Phase transitions in a probabilistic cellular automaton: Growth kinetics and critical properties

Abstract

We investigate a discrete-time kinetic model without detailed balance which simulates the phase segregation of a quenched binary alloy. The model is a variation on the Rothman-Keller cellular automaton in which particles of type A (B) move toward domains of greater concentration of A (B). Modifications include a fully occupied lattice and the introduction of a temperature-like parameter which endows the system with a stochastic evolution. Using computer simulations, we examine domain growth kinetics in the two-dimensional model. For long times after a quench from disorder, we find that the average domain size R(t) ~ l 1/3, in agreement with the prediction of Lifshitz-SlyozovWagner theory. Using a variety of methods, we analyze the critical properties of the associated second-order transition. Our analysis indicates that this model does not fall within either the Ising or mean-field classes.

Long-time tails in lattice gases violating detailed balance

“…(3)]. The above result enables us to calculate the susceptibility in mean field approximation as x~= (c~IG"'(q) Ic~) = 3f (1 -f ), (24) where G, z(q) is the Fourier transform of G;~(r) in (23).…”

“…The basic distinction between the three types of models is the sign of the slope of fo(p) At l. ow densities, model I has on average a deficit of rest particles because (20) (see [5] (19) and using I, GV(r) = G'"(r) =~V~( r o)f*(1 -f*) (23) where f, is the stationary single-particle distribution function. This result is exact for lattice gases satisfying detailed balance [compare Eq.…”

Lattice-gas models of phase separation: interfaces, phase transitions, and multiphase flow

“…For essentially the same model on a square lattice with the maximum density of particles (d = 1) and consequently no hydrodynamics, a considerably more detailed analysis, including comparison with numerical experiments, has been reported by Alexander et al (1992). They discuss not only the dynamical scaling of the structure functions, but also Porod's law, the exponent a in k m~t a , and other related issues.…”

Instabilities and patterns

“…They have also been observed in computer simulations of the negative diffusion model, where an external field F is added, that drives a particle current [Alexander 1992]. The driving force in lattice gases can be implemented by replacing the bias factor in section 3.3 by exp[ (β G(s n.n. )…”

“…A further simplification of the Rothman-Keller model is the negative diffusion model [Alexander 1992]. It is defined on a square lattice.…”