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Europhysics Letters (EPL)

Ising Systems with Conflicting Dynamics: Exact Results for Random Interactions and Fields

A. I. López-Lacomba, Joaquín Marro

Published Jan 20, 1994• 11 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/25/3/003

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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.
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.
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).
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random variables {h} that change randomly with time according to a double-gaussian probability distribution, which consists of two single gaussian distributions, centered at +ho and −ho, with the same width σ. This distribution is very general, and can recover in appropriate limits the bimodal distribution (σ → 0) and the single gaussian one (ho = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32 − 512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities ho, for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ. PACS numbers: 05.10.Ln, 05.50.+q, 64.60.De, 75.10Hk, 75.40.MgFirst-order phase transition in a 2D random-field Ising model with conflicting dynamics2 1.
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbor interactions. In order to generate the random magnetic fields, we have considered random variables { h } that change randomly with time according to a double-Gaussian probability distribution, which consists of two single-Gaussian distributions, centered at + h o and − h o , with the same width σ. This distribution is very general and can recover in appropriate limits the bimodal distribution ( ) and the single-Gaussian one ( h o = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32–512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurrence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities h o for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ.
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.