M Droz

Cellular automata approach to non-equilibrium phase transitions in a surface reaction model: static and dynamic properties

A cellular automata approach to non-equilibrium phase transitions in a surface reaction model is proposed. This surface reaction model describes a simple adsorption-dissociation-desorption on a catalytic surface. This model exhibits two second-order non-equilibrium phase transitions. The stationary critical exponents for the order parameters beta as well as dynamical critical exponents Delta , describing the critical slowing down, are found to be mean-field-like.

Cellular automata approach to non-equilibrium diffusion and gradient percolation

The authors propose a deterministic approach to lattice diffusion in two dimensions. This method is implemented on a cellular automaton special purpose computer in order to study the properties of the interface of particles diffusing from a source to a sink. Fractal properties of the diffusion front of this non-equilibrium process are compared with results from percolation theory. The observed agreement indicates that diffusion fronts and gradient percolation coincide asymptotically and that the cellular automata method is a viable alternative to standard simulations for this class of problems.

Competing two-species directed percolation

The authors study an extension of conventional directed percolation (DP) with two species (A and B) using mean-field and Monte Carlo methods. The densities of the two species in the steady-state exhibit phase transitions which are due both to simple percolation and to competition between the species. In 1+1 dimensions as well as a simple DP transition, there is a line of first-order transitions between pure-A and pure-B phases. The phase diagram in a 2+1 dimensions agrees qualitatively with that obtained from mean-field calculations with second-order transitions between pure and mixed phases. Preliminary studies suggest that the critical exponents are in the same universality class as (one-species) DP.

Critical phenomena at surfaces in a model of nonequilibrium phase transitions

Critical phenomena at surfaces are studied in a simple two-dimensional model of non-equilibrium phase transitions belonging to the class of interacting particle systems. The mean field renormalization group approach and numerical simulations are used to determine the phase diagram for this system and some of the critical exponents associated with ordinary, special, extraordinary and surface phase transitions.

On the critical behaviour of cellular automata models of nonequilibrium phase transitions

Two cellular automata models of nonequilibrium phase transitions with one adsorbing state are studied in one and two dimensions. New evidence is found against the conjecture according to which all the one-component models with a single adsorbing state belong to the universality class of Reggeon field theory or directed percolation.

Nonuniversal critical behaviour in the 1D BEG model with Kawasaki dynamics

The authors investigate the one-dimensional Blume-Emery-Griffiths model with Kawasaki dynamics, using domain-wall arguments (DWA) and Monte Carlo simulations (MCS) (conventional and Gillespie algorithm), after a quench from a disordered state to low temperatures. They observe that this model exhibits domain scaling behaviour, controlled by a universal exponent (x=1/3) as in other dimensions for model B. However, they find also that the critical exponent z is not universal and depends on the coupling constants of the Hamiltonian. The results of DWA are consistent with those of MCS.

On the critical dynamics of one-dimensional disordered Ising models

The critical dynamics of a disordered Ising ferromagnetic chain with two coupling constants (J1>or=J2>0) is studied for Glauber dynamics. Using a domain wall argument the dynamical critical exponent z is found to be non-universal but independent of the disorder, namely z=1+J1/J2. The problem is formulated in terms of diffusion in a random medium. The diffusion is shown to be normal. Relationships with apparently very different diffusion problems, like the diffusion in hierarchically structured media, are established.

Real space renormalisation group for directed systems in arbitrary dimensions

A real space renormalisation group approach to intrinsically anisotropic systems (i.e. systems having two different correlation lengths) is defined. The problems of directed self-avoiding walks (DSAW) and directed percolation in d-dimensions are discussed in detail. For DSAW, one obtains the exact critical fugacity in all dimensions. For directed percolation, the percolation threshold pc obtained is very good in two dimensions, asymptotically exact for large dimensions and quite good in between. The parallel and perpendicular correlation length exponents nu /sub /// and nu perpendicular to are computed for the two problems.

Critical behaviour of a diffusive model with one adsorbing state

The authors study the critical behaviour of a nonequilibrium model for adsorption-desorption with diffusion. They consider a parallel updating rule of the cellular automata type. Without diffusion, it is found that the critical exponents belong to the universality class of directed percolation, as has already been shown for sequential dynamics. When diffusion is present, the critical behaviour can be described in terms of a crossover between the directed percolation regime and a dynamical mean-field regime associated with the case of arbitrarily large diffusion.

On the critical dynamics of one-dimensional Potts models

Simple physical arguments about the movement of domain walls are used to determine the dependence of the dynamical critical exponent z on the transition rates for the one-dimensional q-state Potts model.