1,145 research outputs found
The Lee-Yang theory of equilibrium and nonequilibrium phase transitions
We present a pedagogical account of the Lee-Yang theory of equilibrium phase
transitions and review recent advances in applying this theory to
nonequilibrium systems. Through both general considerations and explicit
studies of specific models, we show that the Lee-Yang approach can be used to
locate and classify phase transitions in nonequilibrium steady states.Comment: 24 pages, 7 papers, invited paper for special issue of The Brazilian
Journal of Physic
Discontinuous Transition in a Boundary Driven Contact Process
The contact process is a stochastic process which exhibits a continuous,
absorbing-state phase transition in the Directed Percolation (DP) universality
class. In this work, we consider a contact process with a bias in conjunction
with an active wall. This model exhibits waves of activity emanating from the
active wall and, when the system is supercritical, propagating indefinitely as
travelling (Fisher) waves. In the subcritical phase the activity is localised
near the wall. We study the phase transition numerically and show that certain
properties of the system, notably the wave velocity, are discontinuous across
the transition. Using a modified Fisher equation to model the system we
elucidate the mechanism by which the the discontinuity arises. Furthermore we
establish relations between properties of the travelling wave and DP critical
exponents.Comment: 14 pages, 9 figure
Exact Solution of the Multi-Allelic Diffusion Model
We give an exact solution to the Kolmogorov equation describing genetic drift
for an arbitrary number of alleles at a given locus. This is achieved by
finding a change of variable which makes the equation separable, and therefore
reduces the problem with an arbitrary number of alleles to the solution of a
set of equations that are essentially no more complicated than that found in
the two-allele case. The same change of variable also renders the Kolmogorov
equation with the effect of mutations added separable, as long as the mutation
matrix has equal entries in each row. Thus this case can also be solved exactly
for an arbitrary number of alleles. The general solution, which is in the form
of a probability distribution, is in agreement with the previously known
results--which were for the cases of two and three alleles only. Results are
also given for a wide range of other quantities of interest, such as the
probabilities of extinction of various numbers of alleles, mean times to these
extinctions, and the means and variances of the allele frequencies. To aid
dissemination, these results are presented in two stages: first of all they are
given without derivations and too much mathematical detail, and then
subsequently derivations and a more technical discussion are provided.Comment: 56 pages. 15 figures. Requires Elsevier document clas
Spatial fluctuations of a surviving particle in the trapping reaction
We consider the trapping reaction, , where and particles
have a diffusive dynamics characterized by diffusion constants and .
The interaction with particles can be formally incorporated in an effective
dynamics for one particle as was recently shown by Bray {\it et al}. [Phys.
Rev. E {\bf 67}, 060102 (2003)]. We use this method to compute, in space
dimension , the asymptotic behaviour of the spatial fluctuation,
, for a surviving particle in the perturbative regime,
, for the case of an initially uniform distribution of
particles. We show that, for , with
. By contrast, the fluctuations of paths constrained to return to
their starting point at time grow with the larger exponent 1/3. Numerical
tests are consistent with these predictions.Comment: 10 pages, 5 figure
Relaxation time in a non-conserving driven-diffusive system with parallel dynamics
We introduce a two-state non-conserving driven-diffusive system in
one-dimension under a discrete-time updating scheme. We show that the
steady-state of the system can be obtained using a matrix product approach. On
the other hand, the steady-state of the system can be expressed in terms of a
linear superposition Bernoulli shock measures with random walk dynamics. The
dynamics of a shock position is studied in detail. The spectrum of the transfer
matrix and the relaxation times to the steady-state have also been studied in
the large-system-size limit.Comment: 10 page
Dyck Paths, Motzkin Paths and Traffic Jams
It has recently been observed that the normalization of a one-dimensional
out-of-equilibrium model, the Asymmetric Exclusion Process (ASEP) with random
sequential dynamics, is exactly equivalent to the partition function of a
two-dimensional lattice path model of one-transit walks, or equivalently Dyck
paths. This explains the applicability of the Lee-Yang theory of partition
function zeros to the ASEP normalization.
In this paper we consider the exact solution of the parallel-update ASEP, a
special case of the Nagel-Schreckenberg model for traffic flow, in which the
ASEP phase transitions can be intepreted as jamming transitions, and find that
Lee-Yang theory still applies. We show that the parallel-update ASEP
normalization can be expressed as one of several equivalent two-dimensional
lattice path problems involving weighted Dyck or Motzkin paths. We introduce
the notion of thermodynamic equivalence for such paths and show that the
robustness of the general form of the ASEP phase diagram under various update
dynamics is a consequence of this thermodynamic equivalence.Comment: Version accepted for publicatio
The Grand-Canonical Asymmetric Exclusion Process and the One-Transit Walk
The one-dimensional Asymmetric Exclusion Process (ASEP) is a paradigm for
nonequilibrium dynamics, in particular driven diffusive processes. It is
usually considered in a canonical ensemble in which the number of sites is
fixed. We observe that the grand-canonical partition function for the ASEP is
remarkably simple. It allows a simple direct derivation of the asymptotics of
the canonical normalization in various phases and of the correspondence with
One-Transit Walks recently observed by Brak et.al.Comment: Published versio
An introduction to phase transitions in stochastic dynamical systems
We give an introduction to phase transitions in the steady states of systems
that evolve stochastically with equilibrium and nonequilibrium dynamics, the
latter defined as those that do not possess a time-reversal symmetry. We try as
much as possible to discuss both cases within the same conceptual framework,
focussing on dynamically attractive `peaks' in state space. A quantitative
characterisation of these peaks leads to expressions for the partition function
and free energy that extend from equilibrium steady states to their
nonequilibrium counterparts. We show that for certain classes of nonequilibrium
systems that have been exactly solved, these expressions provide precise
predictions of their macroscopic phase behaviour.Comment: Pedagogical talk contributed to the "Ageing and the Glass Transition"
Summer School, Luxembourg, September 2005. 12 pages, 8 figures, uses the IOP
'jpconf' document clas
Noise-induced dynamical transition in systems with symmetric absorbing states
We investigate the effect of noise strength on the macroscopic ordering
dynamics of systems with symmetric absorbing states. Using an explicit
stochastic microscopic model, we present evidence for a phase transition in the
coarsening dynamics, from an Ising-like to a voter-like behavior, as the noise
strength is increased past a nontrivial critical value. By mapping to a thermal
diffusion process, we argue that the transition arises due to locally-absorbing
states being entered more readily in the high-noise regime, which in turn
prevents surface tension from driving the ordering process.Comment: v2 with improved introduction and figures, to appear in PRL. 4 pages,
4 figure
Nonequilibrium Stationary States and Phase Transitions in Directed Ising Models
We study the nonequilibrium properties of directed Ising models with non
conserved dynamics, in which each spin is influenced by only a subset of its
nearest neighbours. We treat the following models: (i) the one-dimensional
chain; (ii) the two-dimensional square lattice; (iii) the two-dimensional
triangular lattice; (iv) the three-dimensional cubic lattice. We raise and
answer the question: (a) Under what conditions is the stationary state
described by the equilibrium Boltzmann-Gibbs distribution? We show that for
models (i), (ii), and (iii), in which each spin "sees" only half of its
neighbours, there is a unique set of transition rates, namely with exponential
dependence in the local field, for which this is the case. For model (iv), we
find that any rates satisfying the constraints required for the stationary
measure to be Gibbsian should satisfy detailed balance, ruling out the
possibility of directed dynamics. We finally show that directed models on
lattices of coordination number with exponential rates cannot
accommodate a Gibbsian stationary state. We conjecture that this property
extends to any form of the rates. We are thus led to the conclusion that
directed models with Gibbsian stationary states only exist in dimension one and
two. We then raise the question: (b) Do directed Ising models, augmented by
Glauber dynamics, exhibit a phase transition to a ferromagnetic state? For the
models considered above, the answers are open problems, to the exception of the
simple cases (i) and (ii). For Cayley trees, where each spin sees only the
spins further from the root, we show that there is a phase transition provided
the branching ratio, , satisfies
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