502 research outputs found
Free energy landscapes, dynamics and the edge of chaos in mean-field models of spin glasses
Metastable states in Ising spin-glass models are studied by finding iterative
solutions of mean-field equations for the local magnetizations. Two different
equations are studied: the TAP equations which are exact for the SK model, and
the simpler `naive-mean-field' (NMF) equations. The free-energy landscapes that
emerge are very different. For the TAP equations, the numerical studies confirm
the analytical results of Aspelmeier et al., which predict that TAP states
consist of close pairs of minima and index-one (one unstable direction) saddle
points, while for the NMF equations saddle points with large indices are found.
For TAP the barrier height between a minimum and its nearby saddle point scales
as (f-f_0)^{-1/3} where f is the free energy per spin of the solution and f_0
is the equilibrium free energy per spin. This means that for `pure states', for
which f-f_0 is of order 1/N, the barriers scale as N^{1/3}, but between states
for which f-f_0 is of order one the barriers are finite and also small so such
metastable states will be of limited physical significance. For the NMF
equations there are saddles of index K and we can demonstrate that their
complexity Sigma_K scales as a function of K/N. We have also employed an
iterative scheme with a free parameter that can be adjusted to bring the system
of equations close to the `edge of chaos'. Both for the TAP and NME equations
it is possible with this approach to find metastable states whose free energy
per spin is close to f_0. As N increases, it becomes harder and harder to find
solutions near the edge of chaos, but nevertheless the results which can be
obtained are competitive with those achieved by more time-consuming computing
methods and suggest that this method may be of general utility.Comment: 13 page
Spatiotemporally Complete Condensation in a Non-Poissonian Exclusion Process
We investigate a non-Poissonian version of the asymmetric simple exclusion
process, motivated by the observation that coarse-graining the interactions
between particles in complex systems generically leads to a stochastic process
with a non-Markovian (history-dependent) character. We characterize a large
family of one-dimensional hopping processes using a waiting-time distribution
for individual particle hops. We find that when its variance is infinite, a
real-space condensate forms that is complete in space (involves all particles)
and time (exists at almost any given instant) in the thermodynamic limit. The
mechanism for the onset and stability of the condensate are both rather subtle,
and depends on the microscopic dynamics subsequent to a failed particle hop
attempts.Comment: 5 pages, 5 figures. Version 2 to appear in PR
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
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
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
Discontinuous Phase Transition in an Exactly Solvable One-Dimensional Creation-Annihilation System
An exactly solvable reaction-diffusion model consisting of first-class
particles in the presence of a single second-class particle is introduced on a
one-dimensional lattice with periodic boundary condition. The number of
first-class particles can be changed due to creation and annihilation
reactions. It is shown that the system undergoes a discontinuous phase
transition in contrast to the case where the density of the second-class
particles is finite and the phase transition is continuous.Comment: Revised, 8 pages, 1 EPS figure. Accepted for publication in Journal
of Statistical Mechanics: theory and experimen
Perturbation theory for the one-dimensional trapping reaction
We consider the survival probability of a particle in the presence of a
finite number of diffusing traps in one dimension. Since the general solution
for this quantity is not known when the number of traps is greater than two, we
devise a perturbation series expansion in the diffusion constant of the
particle. We calculate the persistence exponent associated with the particle's
survival probability to second order and find that it is characterised by the
asymmetry in the number of traps initially positioned on each side of the
particle.Comment: 18 pages, no figures. Uses IOP Latex clas
Continued Fractions and the Partially Asymmetric Exclusion Process
We note that a tridiagonal matrix representation of the algebra of the
partially asymmetric exclusion process (PASEP) lends itself to interpretation
as the transfer matrix for weighted Motzkin lattice paths. A continued fraction
("J-Fraction") representation of the lattice path generating function is
particularly well suited to discussing the PASEP, for which the paths have
height dependent weights. We show that this not only allows a succinct
derivation of the normalisation and correlation lengths of the PASEP, but also
reveals how finite-dimensional representations of the PASEP algebra, valid only
along special lines in the phase diagram, relate to the general solution that
requires an infinite-dimensional representation
Ordering in voter models on networks: Exact reduction to a single-coordinate diffusion
We study the voter model and related random-copying processes on arbitrarily
complex network structures. Through a representation of the dynamics as a
particle reaction process, we show that a quantity measuring the degree of
order in a finite system is, under certain conditions, exactly governed by a
universal diffusion equation. Whenever this reduction occurs, the details of
the network structure and random-copying process affect only a single parameter
in the diffusion equation. The validity of the reduction can be established
with considerably less information than one might expect: it suffices to know
just two characteristic timescales within the dynamics of a single pair of
reacting particles. We develop methods to identify these timescales, and apply
them to deterministic and random network structures. We focus in particular on
how the ordering time is affected by degree correlations, since such effects
are hard to access by existing theoretical approaches.Comment: 37 pages, 10 figures. Revised version with additional discussion and
simulation results to appear in J Phys
Dynamical Transition in the Open-boundary Totally Asymmetric Exclusion Process
We revisit the totally asymmetric simple exclusion process with open
boundaries (TASEP), focussing on the recent discovery by de Gier and Essler
that the model has a dynamical transition along a nontrivial line in the phase
diagram. This line coincides neither with any change in the steady-state
properties of the TASEP, nor the corresponding line predicted by domain wall
theory. We provide numerical evidence that the TASEP indeed has a dynamical
transition along the de Gier-Essler line, finding that the most convincing
evidence was obtained from Density Matrix Renormalisation Group (DMRG)
calculations. By contrast, we find that the dynamical transition is rather hard
to see in direct Monte Carlo simulations of the TASEP. We furthermore discuss
in general terms scenarios that admit a distinction between static and dynamic
phase behaviour.Comment: 27 pages, 18 figures. v2 to appear in J Phys A features minor
corrections and better-quality figure
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