47 research outputs found
Convergence in distribution for subcritical 2D oriented percolation seen from its rightmost point
We study subcritical two-dimensional oriented percolation seen from its
rightmost point on the set of infinite configurations which are bounded above.
This a Feller process whose state space is not compact and has no invariant
measures. We prove that it converges in distribution to a measure which charges
only finite configurations.Comment: Published in at http://dx.doi.org/10.1214/13-AOP841 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Extreme paths in oriented 2D Percolation
A useful result about leftmost and rightmost paths in two dimensional bond
percolation is proved. This result was introduced without proof in \cite{G} in
the context of the contact process in continuous time. As discussed here, it
also holds for several related models, including the discrete time contact
process and two dimensional site percolation. Among the consequences are a
natural monotonicity in the probability of percolation between different sites
and a somewhat counter-intuitive correlation inequality
Product Measure Steady States of Generalized Zero Range Processes
We establish necessary and sufficient conditions for the existence of
factorizable steady states of the Generalized Zero Range Process. This process
allows transitions from a site to a site involving multiple particles
with rates depending on the content of the site , the direction of
movement, and the number of particles moving. We also show the sufficiency of a
similar condition for the continuous time Mass Transport Process, where the
mass at each site and the amount transferred in each transition are continuous
variables; we conjecture that this is also a necessary condition.Comment: 9 pages, LaTeX with IOP style files. v2 has minor corrections; v3 has
been rewritten for greater clarit
Nonequilibrium phase transition in a non integrable zero-range process
The present work is an endeavour to determine analytically features of the
stationary measure of a non-integrable zero-range process, and to investigate
the possible existence of phase transitions for such a nonequilibrium model.
The rates defining the model do not satisfy the constraints necessary for the
stationary measure to be a product measure. Even in the absence of a drive,
detailed balance with respect to this measure is violated. Analytical and
numerical investigations on the complete graph demonstrate the existence of a
first-order phase transition between a fluid phase and a condensed phase, where
a single site has macroscopic occupation. The transition is sudden from an
imbalanced fluid where both species have densities larger than the critical
density, to a critical neutral fluid and an imbalanced condensate
Spreading in narrow channels
We study a lattice model for the spreading of fluid films, which are a few
molecular layers thick, in narrow channels with inert lateral walls. We focus
on systems connected to two particle reservoirs at different chemical
potentials, considering an attractive substrate potential at the bottom,
confining side walls, and hard-core repulsive fluid-fluid interactions. Using
kinetic Monte Carlo simulations we find a diffusive behavior. The corresponding
diffusion coefficient depends on the density and is bounded from below by the
free one-dimensional diffusion coefficient, valid for an inert bottom wall.
These numerical results are rationalized within the corresponding continuum
limit.Comment: 16 pages, 10 figure
Structure of the stationary state of the asymmetric target process
We introduce a novel migration process, the target process. This process is
dual to the zero-range process (ZRP) in the sense that, while for the ZRP the
rate of transfer of a particle only depends on the occupation of the departure
site, it only depends on the occupation of the arrival site for the target
process. More precisely, duality associates to a given ZRP a unique target
process, and vice-versa. If the dynamics is symmetric, i.e., in the absence of
a bias, both processes have the same stationary-state product measure. In this
work we focus our interest on the situation where the latter measure exhibits a
continuous condensation transition at some finite critical density ,
irrespective of the dimensionality. The novelty comes from the case of
asymmetric dynamics, where the target process has a nontrivial fluctuating
stationary state, whose characteristics depend on the dimensionality. In one
dimension, the system remains homogeneous at any finite density. An alternating
scenario however prevails in the high-density regime: typical configurations
consist of long alternating sequences of highly occupied and less occupied
sites. The local density of the latter is equal to and their
occupation distribution is critical. In dimension two and above, the asymmetric
target process exhibits a phase transition at a threshold density much
larger than . The system is homogeneous at any density below ,
whereas for higher densities it exhibits an extended condensate elongated along
the direction of the mean current, on top of a critical background with density
.Comment: 30 pages, 16 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]
Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
We review recent progress on the zero-range process, a model of interacting
particles which hop between the sites of a lattice with rates that depend on
the occupancy of the departure site. We discuss several applications which have
stimulated interest in the model such as shaken granular gases and network
dynamics, also we discuss how the model may be used as a coarse-grained
description of driven phase-separating systems. A useful property of the
zero-range process is that the steady state has a factorised form. We show how
this form enables one to analyse in detail condensation transitions, wherein a
finite fraction of particles accumulate at a single site. We review
condensation transitions in homogeneous and heterogeneous systems and also
summarise recent progress in understanding the dynamics of condensation. We
then turn to several generalisations which also, under certain specified
conditions, share the property of a factorised steady state. These include
several species of particles; hop rates which depend on both the departure and
the destination sites; continuous masses; parallel discrete-time updating;
non-conservation of particles and sites.Comment: 54 pages, 9 figures, review articl
Zero-range process with long-range interactions at a T-junction
A generalized zero-range process with a limited number of long-range
interactions is studied as an example of a transport process in which particles
at a T-junction make a choice of which branch to take based on traffic levels
on each branch. The system is analysed with a self-consistent mean-field
approximation which allows phase diagrams to be constructed. Agreement between
the analysis and simulations is found to be very good.Comment: 21 pages, 6 figure
Spectra of non-hermitian quantum spin chains describing boundary induced phase transitions
The spectrum of the non-hermitian asymmetric XXZ-chain with additional
non-diagonal boundary terms is studied. The lowest lying eigenvalues are
determined numerically. For the ferromagnetic and completely asymmetric chain
that corresponds to a reaction-diffusion model with input and outflow of
particles the smallest energy gap which corresponds directly to the inverse of
the temporal correlation length shows the same properties as the spatial
correlation length of the stationary state. For the antiferromagnetic chain
with both boundary terms, we find a conformal invariant spectrum where the
partition function corresponds to the one of a Coulomb gas with only magnetic
charges shifted by a purely imaginary and a lattice-length dependent constant.
Similar results are obtained by studying a toy model that can be diagonalized
analytically in terms of free fermions.Comment: LaTeX, 26 pages, 1 figure, uses ioplppt.st