Biomedical subjects

Ronald Dickman

Publications and source records attributed to Ronald Dickman.

16 recordsLinked to original sources

PubMed2006

Moment ratios for the pair-contact process with diffusion.

We study the continuous absorbing-state phase transition in the one-dimensional pair contact process with diffusion (PCPD). In previous studies [Dickman and de Menezes, Phys. Rev. E 66, 045101(R) (2002)], the critical point moment ratios of the order parameter showed anomalous behavior, growing with system size rather than taking universal values, as expected. Using the quasistationary simulation method we determine the moments of the order parameter up to fourth order at the critical point, in systems of up to 40,960 sites. Due to strong finite-size effects, the ratios converge only for large system sizes. Moment ratios and associated order-parameter histograms are compared with those of directed percolation. We also report an improved estimate [pc=0.077092(1)] for the location of the critical point in the nondiffusive pair contact process.

Journal Article↗

PubMed2006

Critical exponents for the restricted sandpile.

I report large-scale Monte Carlo studies of a one-dimensional height-restricted stochastic sandpile using the quasistationary simulation method. Results for systems of up to 50 000 sites yield estimates for critical exponents that differ significantly from those obtained using smaller systems, but are consistent with recent predictions derived from a Langevin equation for stochastic sandpiles [Ramasco, Phys. Rev. E 69, 045105(R) (2004)]. This suggests that apparent violations of universality in one-dimensional sandpiles are due to strong corrections to scaling and finite-size effects.

Journal Article↗

PubMed2005

Absorbing-state phase transitions with extremal dynamics.

Extremal dynamics represents a path to self-organized criticality in which the order parameter is tuned to a value of zero. The order parameter is associated with a phase transition to an absorbing state. Given a process that exhibits a phase transition to an absorbing state, we define an "extremal absorbing" process, providing the link to the associated extremal (nonabsorbing) process. Stationary properties of the latter correspond to those at the absorbing-state phase transition in the former. Studying the absorbing version of an extremal dynamics model allows to determine certain critical exponents that are not otherwise accessible. In the case of the Bak-Sneppen (BS) model, the absorbing version is closely related to the "f -avalanche" introduced by Paczuski, Maslov, and Bak [Phys. Rev. E 53, 414 (1996)], or, in spreading simulations to the "BS branching process" also studied by these authors. The corresponding nonextremal process belongs to the directed percolation universality class. We revisit the absorbing BS model, obtaining refined estimates for the threshold and critical exponents in one dimension. We also study an extremal version of the usual contact process, using mean-field theory and simulation. The extremal condition slows the spread of activity and modifies the critical behavior radically, defining an "extremal directed percolation" universality class of absorbing-state phase transitions. Asymmetric updating is a relevant perturbation for this class, even though it is irrelevant for the corresponding nonextremal class.

Journal Article↗

PubMed2005

How to simulate the quasistationary state.

For a large class of processes with an absorbing state, statistical properties of the surviving sample attain time-independent values in the quasistationary (QS) regime. We propose a practical simulation method for studying quasistationary properties, based on the equation of motion governing the QS distribution. In applications to the contact process, the method is shown to reproduce exact results (for the process on a complete graph) and known scaling behavior to high precision. At the critical point, our method is about an order of magnitude more efficient than conventional simulation.

Journal Article↗

PubMed2004

Critical behavior of nonequilibrium models in short-time Monte Carlo simulations.

We analyze two alternative methods for determining the dynamic critical exponent z of the contact process and the Domany-Kinzel cellular automaton in Monte Carlo simulations. One method employs mixed initial conditions, as proposed for magnetic models [Phys. Lett. A 298, 325 (2002)]]; the other is based on the growth of the moment ratio m (t) = / 2 starting with all sites occupied. The methods provide reliable estimates for z using the short-time dynamics of the process. Estimates of nu|| are obtained using a method suggested by Grassberger.

Journal Article↗

PubMed2003

Survival-extinction phase transition in a bit-string population with mutation.

A bit-string model for the evolution of a population of haploid organisms, subject to competition, reproduction with mutation, and selection, is studied, using mean-field theory and Monte Carlo simulations. We show that, depending on environmental flexibility and genetic variability, the model exhibits a phase transition between extinction due to random drift and survival. For weak selection the population attains a neutral regime. The mean-field theory describes the infinite-size limit, while simulations are used to study quasistationary properties.

Algorithms↗

PubMed2003

Rain, power laws, and advection.

Localized rain events have been found to follow power-law size and duration distributions over several decades, suggesting parallels between precipitation and seismic activity [O. Peters, Phys. Rev. Lett. 88, 018701 (2002)]]. Similar power laws are generated by treating rain as a passive tracer undergoing advection in a velocity field generated by a two-dimensional system of point vortices.

Journal Article↗

PubMed2003

Phase diagram of a probabilistic cellular automaton with three-site interactions.

We study a (1+1)-dimensional probabilistic cellular automaton that is closely related to the Domany-Kinzel stochastic-cellular automaton (DKCA), but in which the update of a given site depends on the state of three sites at the previous time step. Thus, compared with the DKCA, there is an additional parameter p(3) representing the probability for a site to be active at time t, given that it and its nearest neighbors were active at time t-1. We study phase transitions and critical behavior for the activity and for damage spreading, using one- and two-site mean-field approximations, and simulations, for p(3)=0 and p(3)=1. We find evidence for a line of tricritical points in the (p(1),p(2),p(3)) parameter space, obtained using a mean-field approximation at pair level. To construct the phase diagram in simulations we employ the growth-exponent method in an interface representation. For p(3)=0, the phase diagram is similar to the DKCA, but the damage-spreading transition exhibits a reentrant phase. For p(3)=1, the growth-exponent method reproduces the two absorbing states, first- and second-order phase transitions, bicritical point, and damage-spreading transition recently identified by Bagnoli et al. [Phys. Rev. E 63, 046116 (2001)].

Journal Article↗

PubMed2002

Asymptotic analysis of a random walk with a history-dependent step length.

We study an unbiased, discrete-time random walk on the nonnegative integers, with the origin absorbing, and a history-dependent step length. Letting y denote the maximum distance the walker has ever been from the origin, steps that do not change y have length v, while those that increase y (taking the walker to a site that has never been visited) have length n. The process serves as a simplified model of spreading in systems with an infinite number of absorbing configurations. Asymptotic analysis of the probability generating function shows that, for large t, the survival probability decays as S(t) approximately t(-delta), with delta=v/2n. Our expression for the decay exponent is in agreement with the results obtained via numerical iteration of the transition matrix.

Journal Article↗

PubMed2002

Quasistationary distributions for the Domany-Kinzel stochastic cellular automaton.

We construct the quasistationary (QS) probability distribution for the Domany-Kinzel stochastic cellular automaton (DKCA), a discrete-time Markov process with an absorbing state. QS distributions are derived at both the one-and two-site levels. We characterize the distributions by their mean, and various moment ratios, and analyze the lifetime of the QS state, and the relaxation time to attain this state. Of particular interest are the scaling properties of the QS state along the critical line separating the active and absorbing phases. These exhibit a high degree of similarity to the contact process and the Malthus-Verhulst process (the closest continuous-time analogs of the DKCA), which extends to the scaling form of the QS distribution.

Journal Article↗

PubMed2002

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=rho(2);/rho;(2) grows logarithmically with 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( perpendicular ), and confirm finite size scaling. We also determine the value m(c)=1.334 for the parity-conserving universality class in one dimension.

Journal Article↗

PubMed2002

n-site approximations and coherent-anomaly-method analysis for a stochastic sandpile.

n-site cluster approximations for a stochastic sandpile in one dimension are developed. A height restriction is imposed to limit the number of states: each site can harbor at most two particles (height z(i)< or =2). (This yields a considerable simplification over the unrestricted case, in which the number of states per site is unbounded.) On the basis of results for n< or =11 sites, the critical particle density as zeta(c)=0.930(1) is estimated, in good agreement with simulations. A coherent anomaly analysis yields estimates for the order parameter exponent [beta=0.41(1)] and the relaxation time exponent (nu(//) approximately 2.5).

Journal Article↗

PubMed2002

Scaling exponents of rough surfaces generated by the Domany-Kinzel cellular automaton.

The critical behavior at the frozen-active transition in the Domany-Kinzel stochastic cellular automaton is studied via a surface growth process in (1+1) dimensions. At criticality, this process presents a kinetic roughening transition; we measure the critical exponents in simulations. Two update schemes are considered: in the symmetric scheme, the growth surfaces belong to the directed percolation (DP) universality class, except at one terminal point. At this point, the phase transition is discontinuous and the surfaces belong to the compact directed percolation universality class. The relabeling of space-time points in the nonsymmetric scheme alters significantly the surface growth, changing the values of the critical exponents. The critical behavior of rough surfaces at the nonchaotic-chaotic transition is also studied using the damage spreading technique; the exponents confirm DP values for the symmetric scheme.

Journal Article↗

PubMed2002

Sandpiles with height restrictions.

We study stochastic sandpile models with a height restriction in one and two dimensions. A site can topple if it has a height of two, as in Manna's model, but, in contrast to previously studied sandpiles, here the height (or number of particles per site), cannot exceed two. This yields a considerable simplification over the unrestricted case, in which the number of states per site is unbounded. Two toppling rules are considered: in one, the particles are redistributed independently, while the other involves some cooperativity. We study the fixed-energy system (no input or loss of particles) using cluster approximations and extensive simulations, and find that it exhibits a continuous phase transition to an absorbing state at a critical value zeta(c) of the particle density. The critical exponents agree with those of the unrestricted Manna sandpile.

Journal Article↗

PubMed2002

Numerical analysis of the master equation.

Applied to the master equation, the usual numerical integration methods, such as Runge-Kutta method, become inefficient when the rates associated with various transitions differ by several orders of magnitude. We introduce an integration scheme that remains stable with much larger time increments than can be used in standard methods. When only the stationary distribution is required, a direct iteration method is even more rapid; this method may be extended to construct the quasistationary distribution of a process with an absorbing state. Applications to birth-and-death processes reveal gains in efficiency of two or more orders of magnitude.

Journal Article↗

PubMed2001

Pair contact process with a particle source.

We study the phase diagram and critical behavior of the one-dimensional pair contact process (PCP) with a particle source using cluster approximations and extensive simulations. The source creates isolated particles only, not pairs, and so couples not to the order parameter (the pair density) but to a nonordering field, whose state influences the evolution of the order parameter. While the critical point p(c) shows a singular dependence on the source intensity, the critical exponents appear to be unaffected by the presence of the source, except possibly for a small change in beta. In the course of our paper, we obtain high-precision values for the critical exponents of the standard PCP, confirming directed-percolationlike scaling.

Journal Article↗