168 research outputs found
Metastability in Markov processes
We present a formalism to describe slowly decaying systems in the context of
finite Markov chains obeying detailed balance. We show that phase space can be
partitioned into approximately decoupled regions, in which one may introduce
restricted Markov chains which are close to the original process but do not
leave these regions. Within this context, we identify the conditions under
which the decaying system can be considered to be in a metastable state.
Furthermore, we show that such metastable states can be described in
thermodynamic terms and define their free energy. This is accomplished showing
that the probability distribution describing the metastable state is indeed
proportional to the equilibrium distribution, as is commonly assumed. We test
the formalism numerically in the case of the two-dimensional kinetic Ising
model, using the Wang--Landau algorithm to show this proportionality
explicitly, and confirm that the proportionality constant is as derived in the
theory. Finally, we extend the formalism to situations in which a system can
have several metastable states.Comment: 30 pages, 5 figures; version with one higher quality figure available
at http://www.fis.unam.mx/~dsanders
Refined Simulations of the Reaction Front for Diffusion-Limited Two-Species Annihilation in One Dimension
Extensive simulations are performed of the diffusion-limited reaction
AB in one dimension, with initially separated reagents. The reaction
rate profile, and the probability distributions of the separation and midpoint
of the nearest-neighbour pair of A and B particles, are all shown to exhibit
dynamic scaling, independently of the presence of fluctuations in the initial
state and of an exclusion principle in the model. The data is consistent with
all lengthscales behaving as as . Evidence of
multiscaling, found by other authors, is discussed in the light of these
findings.Comment: Resubmitted as TeX rather than Postscript file. RevTeX version 3.0,
10 pages with 16 Encapsulated Postscript figures (need epsf). University of
Geneva preprint UGVA/DPT 1994/10-85
Occurrence of normal and anomalous diffusion in polygonal billiard channels
From extensive numerical simulations, we find that periodic polygonal
billiard channels with angles which are irrational multiples of pi generically
exhibit normal diffusion (linear growth of the mean squared displacement) when
they have a finite horizon, i.e. when no particle can travel arbitrarily far
without colliding. For the infinite horizon case we present numerical tests
showing that the mean squared displacement instead grows asymptotically as t
log t. When the unit cell contains accessible parallel scatterers, however, we
always find anomalous super-diffusion, i.e. power-law growth with an exponent
larger than 1. This behavior cannot be accounted for quantitatively by a simple
continuous-time random walk model. Instead, we argue that anomalous diffusion
correlates with the existence of families of propagating periodic orbits.
Finally we show that when a configuration with parallel scatterers is
approached there is a crossover from normal to anomalous diffusion, with the
diffusion coefficient exhibiting a power-law divergence.Comment: 9 pages, 15 figures. Revised after referee reports: redrawn figures,
additional comments. Some higher quality figures available at
http://www.fis.unam.mx/~dsander
Static Pairwise Annihilation in Complex Networks
We study static annihilation on complex networks, in which pairs of connected
particles annihilate at a constant rate during time. Through a mean-field
formalism, we compute the temporal evolution of the distribution of surviving
sites with an arbitrary number of connections. This general formalism, which is
exact for disordered networks, is applied to Kronecker, Erd\"os-R\'enyi (i.e.
Poisson) and scale-free networks. We compare our theoretical results with
extensive numerical simulations obtaining excellent agreement. Although the
mean-field approach applies in an exact way neither to ordered lattices nor to
small-world networks, it qualitatively describes the annihilation dynamics in
such structures. Our results indicate that the higher the connectivity of a
given network element, the faster it annihilates. This fact has dramatic
consequences in scale-free networks, for which, once the ``hubs'' have been
annihilated, the network disintegrates and only isolated sites are left.Comment: 7 Figures, 10 page
The Reaction-Diffusion Front for in One Dimension
We study theoretically and numerically the steady state diffusion controlled
reaction , where currents of and particles
are applied at opposite boundaries. For a reaction rate , and equal
diffusion constants , we find that when the
reaction front is well described by mean field theory. However, for , the front acquires a Gaussian profile - a result of
noise induced wandering of the reaction front center. We make a theoretical
prediction for this profile which is in good agreement with simulation.
Finally, we investigate the intrinsic (non-wandering) front width and find
results consistent with scaling and field theoretic predictions.Comment: 11 pages, revtex, 4 separate PostScript figure
Diffusion-Limited Annihilation with Initially Separated Reactants
A diffusion-limited annihilation process, A+B->0, with species initially
separated in space is investigated. A heuristic argument suggests the form of
the reaction rate in dimensions less or equal to the upper critical dimension
. Using this reaction rate we find that the width of the reaction front
grows as in one dimension and as in two
dimensions.Comment: 9 pages, Plain Te
Localisation Transition of A Dynamic Reaction Front
We study the reaction-diffusion process with injection of
each species at opposite boundaries of a one-dimensional lattice and bulk
driving of each species in opposing directions with a hardcore interaction. The
system shows the novel feature of phase transitions between localised and
delocalised reaction zones as the injection rate or reaction rate is varied. An
approximate analytical form for the phase diagram is derived by relating both
the domain of reactants and the domain of reactants to asymmetric
exclusion processes with open boundaries, a system for which the phase diagram
is known exactly, giving rise to three phases. The reaction zone width is
described by a finite size scaling form relating the early time growth,
relaxation time and saturation width exponents. In each phase the exponents are
distinct from the previously studied case where the reactants diffuse
isotropically.Comment: 13 pages, latex, uses eps
Memory Effects in Nonequilibrium Transport for Deterministic Hamiltonian Systems
We consider nonequilibrium transport in a simple chain of identical
mechanical cells in which particles move around. In each cell, there is a
rotating disc, with which these particles interact, and this is the only
interaction in the model. It was shown in \cite{eckmann-young} that when the
cells are weakly coupled, to a good approximation, the jump rates of particles
and the energy-exchange rates from cell to cell follow linear profiles. Here,
we refine that study by analyzing higher-order effects which are induced by the
presence of external gradients for situations in which memory effects, typical
of Hamiltonian dynamics, cannot be neglected. For the steady state we propose a
set of balance equations for the particle number and energy in terms of the
reflection probabilities of the cell and solve it phenomenologically. Using
this approximate theory we explain how these asymmetries affect various aspects
of heat and particle transport in systems of the general type described above
and obtain in the infinite volume limit the deviation from the theory in
\cite{eckmann-young} to first-order. We verify our assumptions with extensive
numerical simulations.Comment: Several change
Scaling exponents for fracture surfaces in homogenous glass and glassy ceramics
We investigate the scaling properties of post-mortem fracture surfaces in
silica glass and glassy ceramics. In both cases, the 2D height-height
correlation function is found to obey Family-Viseck scaling properties, but
with two sets of critical exponents, in particular a roughness exponent
in homogeneous glass and in glassy
ceramics. The ranges of length-scales over which these two scalings are
observed are shown to be below and above the size of process zone respectively.
A model derived from Linear Elastic Fracture Mechanics (LEFM) in the
quasistatic approximation succeeds to reproduce the scaling exponents observed
in glassy ceramics. The critical exponents observed in homogeneous glass are
conjectured to reflect damage screening occurring for length-scales below the
size of the process zone
Order statistics for d-dimensional diffusion processes
We present results for the ordered sequence of first passage times of arrival
of N random walkers at a boundary in Euclidean spaces of d dimensions
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