7,187 research outputs found
Some Exact Results for the Exclusion Process
The asymmetric simple exclusion process (ASEP) is a paradigm for
non-equilibrium physics that appears as a building block to model various
low-dimensional transport phenomena, ranging from intracellular traffic to
quantum dots. We review some recent results obtained for the system on a
periodic ring by using the Bethe Ansatz. We show that this method allows to
derive analytically many properties of the dynamics of the model such as the
spectral gap and the generating function of the current. We also discuss the
solution of a generalized exclusion process with -species of particles and
explain how a geometric construction inspired from queuing theory sheds light
on the Matrix Product Representation technique that has been very fruitful to
derive exact results for the ASEP.Comment: 21 pages; Proceedings of STATPHYS24 (Cairns, Australia, July 2010
Nonequilibrium stationary states with Gibbs measure for two or three species of interacting particles
We construct explicit examples of one-dimensional driven diffusive systems
for two and three species of interacting particles, defined by asymmetric
dynamical rules which do not obey detailed balance, but whose nonequilibrium
stationary-state measure coincides with a prescribed equilibrium Gibbs measure.
For simplicity, the measures considered in this construction only involve
nearest-neighbor interactions. For two species, the dynamics thus obtained
generically has five free parameters, and does not obey pairwise balance in
general. The latter property is satisfied only by the totally asymmetric
dynamics and the partially asymmetric dynamics with uniform bias, i.e., the
cases originally considered by Katz, Lebowitz, and Spohn. For three species of
interacting particles, with nearest-neighbor interactions between particles of
the same species, the totally asymmetric dynamics thus obtained has two free
parameters, and obeys pairwise balance. These models are put in perspective
with other examples of driven diffusive systems. The emerging picture is that
asymmetric (nonequilibrium) stochastic dynamics leading to a given
stationary-state measure are far more constrained (in terms of numbers of free
parameters) than the corresponding symmetric (equilibrium) dynamics.Comment: 18 pages, 8 tables, 1 figure. Stylistic and other minor improvement
From interacting particle systems to random matrices
In this contribution we consider stochastic growth models in the
Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large
time distribution and processes and their dependence on the class on initial
condition. This means that the scaling exponents do not uniquely determine the
large time surface statistics, but one has to further divide into subclasses.
Some of the fluctuation laws were first discovered in random matrix models.
Moreover, the limit process for curved limit shape turned out to show up in a
dynamical version of hermitian random matrices, but this analogy does not
extend to the case of symmetric matrices. Therefore the connections between
growth models and random matrices is only partial.Comment: 18 pages, 8 figures; Contribution to StatPhys24 special issue; minor
corrections in scaling of section 2.
Open two-species exclusion processes with integrable boundaries
We give a complete classification of integrable Markovian boundary conditions
for the asymmetric simple exclusion process with two species (or classes) of
particles. Some of these boundary conditions lead to non-vanishing particle
currents for each species. We explain how the stationary state of all these
models can be expressed in a matrix product form, starting from two key
components, the Zamolodchikov-Faddeev and Ghoshal-Zamolodchikov relations. This
statement is illustrated by studying in detail a specific example, for which
the matrix Ansatz (involving 9 generators) is explicitly constructed and
physical observables (such as currents, densities) calculated.Comment: 19 pages; typos corrected, more details on the Matrix Ansatz algebr
TASEP hydrodynamics using microscopic characteristics
The convergence of the totally asymmetric simple exclusion process to the
solution of the Burgers equation is a classical result. In his seminal 1981
paper, Herman Rost proved the convergence of the density fields and local
equilibrium when the limiting solution of the equation is a rarefaction fan. An
important tool of his proof is the subadditive ergodic theorem. We prove his
results by showing how second class particles transport the rarefaction-fan
solution, as characteristics do for the Burgers equation, avoiding
subadditivity. In the way we show laws of large numbers for tagged particles,
fluxes and second class particles, and simplify existing proofs in the shock
cases. The presentation is self contained.Comment: 20 pages, 13 figures. This version is accepted for publication in
Probability Surveys, February 20 201
Exact connections between current fluctuations and the second class particle in a class of deposition models
We consider a large class of nearest neighbor attractive stochastic
interacting systems that includes the asymmetric simple exclusion, zero range,
bricklayers' and the symmetric K-exclusion processes. We provide exact formulas
that connect particle flux (or surface growth) fluctuations to the two-point
function of the process and to the motion of the second class particle. Such
connections have only been available for simple exclusion where they were of
great use in particle current fluctuation investigations.Comment: Second version, results a bit more clear; 23 page
Stochastic interacting particle systems out of equilibrium
This paper provides an introduction to some stochastic models of lattice
gases out of equilibrium and a discussion of results of various kinds obtained
in recent years. Although these models are different in their microscopic
features, a unified picture is emerging at the macroscopic level, applicable,
in our view, to real phenomena where diffusion is the dominating physical
mechanism. We rely mainly on an approach developed by the authors based on the
study of dynamical large fluctuations in stationary states of open systems. The
outcome of this approach is a theory connecting the non equilibrium
thermodynamics to the transport coefficients via a variational principle. This
leads ultimately to a functional derivative equation of Hamilton-Jacobi type
for the non equilibrium free energy in which local thermodynamic variables are
the independent arguments. In the first part of the paper we give a detailed
introduction to the microscopic dynamics considered, while the second part,
devoted to the macroscopic properties, illustrates many consequences of the
Hamilton-Jacobi equation. In both parts several novelties are included.Comment: 36 page
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