9,011 research outputs found
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
Coarse graining of master equations with fast and slow states
We propose a general method for simplifying master equations by eliminating
from the description rapidly evolving states. The physical recipe we impose is
the suppression of these states and a renormalization of the rates of all the
surviving states. In some cases, this decimation procedure can be analytically
carried out and is consistent with other analytical approaches, like in the
problem of the random walk in a double-well potential. We discuss the
application of our method to nontrivial examples: diffusion in a lattice with
defects and a model of an enzymatic reaction outside the steady state regime.Comment: 9 pages, 9 figures, final version (new subsection and many minor
improvements
Strong disorder RG approach of random systems
There is a large variety of quantum and classical systems in which the
quenched disorder plays a dominant r\^ole over quantum, thermal, or stochastic
fluctuations : these systems display strong spatial heterogeneities, and many
averaged observables are actually governed by rare regions. A unifying approach
to treat the dynamical and/or static singularities of these systems has emerged
recently, following the pioneering RG idea by Ma and Dasgupta and the detailed
analysis by Fisher who showed that the Ma-Dasgupta RG rules yield asymptotic
exact results if the broadness of the disorder grows indefinitely at large
scales. Here we report these new developments by starting with an introduction
of the main ingredients of the strong disorder RG method. We describe the basic
properties of infinite disorder fixed points, which are realized at critical
points, and of strong disorder fixed points, which control the singular
behaviors in the Griffiths-phases. We then review in detail applications of the
RG method to various disordered models, either (i) quantum models, such as
random spin chains, ladders and higher dimensional spin systems, or (ii)
classical models, such as diffusion in a random potential, equilibrium at low
temperature and coarsening dynamics of classical random spin chains, trap
models, delocalization transition of a random polymer from an interface, driven
lattice gases and reaction diffusion models in the presence of quenched
disorder. For several one-dimensional systems, the Ma-Dasgupta RG rules yields
very detailed analytical results, whereas for other, mainly higher dimensional
problems, the RG rules have to be implemented numerically. If available, the
strong disorder RG results are compared with another, exact or numerical
calculations.Comment: review article, 195 pages, 36 figures; final version to be published
in Physics Report
Quasi-stationary regime of a branching random walk in presence of an absorbing wall
A branching random walk in presence of an absorbing wall moving at a constant
velocity undergoes a phase transition as the velocity of the wall
varies. Below the critical velocity , the population has a non-zero
survival probability and when the population survives its size grows
exponentially. We investigate the histories of the population conditioned on
having a single survivor at some final time . We study the quasi-stationary
regime for when is large. To do so, one can construct a modified
stochastic process which is equivalent to the original process conditioned on
having a single survivor at final time . We then use this construction to
show that the properties of the quasi-stationary regime are universal when
. We also solve exactly a simple version of the problem, the
exponential model, for which the study of the quasi-stationary regime can be
reduced to the analysis of a single one-dimensional map.Comment: 2 figures, minor corrections, one reference adde
Diffusion-limited reactions and mortal random walkers in confined geometries
Motivated by the diffusion-reaction kinetics on interstellar dust grains, we
study a first-passage problem of mortal random walkers in a confined
two-dimensional geometry. We provide an exact expression for the encounter
probability of two walkers, which is evaluated in limiting cases and checked
against extensive kinetic Monte Carlo simulations. We analyze the continuum
limit which is approached very slowly, with corrections that vanish
logarithmically with the lattice size. We then examine the influence of the
shape of the lattice on the first-passage probability, where we focus on the
aspect ratio dependence: Distorting the lattice always reduces the encounter
probability of two walkers and can exhibit a crossover to the behavior of a
genuinely one-dimensional random walk. The nature of this transition is also
explained qualitatively.Comment: 18 pages, 16 figure
Wigner Surmise For Domain Systems
In random matrix theory, the spacing distribution functions are
well fitted by the Wigner surmise and its generalizations. In this
approximation the spacing functions are completely described by the behavior of
the exact functions in the limits s->0 and s->infinity. Most non equilibrium
systems do not have analytical solutions for the spacing distribution and
correlation functions. Because of that, we explore the possibility to use the
Wigner surmise approximation in these systems. We found that this approximation
provides a first approach to the statistical behavior of complex systems, in
particular we use it to find an analytical approximation to the nearest
neighbor distribution of the annihilation random walk
Anomalous diffusion: A basic mechanism for the evolution of inhomogeneous systems
In this article we review classical and recent results in anomalous diffusion
and provide mechanisms useful for the study of the fundamentals of certain
processes, mainly in condensed matter physics, chemistry and biology. Emphasis
will be given to some methods applied in the analysis and characterization of
diffusive regimes through the memory function, the mixing condition (or
irreversibility), and ergodicity. Those methods can be used in the study of
small-scale systems, ranging in size from single-molecule to particle clusters
and including among others polymers, proteins, ion channels and biological
cells, whose diffusive properties have received much attention lately.Comment: Review article, 20 pages, 7 figures. arXiv admin note: text overlap
with arXiv:cond-mat/0201446 by other author
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