465 research outputs found
ERROR PROPAGATION IN EXTENDED CHAOTIC SYSTEMS
A strong analogy is found between the evolution of localized disturbances in
extended chaotic systems and the propagation of fronts separating different
phases. A condition for the evolution to be controlled by nonlinear mechanisms
is derived on the basis of this relationship. An approximate expression for the
nonlinear velocity is also determined by extending the concept of Lyapunov
exponent to growth rate of finite perturbations.Comment: Tex file without figures- Figures and text in post-script available
via anonymous ftp at ftp://wpts0.physik.uni-wuppertal.de/pub/torcini/jpa_le
Space-time estimation of a particle system model
13 pagesLet X be a discrete time contact process (CP) on the discrete bidimensional lattice as define by Durett - Levin (1994) . We study estimation of the model based on space-time evolution on a finite subset of sites. For this, we make use of a marginal pseudo-likelihood. The estimator obtained is consistent and asymptoticaly normal for non-vanishing supercritical CP. Numerical studies confirm these results
Contact process with long-range interactions: a study in the ensemble of constant particle number
We analyze the properties of the contact process with long-range interactions
by the use of a kinetic ensemble in which the total number of particles is
strictly conserved. In this ensemble, both annihilation and creation processes
are replaced by an unique process in which a particle of the system chosen at
random leaves its place and jumps to an active site. The present approach is
particularly useful for determining the transition point and the nature of the
transition, whether continuous or discontinuous, by evaluating the fractal
dimension of the cluster at the emergence of the phase transition. We also
present another criterion appropriate to identify the phase transition that
consists of studying the system in the supercritical regime, where the presence
of a "loop" characterizes the first-order transition. All results obtained by
the present approach are in full agreement with those obtained by using the
constant rate ensemble, supporting that, in the thermodynamic limit the results
from distinct ensembles are equivalent
Rare Events Statistics in Reaction--Diffusion Systems
We develop an efficient method to calculate probabilities of large deviations
from the typical behavior (rare events) in reaction--diffusion systems. The
method is based on a semiclassical treatment of underlying "quantum"
Hamiltonian, encoding the system's evolution. To this end we formulate
corresponding canonical dynamical system and investigate its phase portrait.
The method is presented for a number of pedagogical examples.Comment: 12 pages, 6 figure
Non-equilibrium Phase Transitions with Long-Range Interactions
This review article gives an overview of recent progress in the field of
non-equilibrium phase transitions into absorbing states with long-range
interactions. It focuses on two possible types of long-range interactions. The
first one is to replace nearest-neighbor couplings by unrestricted Levy flights
with a power-law distribution P(r) ~ r^(-d-sigma) controlled by an exponent
sigma. Similarly, the temporal evolution can be modified by introducing waiting
times Dt between subsequent moves which are distributed algebraically as P(Dt)~
(Dt)^(-1-kappa). It turns out that such systems with Levy-distributed
long-range interactions still exhibit a continuous phase transition with
critical exponents varying continuously with sigma and/or kappa in certain
ranges of the parameter space. In a field-theoretical framework such
algebraically distributed long-range interactions can be accounted for by
replacing the differential operators nabla^2 and d/dt with fractional
derivatives nabla^sigma and (d/dt)^kappa. As another possibility, one may
introduce algebraically decaying long-range interactions which cannot exceed
the actual distance to the nearest particle. Such interactions are motivated by
studies of non-equilibrium growth processes and may be interpreted as Levy
flights cut off at the actual distance to the nearest particle. In the
continuum limit such truncated Levy flights can be described to leading order
by terms involving fractional powers of the density field while the
differential operators remain short-ranged.Comment: LaTeX, 39 pages, 13 figures, minor revision
Branching and annihilating Levy flights
We consider a system of particles undergoing the branching and annihilating
reactions A -> (m+1)A and A + A -> 0, with m even. The particles move via
long-range Levy flights, where the probability of moving a distance r decays as
r^{-d-sigma}. We analyze this system of branching and annihilating Levy flights
(BALF) using field theoretic renormalization group techniques close to the
upper critical dimension d_c=sigma, with sigma<2. These results are then
compared with Monte-Carlo simulations in d=1. For sigma close to unity in d=1,
the critical point for the transition from an absorbing to an active phase
occurs at zero branching. However, for sigma bigger than about 3/2 in d=1, the
critical branching rate moves smoothly away from zero with increasing sigma,
and the transition lies in a different universality class, inaccessible to
controlled perturbative expansions. We measure the exponents in both
universality classes and examine their behavior as a function of sigma.Comment: 9 pages, 4 figure
Spreading with immunization in high dimensions
We investigate a model of epidemic spreading with partial immunization which
is controlled by two probabilities, namely, for first infections, , and
reinfections, . When the two probabilities are equal, the model reduces to
directed percolation, while for perfect immunization one obtains the general
epidemic process belonging to the universality class of dynamical percolation.
We focus on the critical behavior in the vicinity of the directed percolation
point, especially in high dimensions . It is argued that the clusters of
immune sites are compact for . This observation implies that a
recently introduced scaling argument, suggesting a stretched exponential decay
of the survival probability for , in one spatial dimension,
where denotes the critical threshold for directed percolation, should
apply in any dimension and maybe for as well. Moreover, we
show that the phase transition line, connecting the critical points of directed
percolation and of dynamical percolation, terminates in the critical point of
directed percolation with vanishing slope for and with finite slope for
. Furthermore, an exponent is identified for the temporal correlation
length for the case of and , , which
is different from the exponent of directed percolation. We also
improve numerical estimates of several critical parameters and exponents,
especially for dynamical percolation in .Comment: LaTeX, IOP-style, 18 pages, 9 eps figures, minor changes, additional
reference
Two-dimensional SIR epidemics with long range infection
We extend a recent study of susceptible-infected-removed epidemic processes
with long range infection (referred to as I in the following) from
1-dimensional lattices to lattices in two dimensions. As in I we use hashing to
simulate very large lattices for which finite size effects can be neglected, in
spite of the assumed power law for the
probability that a site can infect another site a distance vector
apart. As in I we present detailed results for the critical case, for the
supercritical case with , and for the supercritical case with . For the latter we verify the stretched exponential growth of the
infected cluster with time predicted by M. Biskup. For we find
generic power laws with dependent exponents in the supercritical
phase, but no Kosterlitz-Thouless (KT) like critical point as in 1-d. Instead
of diverging exponentially with the distance from the critical point, the
correlation length increases with an inverse power, as in an ordinary critical
point. Finally we study the dependence of the critical exponents on in
the regime , and compare with field theoretic predictions. In
particular we discuss in detail whether the critical behavior for
slightly less than 2 is in the short range universality class, as conjectured
recently by F. Linder {\it et al.}. As in I we also consider a modified version
of the model where only some of the contacts are long range, the others being
between nearest neighbors. If the number of the latter reaches the percolation
threshold, the critical behavior is changed but the supercritical behavior
stays qualitatively the same.Comment: 14 pages, including 29 figure
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