3,290 research outputs found
Entropy of Open Lattice Systems
We investigate the behavior of the Gibbs-Shannon entropy of the stationary
nonequilibrium measure describing a one-dimensional lattice gas, of L sites,
with symmetric exclusion dynamics and in contact with particle reservoirs at
different densities. In the hydrodynamic scaling limit, L to infinity, the
leading order (O(L)) behavior of this entropy has been shown by Bahadoran to be
that of a product measure corresponding to strict local equilibrium; we compute
the first correction, which is O(1). The computation uses a formal expansion of
the entropy in terms of truncated correlation functions; for this system the
k-th such correlation is shown to be O(L^{-k+1}). This entropy correction
depends only on the scaled truncated pair correlation, which describes the
covariance of the density field. It coincides, in the large L limit, with the
corresponding correction obtained from a Gaussian measure with the same
covariance.Comment: Latex, 28 pages, 4 figures as eps file
Shift Equivalence of Measures and the Intrinsic Structure of Shocks in the Asymmetric Simple Exclusion Process
We investigate properties of non-translation-invariant measures, describing
particle systems on \bbz, which are asymptotic to different translation
invariant measures on the left and on the right. Often the structure of the
transition region can only be observed from a point of view which is
random---in particular, configuration dependent. Two such measures will be
called shift equivalent if they differ only by the choice of such a viewpoint.
We introduce certain quantities, called translation sums, which, under some
auxiliary conditions, characterize the equivalence classes. Our prime example
is the asymmetric simple exclusion process, for which the measures in question
describe the microscopic structure of shocks. In this case we compute
explicitly the translation sums and find that shocks generated in different
ways---in particular, via initial conditions in an infinite system or by
boundary conditions in a finite system---are described by shift equivalent
measures. We show also that when the shock in the infinite system is observed
from the location of a second class particle, treating this particle either as
a first class particle or as an empty site leads to shift equivalent shock
measures.Comment: Plain TeX, 2 figures; [email protected], [email protected],
[email protected], [email protected]
Spontaneous symmetry breaking: exact results for a biased random walk model of an exclusion process
It has been recently suggested that a totally asymmetric exclusion process
with two species on an open chain could exhibit spontaneous symmetry breaking
in some range of the parameters defining its dynamics. The symmetry breaking is
manifested by the existence of a phase in which the densities of the two
species are not equal. In order to provide a more rigorous basis to these
observations we consider the limit of the process when the rate at which
particles leave the system goes to zero. In this limit the process reduces to a
biased random walk in the positive quarter plane, with specific boundary
conditions. The stationary probability measure of the position of the walker in
the plane is shown to be concentrated around two symmetrically located points,
one on each axis, corresponding to the fact that the system is typically in one
of the two states of broken symmetry in the exclusion process. We compute the
average time for the walker to traverse the quarter plane from one axis to the
other, which corresponds to the average time separating two flips between
states of broken symmetry in the exclusion process. This time is shown to
diverge exponentially with the size of the chain.Comment: 42 page
Exact solutions for a mean-field Abelian sandpile
We introduce a model for a sandpile, with N sites, critical height N and each
site connected to every other site. It is thus a mean-field model in the
spin-glass sense. We find an exact solution for the steady state probability
distribution of avalanche sizes, and discuss its asymptotics for large N.Comment: 10 pages, LaTe
On the Two Species Asymmetric Exclusion Process with Semi-Permeable Boundaries
We investigate the structure of the nonequilibrium stationary state (NESS) of
a system of first and second class particles, as well as vacancies (holes), on
L sites of a one-dimensional lattice in contact with first class particle
reservoirs at the boundary sites; these particles can enter at site 1, when it
is vacant, with rate alpha, and exit from site L with rate beta. Second class
particles can neither enter nor leave the system, so the boundaries are
semi-permeable. The internal dynamics are described by the usual totally
asymmetric exclusion process (TASEP) with second class particles. An exact
solution of the NESS was found by Arita. Here we describe two consequences of
the fact that the flux of second class particles is zero. First, there exist
(pinned and unpinned) fat shocks which determine the general structure of the
phase diagram and of the local measures; the latter describe the microscopic
structure of the system at different macroscopic points (in the limit L going
to infinity in terms of superpositions of extremal measures of the infinite
system. Second, the distribution of second class particles is given by an
equilibrium ensemble in fixed volume, or equivalently but more simply by a
pressure ensemble, in which the pair potential between neighboring particles
grows logarithmically with distance. We also point out an unexpected feature in
the microscopic structure of the NESS for finite L: if there are n second class
particles in the system then the distribution of first class particles
(respectively holes) on the first (respectively last) n sites is exchangeable.Comment: 28 pages, 4 figures. Changed title and introduction for clarity,
added reference
Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case
We consider the steady state of an open system in which there is a flux of
matter between two reservoirs at different chemical potentials. For a large
system of size , the probability of any macroscopic density profile
is ; thus generalizes to
nonequilibrium systems the notion of free energy density for equilibrium
systems. Our exact expression for is a nonlocal functional of ,
which yields the macroscopically long range correlations in the nonequilibrium
steady state previously predicted by fluctuating hydrodynamics and observed
experimentally.Comment: 4 pages, RevTeX. Changes: correct minor errors, add reference, minor
rewriting requested by editors and refere
Numerical study of a non-equilibrium interface model
We have carried out extensive computer simulations of one-dimensional models
related to the low noise (solid-on-solid) non-equilibrium interface of a two
dimensional anchored Toom model with unbiased and biased noise. For the
unbiased case the computed fluctuations of the interface in this limit provide
new numerical evidence for the logarithmic correction to the subnormal L^(1/2)
variance which was predicted by the dynamic renormalization group calculations
on the modified Edwards-Wilkinson equation. In the biased case the simulations
are in close quantitative agreement with the predictions of the Collective
Variable Approximation (CVA), which gives the same L^(2/3) behavior of the
variance as the KPZ equation.Comment: 15 pages revtex, 4 Postscript Figure
Shock Profiles for the Asymmetric Simple Exclusion Process in One Dimension
The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice
is a system of particles which jump at rates and (here ) to
adjacent empty sites on their right and left respectively. The system is
described on suitable macroscopic spatial and temporal scales by the inviscid
Burgers' equation; the latter has shock solutions with a discontinuous jump
from left density to right density , , which
travel with velocity . In the microscopic system we
may track the shock position by introducing a second class particle, which is
attracted to and travels with the shock. In this paper we obtain the time
invariant measure for this shock solution in the ASEP, as seen from such a
particle. The mean density at lattice site , measured from this particle,
approaches at an exponential rate as , with a
characteristic length which becomes independent of when
. For a special value of the
asymmetry, given by , the measure is
Bernoulli, with density on the left and on the right. In the
weakly asymmetric limit, , the microscopic width of the shock
diverges as . The stationary measure is then essentially a
superposition of Bernoulli measures, corresponding to a convolution of a
density profile described by the viscous Burgers equation with a well-defined
distribution for the location of the second class particle.Comment: 34 pages, LaTeX, 2 figures are included in the LaTeX file. Email:
[email protected], [email protected], [email protected]
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