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Physical review. B, Condensed matter

Nonequilibrium phase transitions in lattice systems with random-field competing kinetics

A. I. López-Lacomba, Joaquín Marro

Published Oct 1, 1992• 16 citations
Publisher

Abstract

We study a class of lattice interacting-spin systems evolving stochastically under the simultaneous operation of several spin-flip mechanisms, each acting independently and responding to a different applied magnetic field. This induces an extra randomness which may occur in real systems, e.g., a magnetic system under the action of a field varying with a much shorter period than the mean time between successive transitions. Such a situation--in which one may say in some sense that frustration has a dynamical origin-- may also be viewed as a nonequilibrium version of the random-field Ising model. By following a method of investigating stationary probability distributions in systems with competing kinetics [P. L. Garrido and J. Marro, Phys. Rev. Lett. 62, 1929 (1989)], we solve one-dimensional lattices supporting different field distributions and transition rates for the elementary kinetical processes, thus revealing a rich variety of phase transitions and critical phenomena. Some exact results for lattices of arbitrary dimension, and comparisons with the standard quenched and annealed random-field models, and with a nonequilibrium diluted antiferromagnetic system, are also reported.

DOI
10.1103/physrevb.46.8244

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We study a class of lattice interacting-spin systems evolving stochastically under the simultaneous operation of several spin-flip mechanisms, each acting independently and responding to a different applied magnetic field. This induces an extra randomness which may occur in real systems, e.g., a magnetic system under the action of a field varying with a much shorter period than the mean time between successive transitions. Such a situation--in which one may say in some sense that frustration has a dynamical origin-- may also be viewed as a nonequilibrium version of the random-field Ising model. By following a method of investigating stationary probability distributions in systems with competing kinetics [P. L. Garrido and J. Marro, Phys. Rev. Lett. 62, 1929 (1989)], we solve one-dimensional lattices supporting different field distributions and transition rates for the elementary kinetical processes, thus revealing a rich variety of phase transitions and critical phenomena. Some exact results for lattices of arbitrary dimension, and comparisons with the standard quenched and annealed random-field models, and with a nonequilibrium diluted antiferromagnetic system, are also reported.
We report on exact results for stochastic Ising systems that evolve in time due to a simultaneous action of several independent spin-flip mechanisms. The transition probability describes a competition between different values for the two parameters characterizing the involved (nearest-neighbour) Hamiltonian, namely, the interaction strength and the applied magnetic field. Such a conflicting dynamics aims to represent the sort of diffusing microscopic disorder that may occur in real systems. We determine the steady states for a class of transition probabilities for one-dimensional lattices; they exhibit amazing critical behaviour. Furthermore, it is shown that the steady state for a singular class of transition probabilities may be represented by an effective Hamiltonian that has the Ising structure with complex parameters for any lattice dimension.
The ordinary kinetic versions of the Ising model may be modified to exhibit steady nonequilibrium states. This is illustrated in chapter 4 where a conflict between two canonical mechanisms ( diffusion and reaction ) drives the configuration away from equilibrium. A more systematic investigation of this possibility, when the conflict is between different reaction processes only, is described here. We focus on spin systems evolving by a superposition of independent local processes of the kind variously known as spin flips, birth/death or creation/annihilation. The restriction to spin flip dynamics does not prevent the systems in this class from exhibiting a variety of nonequilibrium phase transitions and critical phenomena. Their consideration may therefore help in developing nonequilibrium theory. In addition, they have some practical interest, e.g., conflicting dynamics may occur in disordered materials such as dilute magnetic systems, and some of these situations can be implemented in the laboratory. The present chapter describes some exact, mean-field and MC results that together render an intriguing picture encouraging further study. It is argued in §7.1 that some of the peculiar, emergent, macroscopic behavior of microscopically disordered materials may be related to diffusion of disorder. This provides a physical motivation for the nonequilibrium random-field Ising model (NRFM).
We report first-order mean-field results for nonequilibrium random-exchange lattice systems, namely, Ising-like models whose kinetics involve a simultaneous, random competition between ferromagnetic and antiferromagnetic interactions that generally induces nonequilibrium steady states. We consider the competition between symmetric bonds. \ifmmode\pm\else\textpm\fi{}J0{\mathit{J}}_{0}J0​, symmetric bonds competing with broken ones, \ifmmode\pm\else\textpm\fi{}J0{\mathit{J}}_{0}J0​ and J=0, nonsymmetric bonds, \ensuremath{\Vert}J1{\mathit{J}}_{1}J1​\ensuremath{\Vert} and -\ensuremath{\Vert}J2{\mathit{J}}_{2}J2​\ensuremath{\Vert}, and some of those systems under an external magnetic field h. The time evolution and steady-state properties, including phase diagrams, are obtained for several lattice coordination numbers, e.g., the case of simple-cubic lattices of dimension d\ensuremath{\le}3. A comparison is made with existing exact results for d=1, h=0, and \ifmmode\pm\else\textpm\fi{}J0{\mathit{J}}_{0}J0​.
Nonequilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and Kawasaki-type spin-exchange kinetics at infinite temperature T are investigated here in one dimension from the point of view of phase transition and critical behaviour. Branching annihilating random walks with an even number of offspring (on the part of the ferromagnetic domain boundaries), is a decisive process in forming the steady state of the system for a range of parameters, in the family of models considered. A wide variety of quantities characterize the critical behaviour of the system. Results of computer simulations and of a generalized mean field theory are presented and discussed.
The authors study d-dimensional lattice model systems whose dynamics consist of competing independent processes which may induce nonequilibrium steady states. Interesting phase transitions and critical phenomena ensue which are influenced, even dominated, by a sort of dynamical bond disorder. The latter, which may occur in natural systems, reminds one of the disorder in more familiar bond-diluted and other impure models, for example, percolation-like phenomena arise in some of the model cases. Exact solutions for d=1, some exact results for d<1, and a comparison with more standard, either quenched or annealed, magnetically diluted models are reported.