.  Suprisingly, a
closer examination of the density near the block indicates that the excess
density appears to decay with the distance from the blockage $x$ like
$1/|x+{\rm const.}|$, as opposed to exponentially.  A close-up of the density
near one side of the blockage (from a long run with good statistics---$10^7$
samples) is presented in figure~\ref{1overR}. 
\begin{figure}[htb]
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\end{center}
\caption{Density profile near the block: 1200 sites, blockage
transmission=.40, average density=0.5}\label{1overR}
\end{figure}
We see the excellent agreement between the data and the function exhibiting
the $1/|x|$ behavior.

Note that the profile is symmetric about density $\rho=0.5$; the symmetry
between the high and low density phases is an important part of the dynamics
and the stationary state: instead of moving particles we can just as well
consider the situation where when the site $i$ is picked, and it contains a
hole ({\em i.e.}\ is empty), that hole jumps to the left if the site $i-1$
does not already contain a hole, or jumps with probability $r$ to site $L$ if
$i=1$.

Since the average current through the bond $k\rightarrow k+1$ is given by
$\langle \eta(k)[1 - \eta(k+1)]\rangle$, we see that the
current in a state $\nu_\rho$ of uniform density $\rho$ without correlations
is $\rho(1-\rho)$, so if the measure far from the blockage and the shock is
just a product measure, current conservation requires
\begin{equation}\label{lh}
{\rho_{\rm low}}(1-{\rho_{\rm low}}) = {\rho_{\rm high}}(1-{\rho_{\rm high}});
\end{equation}
equivalently,
\begin{equation}\label{lhdens}
{\rho_{\rm low}} = {\rho_{\rm high}}
\qquad {\rm or} \qquad
{\rho_{\rm low}} = 1-{\rho_{\rm high}}.
\end{equation}
If the low density current were not equal to the high density current,
particle conservation would require that the (infinite system) interface move
with velocity \(v_{\rm shock} = 1 - {\rho_{\rm low}} - {\rho_{\rm high}}\);
the requirement that this velocity be zero if the densities are not equal may
also be used to derive (\ref{lhdens}).

Since we have a deviation from uniform product measure near the
blockage, the exact values of ${\rho_{\rm low}}$, ${\rho_{\rm high}}$
can not be computed directly but must be determined by solving for the
stationary state, or lacking that, empirically.  We can obtain
approximate values for these densities by neglecting correlations
between all sites, not just sites far from the blockage.  Then, (since
the average current must be constant)

\begin{equation}\label{Japprox}
r{\rho_{\rm high}} (1-{\rho_{\rm low}}) \approx J_{\rm block} = J_{\rm bulk} =
{\rho_{\rm high}} (1-{\rho_{\rm high}}),
\end{equation}
so that
\begin{equation}\label{rhoapprox}
{\rho_{\rm high}} \approx \frac{1}{r+1},
\qquad
{\rho_{\rm low}} \approx \frac{r}{r+1}.
\end{equation}
For the system of figure~\ref{prof35}, this computation yields the result
${\rho_{\rm low}} \approx 0.26$; compare it with the actual value ${\rho_{\rm
low}} = 0.29$.  Clearly the system manages to increase the current (and thus
$\rho_{\rm low}$) by making the density non-uniform and introducing
correlations near the block.  The specifics of how this occurs, {\em i.e.}\
the structure of the stationary state near the blocked site, will be the
subject of a separate inquiry.

For density different from 1/2, the average shock location is displaced from
the center, but the basic structure remains the same---provided that the
density is such that the shock would not need to be shifted all the way to (or
past) the blockage.  In other words, ${\rho_{\rm low}}$ and ${\rho_{\rm
high}}$ remain the same as before provided that we have a solution to
\begin{equation}
\rho_{\rm avg} = a_l{\rho_{\rm low}} + a_h{\rho_{\rm high}},
\qquad a_l + a_h = 1,
\end{equation}
with $a_l, a_h>0$.  If we use the approximation (\ref{rhoapprox}), we must
have $|\rho_{\rm avg} - 1/2| < (1-r)/(2(1+r))$.  If this is not the case we no
longer have the basic structure of two near-uniform regions meeting at a shock
front, with local perturbations near the blockage; instead we have only one
approximately uniform phase.  Apparently, if the current in the uniform phase
is sufficiently small, there is no need for phase segregation to reduce the
bulk current to that at the blockage~\cite{Krug}.  Similar behavior is
observed when we examine an infinite system with one site blocked (see
Section~\ref{infinite}).

\subsection*{Tracking the Shock}
While the average shock location is easy to determine, finding its microscopic
position or even defining it precisely at a given time is nontrivial.  Instead
of determining the shock position directly, we track it through the use of a
{\bf second class particle}~\cite{BCFG}.  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.  In other words, if
we select site $i$ and it is occupied by an ordinary particle, it moves to
site $i+1$ either if that site is empty or if it is occupied by the second
class particle; in the latter case the second class particle simultaneously
jumps (to the left) to site $i$, {\em i.e.}\ there is an exchange between the
first and second class particles.  If site $i$ is occupied by a second class
particle, it moves to site $i+1$ only if site $i+1$ is unoccupied; if site
$i+1$ is occupied it stays put.  Therefore, if the second class particle is in
a high density region (of ordinary particles), it will be forced to the left,
while if it is in a low density region it will be able to move to the right.

A shock consists of an abrupt change from a low density region to a high
density region.  To the left of a shock (a low density region) a second class
particle moves predominantly to the right, while on the right side (a high
density region) it moves predominantly to the left.  The second class particle
therefore executes a biased random walk with drift towards the shock position.
Simple analysis shows and computer simulations confirm that the inherent
fluctuations of this random walk are small compared with the movement of the
shock---namely that if we fix the densities on either side of the shock
(${\rho_{\rm low}}, {\rho_{\rm high}}$) and consider different values of the
system size $L$, the motion of the second class particle about the shock
position will have fixed variance while the variance of the shock position
itself will grow with system size.  Thus analyzing the motion of the second
class particle will allow us to indirectly examine the shock motion.

An alternative method of finding the shock is suggested by the surface growth
model. Recalling equation~(\ref{surfaceheight}), we see that surface height
increases in low (particle) density regions and decreases in high density
regions, so that the shock location should correspond intuitively to the
location of the maximum height.  Unfortunately this position need not be
(microscopically) unique, so we prefer the use of the second class particle.
It suggests, however, that in addition to the shock position we study other
features from the growth model, such as the difference between the maximum and
minimum surface heights, which is equivalent to studying the number of
particles on either side of the interface in the ASEP.

As a third possibility we could ignore the microscopic shock position entirely
and simply study the width of the time-averaged shock density profile, such as
that in figure~\ref{prof35}.  However, this method obscures the distinction
between the instantaneous microscopic width of the shock and the width due to
fluctuations in the shock's position (particularly relevant in the extension
to two-dimensional models), and the width is difficult to determine accurately
because of lattice effects.  In any case, we expect that as the system size
goes to infinity that rescaling the shock profile by the standard deviation of
the shock fluctuation (determined by any of the three methods) should yield a
well-defined limiting shape.

We sample the position of the second class particle after allowing the system
to reach a steady state.  For each system we obtain a distribution (histogram)
of shock (second class particle) positions and compute the variance of the
obtained distribution; a typical run consisted of approximately 25,000 samples
where 15 sweeps ($15\times L$ monte carlo steps) were made between samples.
We then compare the variances obtained from systems with different values of
the parameters.  We also perform the same set of computations for the number
of particles found between site~1 and the second class particle position.

In figure~\ref{var55} we plot the standard deviation of the interface position
{\em vs.}\ system size for average density 0.55 and blocking rate $r=0.35$.
\begin{figure}[hbt]
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\end{center}
\caption{Interface fluctuations with average density 0.55, $r=0.35$.  The solid
line has slope 1/2.}
\label{var55}
\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.  The line drawn through the
data points has a slope of 1/2, so that fluctuations grow like the square root
of the system size, indicating standard finite size behavior.

In figure~\ref{var35w-1} we plot the same quantities as in figure~\ref{var55},
but the average particle density is 0.5.
\begin{figure}[htb]
\begin{center}
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\end{center}
\caption{Interface fluctuations with average density 0.5, $r=0.35, 0.5$.  The
solid lines have slope 1/3.}
\label{var35w-1}
\end{figure}
We examined two sets of systems, with blocking rates $r=0.35, 0.5$.  Here the
lines drawn through the data points have slope 1/3, so that fluctuations are
suppressed compared with systems whose densities are different from 0.5,
yielding growth only like the cube root of the system size.  (Note that
changing $r$ does not affect this conclusion; the relevant variable is whether
or not the density is 0.5.)

The reduction in the fluctuations of the interface when the average density is
1/2 is clearly caused by some type of cancellation since other fluctuations in
the system still scale like $L^{1/2}$---in figure~\ref{var35h} we plot the
standard deviation of the number of particles found to the left of the
interface, {\em i.e.}\ between site~1 and the second class particle;
\begin{figure}[hbt]
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\caption{Fluctuations in the number of particles to the left of the interface:
average density 0.5, $r=0.35$.  The solid line, which is a best linear fit to
the data, has slope 0.46.}
\label{var35h}
\end{figure}
this data is consistent with scaling exponent 1/2 (and inconsistent with 1/3).
An explanation of this behavior is given in the next section.

\section{Analysis}
To make sense of our results, it is convenient to decompose the stochastic
nature of our system into two pieces: the random flow of particles through the
blockage, and the random dynamics everywhere else.  We hypothesize that the
``blockage randomness'' is responsible for the $L^{1/2}$ behavior and that the
``dynamical randomness'' is responsible for the $L^{1/3}$ behavior.

A particle traversing the blockage can be thought of as two simultaneous
events: a particle being released at site~$1$ while a hole is released at
site~$L$.  These two events are of course completely correlated, but the
motions of the particle and hole {\em after} release are nearly independent.

The position of the interface is determined by the {\em difference} between
the number of particles which arrive from the left and the number of holes
which come from the right.  (It may be convenient for visualization purposes
to consider the situation where the density to the right of the shock is very
close to one and the density on the left is near zero.)  If we neglect the
dynamical randomness---{\em i.e.}\ if we assume that particles in the low
density region move at their average velocity $1-{\rho_{\rm low}}$, and holes
in the high density region move at their average velocity $-{\rho_{\rm high}}
= -(1-{\rho_{\rm low}})$, and consider a shock in the center of the system,
then no matter what the nature of the creation of excitations at the blockage,
particles and holes will always impinge upon the shock in pairs: when a pair
of excitations is created, the hole and particle will travel with equal speeds
and opposite directions, meeting (and annihilating) when they have each
traveled a distance $L/2$ and reach the shock position.  Since the ``blockage
randomness'' plays no role here, a system with density~0.5, and therefore
particle-hole symmetry (figure~\ref{var35w-1}), has fluctuations determined
only by the ``dynamical randomness''---the deviation of the particle
velocities from their average.

On the other hand, if the shock is not in the center of the system, the nature
of ``blockage randomness'' becomes relevant as the times for particles and
holes to reach the interface are no longer the same.  For example, if the
average density is greater than 1/2 the (average) shock position will be less
than $L/2$, and particles which start at the blockage and travel to the right
will reach the shock before holes (starting at the same time and place)
traveling to the left---the arrival times of the particles no longer match
those of the holes because the distances they need traverse are different, and
the resulting motion of the shock position will depend upon exactly how the
excitations are created.  So a system with density different from 0.5
(figure~\ref{var55}) and thus with an average shock position shifted from the
center will have fluctuations from the ``blockage randomness'' proportional to
the amount of the shift.

The $L^{1/2}$ scaling of the ``blockage randomness'' results from the rapid
decay of correlations in both time and space in the stationary state of the
ASEP, as well as the use of an independent random choice for each attempt to
cross the blockage. Thus the creation of one excitation is very nearly
independent from the creation of another.  Therefore the fluctuation in the
number of excitations will scale as the square root of the average number.
There is a finite density of excitations, so their number is $O(L)$, yielding
fluctuations that scale like $O(L^{1/2})$.
In fact, if we neglect the dynamical randomness and treat particle flow
through the blockage as a Poisson process, we can determine through
straightforward analysis the variance of the shock position.
Each particle or hole added to the shock moves it by
$(\rho_{\rm high} -\rho_{\rm low})^{-1}$.  Thus the shock is shifted by an
amount $|\rho_{\rm avg} - 1/2| L/(\rho_{\rm high} -\rho_{\rm low})$,
which is half the difference in path length between particles and holes.
Therefore the time differential is $2|\rho_{\rm avg} - 1/2| L\big/
\left(\rho_{\rm high}(\rho_{\rm high} - \rho_{\rm low})\right)$.  The
probability of the formation of an excitation is $\rho_{\rm high}\rho_{\rm
low}$ so that the standard deviation of the shock position should be
\begin{equation}\label{STDSP}
(\rho_{\rm high} -\rho_{\rm low})^{-1}\left(
	\frac{2|\rho_{\rm avg} - 1/2| L}{\rho_{\rm high}(\rho_{\rm high} -
		\rho_{\rm low})}
	\rho_{\rm high}\rho_{\rm low}\right)^{1/2} =
\left(\frac{2{\rho_{\rm low}} |\rho_{\rm avg} - 1/2| L}{({\rho_{\rm high}} -
{\rho_{\rm low}})^3}\right)^{1/2}.
\end{equation}
This calculation should be accurate for $r$ near zero but will be too large
otherwise, as the actual variance of the particle flow is less than that of a
Poisson process since we do have (small) memory effects.  For the system of
figure~\ref{var55}, the measured behavior is $0.50L^{1/2}$ while
eq.~(\ref{STDSP}) yields $0.63L^{1/2}$.

What is the nature of the randomness in the dynamics?  Recent work by van
Beijeren~\cite{vB} indicates that density fluctuations grow in time like
$t^{1/3}$, while since the distance is of order $L$ and the speed is $O(1)$
the time required for an excitation to reach the interface from the blockage
is of order $L$.  Thus an initially nonrandom arrangement (or a random
arrangement with number particles = number holes) of excitations will have
fluctuations of order $L^{1/3}$ by the time it reaches the interface.  We
confirm this by examining a system where we alter the behavior near the
blockage so that particles pass through the blockage in a near-uniform manner.
Of course we can not have a completely uniform current since the occupation
numbers of each site are limited to 0 and 1, but we try to make the flow as
uniform and non-random as possible: if $n(t)$ is the number of particles
already having passed through the blockage at time $t$, we specify the current
exactly by opening the blockage if $n(t) < Jt$ and closing it otherwise.  If a
particle is immediately to the left of the blockage while it is open and the
site to the right is vacant, the particle jumps through the blockage.  Thus we
attempt to schedule particle jumps at intervals of $1/J$; if a jump is not
possible because the appropriate occupancy conditions are not satisfied we
``make up'' that jump as soon as possible.

By altering the nature of the blockage, we alter the behavior of the shock
fluctuations. For this uniform-block model, the lack of ``blockage
randomness'' means that even without particle-hole symmetry, {\em i.e.}\ even
when the shock is displaced from the center we have fluctuations that scale
like $L^{1/3}$ for {\em both} the interface position {\em and} the number of
particles to the left of the interface---see figure~\ref{varu}.  Thus we have
(partial) confirmation of our hypothesis.
\begin{figure}[htb]
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\end{picture}
\end{center}
\caption{Fluctuations with uniform (non-random) flow through the block.
Average density=0.55.  Solid lines have slope 1/3.}\label{varu}
\end{figure}
\section{Infinite System with Block}\label{infinite}
We have been considering particles evolving according to the ($p=1$) ASEP
dynamics on a circle of $L$ sites with a blockage parameterized by $r$.  For
each such system with a fixed number of particles $N$ (or average density
$\rho_{\rm avg} \equiv N/L$) the stationary state is unique.  The question
naturally arises, ``What happens to these states in the infinite volume limit
$L\rightarrow\infty$, $N\rightarrow\infty$, $N/L\rightarrow\rho$ fixed?''  To
make this question precise we have to specify the position of our domain of
observation relative to the blocked site as $L\rightarrow\infty$.  This is
unlike the case in the finite system where except for labeling the state is
independent of the position of the blocked bond, since the domain of
observation corresponds to all $L$ sites.

Let us label the sites of our system (relative to the observation domain) from
$-L/2$ to $L/2$, and put the left end of the blocked bond at $K_L$.  Then
if $K_L\rightarrow\pm\infty$ as $L\rightarrow\infty$ we are in the situation
of an infinite uniform system, and the stationary states are superpositions of
product states.  The product states are of two kinds: for $0<J\le 1/4$ they
are translation invariant with densities $\rho_\pm =
\left(1\pm\sqrt{1-4J}\right)/2$, while for $J=0$ the density is zero to the
left of some site $i_0$ and one to the right of it.

As $L, K_L\rightarrow\infty$, the finite volume current $J_L$ will converge to
a limit $J$ and the stationary state $\mu_s$ will be a superposition of two
product measures $\nu_{\rho_+}$ and $\nu_{\rho_-}$,
\begin{equation}\label{superposition}
\mu_s = \alpha \nu_{\rho_+} + (1-\alpha) \nu_{\rho_-},
	\qquad 0\le\alpha\le1;
\end{equation}
$\alpha$ depends on the average position of the shock as $L\rightarrow\infty$,
{\em i.e.}\ if the distance to the shock goes to infinity sufficiently fast
compared to the fluctuations in the shock position ($L^{1/2}$ or $L^{1/3}$)
then $\alpha$ will be zero or one, but it will be non-trivial if the distance
between the shock and the origin grows more slowly~\cite{ABL}.

The situation is different if $K_L$ remains finite as $L\rightarrow\infty$.
In this case we can just as well take $K_L = 0$ and the class of stationary
states with $J>0$ will then depend on $r$ (for $J=0$ we have again the fully
blocked states and $r$ is irrelevant).

We believe that it can be shown that for any $J>0$ the asymptotic distribution
to the left and right of the origin is a product measure with densities
$\rho_{\rm left}$ and $\rho_{\rm right}$ where either
\begin{equation}\label{infequal}
\rho_{\rm left} = \rho_{\rm right} =
\left(1\pm\sqrt{1-4J}\right)/2
\end{equation}
or
\begin{equation}\label{infunequal}
\rho_{\rm right} = \left(1-\sqrt{1-4J}\right)/2 <
\rho_{\rm left} = \left(1+\sqrt{1-4J}\right)/2.
\end{equation}
The problem is then to find the phase diagram in the $J$--$r$ plane.  We have
used the results from our finite system studies as well as numerical
simulations intended to imitate the infinite situation (we use boundary
conditions of fixed density) to obtain the following results:

Away from the blockage the infinite stationary state resembles a product
measure, so that the need for the current to be the same on both sides
requires (as in equations~(\ref{lh})--(\ref{lhdens})) that the densities on
either side of the block be equal or be symmetric about $1/2$, in full
correspondence with (\ref{infequal})--(\ref{infunequal}).

Let $\rho_{\rm crit} = \rho_{\rm low} = 1 - \rho_{\rm high}$ refer to the
lower density of the segregated periodic system with the same blockage
transmission $r$ as the infinite system.  Then there are three nontrivial
classes of extremal stationary states for the infinite system in addition to
the zero-current measures (we can also have superposition measures as in
(\ref{superposition})):
\begin{enumerate}
\item $\rho_{\rm left} = \rho_{\rm right} < \rho_{\rm crit}$,\label{classlow}
\item $\rho_{\rm left} = \rho_{\rm right} > 1 - \rho_{\rm crit}$,
\label{classhigh}
\item $1 - \rho_{\rm left} = \rho_{\rm right} = \rho_{\rm crit}$.
\label{classhighlow}
\end{enumerate}
We see that we are justified in referring to $\rho_{\rm crit}$ as a critical
density; the system cannot be stationary at densities between $\rho_{\rm
crit}$ and $1 - \rho_{\rm crit}$ because the required current is too high.
We can illustrate this in figure~\ref{J-r} where the critical line is
approximated by the no-correlation formulae (\ref{Japprox})--(\ref{rhoapprox}).
\begin{figure}[hbt]
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\put(750,380){\makebox(0,0)[l]{Uniform asymptotic density}}
\put(750,330){\makebox(0,0)[l]{(type 1,2)}}
\end{picture}
\end{center}
\caption{Phase diagram for infinite system with block; current {\em vs.}\
blockage rate.}\label{J-r}
\end{figure}

The first two families of (near) uniform states correspond to the (near)
uniform states in the periodic system, when $\rho_{\rm avg} < \rho_{\rm crit}$
or $\rho_{\rm avg} > 1 - \rho_{\rm crit}$.  The nonuniform state corresponds
to segregated systems with $\rho_{\rm crit} < \rho_{\rm avg} < 1 - \rho_{\rm
crit}$.

An apparent fourth type of state with $1 - \rho_{\rm right} = \rho_{\rm left}
< \rho_{\rm crit}$ is not stationary since the interface is not bound to the
block and thus will eventually drift to infinity, even though its average
velocity is zero.

Nearly as interesting as the stationary states themselves are their basins of
attraction.  Choosing as initial conditions $\rho= \rho_{\rm left,0}$ to
the left of the blockage and $\rho = \rho_{\rm right,0}$ on the right we found
that
\begin{itemize}
\item For $\rho_{\rm left,0} < \rho_{\rm crit}$, $\rho_{\rm left,0} <
1-\rho_{\rm right}$, the system approaches a state from class~\ref{classlow}
above with $\rho_{\rm left} = \rho_{\rm right} = \rho_{\rm left,0}$.
\item For $\rho_{\rm right,0} > 1-\rho_{\rm crit}$, $\rho_{\rm right,0} >
1-\rho_{\rm left}$, the system approaches a state from class~\ref{classhigh}
above with $\rho_{\rm left} = \rho_{\rm right} = \rho_{\rm right,0}$.
\item For $\rho_{\rm left,0} \ge \rho_{\rm crit}$, $\rho_{\rm right,0} \leq
1-\rho_{\rm crit}$, the system approaches a state from class~\ref{classhighlow}
with $1-\rho_{\rm left} = \rho_{\rm right} = \rho_{\rm crit}$.
\end{itemize}
This behavior is presented graphically in figure~\ref{phase}.
\begin{figure}[hbt]
\begin{center}
\setlength{\unitlength}{0.24pt}
\begin{picture}(1200,1100)(-50,-50)
\put(0,0){\line(1,0){1000}}
\put(0,1000){\line(1,0){1000}}
\put(0,0){\line(0,1){1000}}
\put(1000,0){\line(0,1){1000}}

\put(-12,0){\makebox(0,0)[r]{0}}
\put(-12,1000){\makebox(0,0)[r]{1}}
\put(-12,300){\makebox(0,0)[r]{$\rho_{\rm crit}$}}
\put(-85,500){\makebox(0,0){$\rho_{\rm left,0}$}}

\put(0,-27){\makebox(0,0){0}}
\put(1000,-27){\makebox(0,0){1}}
\put(700,-27){\makebox(0,0){$1-\rho_{\rm crit}$}}
\put(700,0){\line(0,1){20}}
\put(500,-65){\makebox(0,0){$\rho_{\rm right,0}$}}

\put(0,300){\line(1,0){700}}
\put(700,300){\line(0,1){700}}
\put(700,300){\line(1,-1){300}}

\put(145,800){\makebox(0,0)[l]{$1-\rho_{\rm left} = \rho_{\rm right} =
	\rho_{\rm crit}$}}
\put(145,750){\makebox(0,0)[l]{(type 3)}}
\put(195,150){\makebox(0,0)[l]{$\rho_{\rm left} = \rho_{\rm right} =
	\rho_{\rm left,0}$}}
\put(195,100){\makebox(0,0)[l]{(type 1)}}
\put(725,700){\makebox(0,0)[l]{$\rho_{\rm left} = \rho_{\rm right}$}}
\put(760,650){\makebox(0,0)[l]{$= \rho_{\rm right,0}$}}
\put(725,580){\makebox(0,0)[l]{(type 2)}}

\end{picture}
\end{center}
\caption{Phase diagram for infinite system with block; the axes represent the
initial densities on each side of the block.}\label{phase}
\end{figure}

In the first two cases the system chooses whichever initial state has the
lowest current.  In the last case the system forces itself critical whenever
the initial current on both sides of the block is greater than that which can
be sustained through the block.

\section{Conclusion}
We have examined a version of the asymmetric simple exclusion process on a
finite periodic lattice.  Our model has many of the features of the time
dependent system but none of the sensitivity to initial conditions, and one
can use the presence or absence of particle-hole symmetry to indicate the
model's behavior.  It thus provides a more convenient platform for the study
of the asymmetric simple exclusion process and similar models.

We have used the presence of a ``slow bond'' as a technique to introduce a
shock into the finite system, but the study of such an impurity is of
intrinsic interest as well.  One can examine the effects of inserting one
blockage, or a periodic or random array of blockages, into the infinite
system.  We have identified some of the gross features of such systems, but
much of the determination of the local and global properties remains an
interesting open problem.

Our investigations can also be extended to dimension greater than one.
Preliminary results~\cite{ACJL} indicate the importance of particle-hole
symmetry in the two-dimensional system and illustrate the utility of studying
interface behavior by forcing the formation of a shock in a stationary state.

\subsection*{Acknowledgments}
We thank Maury Bramson, Zheming Cheng, Herbert Spohn and particularly Henk van
Beijeren for their contributions to this work.
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\end{document}