45 research outputs found
The asymmetric Exclusion Process and Brownian Excursions
We consider the totally asymmetric exclusion process (TASEP) in one dimension
in its maximal current phase. We show, by an exact calculation, that the
non-Gaussian part of the fluctuations of density can be described in terms of
the statistical properties of a Brownian excursion. Numerical simulations
indicate that the description in terms of a Brownian excursion remains valid
for more general one dimensional driven systems in their maximal current phase.Comment: 23 pages, 1 figure, in latex, e-mail addresses: [email protected],
[email protected], [email protected]
Large deviation functional of the weakly asymmetric exclusion process
We obtain the large deviation functional of a density profile for the
asymmetric exclusion process of L sites with open boundary conditions when the
asymmetry scales like 1/L. We recover as limiting cases the expressions derived
recently for the symmetric (SSEP) and the asymmetric (ASEP) cases. In the ASEP
limit, the non linear differential equation one needs to solve can be analysed
by a method which resembles the WKB method
Fluctuations in the weakly asymmetric exclusion process with open boundary conditions
accepted in Journal of Statistical PhysicsWe investigate the fluctuations around the average density profile in the weakly asymmetric exclusion process with open boundaries in the steady state. We show that these fluctuations are given, in the macroscopic limit, by a centered Gaussian field and we compute explicitly its covariance function. We use two approaches. The first method is dynamical and based on fluctuations around the hydrodynamic limit. We prove that the density fluctuations evolve macroscopically according to an autonomous stochastic equation, and we search for the stationary distribution of this evolution. The second approach, which is based on a representation of the steady state as a sum over paths, allows one to write the density fluctuations in the steady state as a sum over two independent processes, one of which is the derivative of a Brownian motion, the other one being related to a random path in a potential
Sample-Dependent Phase Transitions in Disordered Exclusion Models
We give numerical evidence that the location of the first order phase
transition between the low and the high density phases of the one dimensional
asymmetric simple exclusion process with open boundaries becomes sample
dependent when quenched disorder is introduced for the hopping rates.Comment: accepted in Europhysics Letter
Single-Bottleneck Approximation for Driven Lattice Gases with Disorder and Open Boundary Conditions
We investigate the effects of disorder on driven lattice gases with open
boundaries using the totally asymmetric simple exclusion process as a
paradigmatic example. Disorder is realized by randomly distributed defect sites
with reduced hopping rate. In contrast to equilibrium, even macroscopic
quantities in disordered non-equilibrium systems depend sensitively on the
defect sample. We study the current as function of the entry and exit rates and
the realization of disorder and find that it is, in leading order, determined
by the longest stretch of consecutive defect sites (single-bottleneck
approximation, SBA). Using results from extreme value statistics the SBA allows
to study ensembles with fixed defect density which gives accurate results, e.g.
for the expectation value of the current. Corrections to SBA come from
effective interactions of bottlenecks close to the longest one. Defects close
to the boundaries can be described by effective boundary rates and lead to
shifts of the phase transitions. Finally it is shown that the SBA also works
for more complex models. As an example we discuss a model with internal states
that has been proposed to describe transport of the kinesin KIF1A.Comment: submitted to J. Stat. Mec
Nonequilibrium stationary states and equilibrium models with long range interactions
It was recently suggested by Blythe and Evans that a properly defined steady
state normalisation factor can be seen as a partition function of a fictitious
statistical ensemble in which the transition rates of the stochastic process
play the role of fugacities. In analogy with the Lee-Yang description of phase
transition of equilibrium systems, they studied the zeroes in the complex plane
of the normalisation factor in order to find phase transitions in
nonequilibrium steady states. We show that like for equilibrium systems, the
``densities'' associated to the rates are non-decreasing functions of the rates
and therefore one can obtain the location and nature of phase transitions
directly from the analytical properties of the ``densities''. We illustrate
this phenomenon for the asymmetric exclusion process. We actually show that its
normalisation factor coincides with an equilibrium partition function of a walk
model in which the ``densities'' have a simple physical interpretation.Comment: LaTeX, 23 pages, 3 EPS figure
Strong asymmetric limit of the quasi-potential of the boundary driven weakly asymmetric exclusion process
We consider the weakly asymmetric exclusion process on a bounded interval
with particles reservoirs at the endpoints. The hydrodynamic limit for the
empirical density, obtained in the diffusive scaling, is given by the viscous
Burgers equation with Dirichlet boundary conditions. In the case in which the
bulk asymmetry is in the same direction as the drift due to the boundary
reservoirs, we prove that the quasi-potential can be expressed in terms of the
solution to a one-dimensional boundary value problem which has been introduced
by Enaud and Derrida \cite{de}. We consider the strong asymmetric limit of the
quasi-potential and recover the functional derived by Derrida, Lebowitz, and
Speer \cite{DLS3} for the asymmetric exclusion process
Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide
We consider the general problem of determining the steady state of stochastic
nonequilibrium systems such as those that have been used to model (among other
things) biological transport and traffic flow. We begin with a broad overview
of this class of driven diffusive systems - which includes exclusion processes
- focusing on interesting physical properties, such as shocks and phase
transitions. We then turn our attention specifically to those models for which
the exact distribution of microstates in the steady state can be expressed in a
matrix product form. In addition to a gentle introduction to this matrix
product approach, how it works and how it relates to similar constructions that
arise in other physical contexts, we present a unified, pedagogical account of
the various means by which the statistical mechanical calculations of
macroscopic physical quantities are actually performed. We also review a number
of more advanced topics, including nonequilibrium free energy functionals, the
classification of exclusion processes involving multiple particle species,
existence proofs of a matrix product state for a given model and more
complicated variants of the matrix product state that allow various types of
parallel dynamics to be handled. We conclude with a brief discussion of open
problems for future research.Comment: 127 pages, 31 figures, invited topical review for J. Phys. A (uses
IOP class file