13 research outputs found
Mean-Field and Non-Mean-Field Behaviors in Scale-free Networks with Random Boolean Dynamics
We study two types of simplified Boolean dynamics over scale-free networks,
both with synchronous update. Assigning only Boolean functions AND and XOR to
the nodes with probability and , respectively, we are able to analyze
the density of 1's and the Hamming distance on the network by numerical
simulations and by a mean-field approximation (annealed approximation). We show
that the behavior is quite different if the node always enters in the dynamic
as its own input (self-regulation) or not. The same conclusion holds for the
Kauffman KN model. Moreover, the simulation results and the mean-field ones (i)
agree well when there is no self-regulation, and (ii) disagree for small
when self-regulation is present in the model.Comment: 12 pages, 7 figure
Glauber dynamics in a single-chain magnet: From theory to real systems
The Glauber dynamics is studied in a single-chain magnet. As predicted, a
single relaxation mode of the magnetization is found. Above 2.7 K, the
thermally activated relaxation time is mainly governed by the effect of
magnetic correlations and the energy barrier experienced by each magnetic unit.
This result is in perfect agreement with independent thermodynamical
measurements. Below 2.7 K, a crossover towards a relaxation regime is observed
that is interpreted as the manifestation of finite-size effects. The
temperature dependences of the relaxation time and of the magnetic
susceptibility reveal the importance of the boundary conditions.Comment: Submitted to PRL 10 May 2003. Submitted to PRB 12 December 2003;
published 15 April 200
Nonuniversality in the pair contact process with diffusion
We study the static and dynamic behavior of the one dimensional pair contact
process with diffusion. Several critical exponents are found to vary with the
diffusion rate, while the order-parameter moment ratio m=\bar{rho^2}
/\bar{rho}^2 grows logarithmically with the system size. The anomalous behavior
of m is traced to a violation of scaling in the order parameter probability
density, which in turn reflects the presence of two distinct sectors, one
purely diffusive, the other reactive, within the active phase. Studies
restricted to the reactive sector yield precise estimates for exponents beta
and nu_perp, and confirm finite size scaling of the order parameter. In the
course of our study we determine, for the first time, the universal value m_c =
1.334 associated with the parity-conserving universality class in one
dimension.Comment: 9 pages, 5 figure
Activated Random Walkers: Facts, Conjectures and Challenges
We study a particle system with hopping (random walk) dynamics on the integer
lattice . The particles can exist in two states, active or
inactive (sleeping); only the former can hop. The dynamics conserves the number
of particles; there is no limit on the number of particles at a given site.
Isolated active particles fall asleep at rate , and then remain
asleep until joined by another particle at the same site. The state in which
all particles are inactive is absorbing. Whether activity continues at long
times depends on the relation between the particle density and the
sleeping rate . We discuss the general case, and then, for the
one-dimensional totally asymmetric case, study the phase transition between an
active phase (for sufficiently large particle densities and/or small )
and an absorbing one. We also present arguments regarding the asymptotic mean
hopping velocity in the active phase, the rate of fixation in the absorbing
phase, and survival of the infinite system at criticality. Using mean-field
theory and Monte Carlo simulation, we locate the phase boundary. The phase
transition appears to be continuous in both the symmetric and asymmetric
versions of the process, but the critical behavior is very different. The
former case is characterized by simple integer or rational values for critical
exponents (, for example), and the phase diagram is in accord with
the prediction of mean-field theory. We present evidence that the symmetric
version belongs to the universality class of conserved stochastic sandpiles,
also known as conserved directed percolation. Simulations also reveal an
interesting transient phenomenon of damped oscillations in the activity
density
Universal finite-size scaling amplitudes in anisotropic scaling
Phenomenological scaling arguments suggest the existence of universal
amplitudes in the finite-size scaling of certain correlation lengths in
strongly anisotropic or dynamical phase transitions. For equilibrium systems,
provided that translation invariance and hyperscaling are valid, the
Privman-Fisher scaling form of isotropic equilibrium phase transitions is
readily generalized. For non-equilibrium systems, universality is shown
analytically for directed percolation and is tested numerically in the
annihilation-coagulation model and in the pair contact process with diffusion.
In these models, for both periodic and free boundary conditions, the
universality of the finite-size scaling amplitude of the leading relaxation
time is checked. Amplitude universality reveals strong transient effects along
the active-inactive transition line in the pair contact process.Comment: 16 pages, Latex, 2 figures, final version, to appear in J. Phys.
The non-equilibrium phase transition of the pair-contact process with diffusion
The pair-contact process 2A->3A, 2A->0 with diffusion of individual particles
is a simple branching-annihilation processes which exhibits a phase transition
from an active into an absorbing phase with an unusual type of critical
behaviour which had not been seen before. Although the model has attracted
considerable interest during the past few years it is not yet clear how its
critical behaviour can be characterized and to what extent the diffusive
pair-contact process represents an independent universality class. Recent
research is reviewed and some standing open questions are outlined.Comment: Latexe2e, 53 pp, with IOP macros, some details adde
Universal scaling behavior of non-equilibrium phase transitions
One of the most impressive features of continuous phase transitions is the
concept of universality, that allows to group the great variety of different
critical phenomena into a small number of universality classes. All systems
belonging to a given universality class have the same critical exponents, and
certain scaling functions become identical near the critical point. It is the
aim of this work to demonstrate the usefulness of universal scaling functions
for the analysis of non-equilibrium phase transitions. In order to limit the
coverage of this article, we focus on a particular class of non-equilibrium
critical phenomena, the so-called absorbing phase transitions. These phase
transitions arise from a competition of opposing processes, usually creation
and annihilation processes. The transition point separates an active phase and
an absorbing phase in which the dynamics is frozen. A systematic analysis of
universal scaling functions of absorbing phase transitions is presented,
including static, dynamical, and finite-size scaling measurements. As a result
a picture gallery of universal scaling functions is presented which allows to
identify and to distinguish universality classes.Comment: review article, 160 pages, 60 figures include
Role-separating ordering in social dilemmas controlled by topological frustration.
Three is a crowd? is an old proverb that applies as much to social interactions, as it does to frustrated
configurations in statistical physics models. Accordingly, social relations within a triangle deserve special attention.
With this motivation, we explore the impact of topological frustration on the evolutionary dynamics of
the snowdrift game on a triangular lattice. This topology provides an irreconcilable frustration, which prevents
anti-coordination of competing strategies that would be needed for an optimal outcome of the game. By using
different strategy updating protocols, we observe complex spatial patterns in dependence on payoff values that
are reminiscent to a honeycomb-like organization, which helps to minimize the negative consequence of the
topological frustration. We relate the emergence of these patterns to the microscopic dynamics of the evolutionary
process, both by means of mean-field approximations and Monte Carlo simulations. For comparison, we
also consider the same evolutionary dynamics on the square lattice, where of course the topological frustration
is absent. However, with the deletion of diagonal links of the triangular lattice, we can gradually bridge the
gap to the square lattice. Interestingly, in this case the level of cooperation in the system is a direct indicator
of the level of topological frustration, thus providing a method to determine frustration levels in an arbitrary
interaction network