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Journal of Physics A Mathematical and General

Nonequilibrium impure lattice systems

Pedro L. Garrido, Joaquín Marro

Published Mar 21, 1992• 17 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/25/6/008

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•Jan 1, 2021•0 citations
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.
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).
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​.
An overview is presented of non-equilibrium systems in the presence of disorder, which typically strongly affects their behaviour. The examples considered here are those for which most progress is currently being made, namely low-dimensional collective models of the lattice-based exclusion type with substitutional disorder. Discussions are given of such effects as localization, and of disorder-induced crossovers to new critical or singular behaviour at steady state non-equilibrium transitions, and to new asymptotic or critical dynamics. Mappings to disordered quantum spin systems are used to provide some exact results (exploiting Goldstone symmetries or free fermion equivalences) and to introduce non-equilibrium/disorder effects analogous to Griffiths phases and McCoy-Wu singularities. Among other topics also discussed here are criteria for crossovers, and non-equilibrium transitions and dynamics on diluted and possibly scale-invariant backgrounds. As well as mappings, direct approaches of various sorts are used, such as mean field methods, and scaling procedures typically scaling a disorder distribution. Scaling for the non-equilibrium dynamics is relatively simple and very effective when the equations of motion are linear or linearizable by certain transformations presented here. The current situation regarding scaling approaches to the more resistant non-linear cases is briefly discussed.
Clearly, in nature, but also in technological applications, complex systems built in an entirely ordered and regular fashion are the exception rather than the rule. In this thesis we explore how critical phenomena are influenced by quenched spatial randomness. Specifically, we consider physical systems undergoing a continuous phase transition in the presence of topological disorder, where the underlying structure, on which the system evolves, is given by a non-regular, discrete lattice. We therefore endeavour to achieve a thorough understanding of the interplay between collective dynamics and quenched randomness. According to the intriguing concept of universality, certain laws emerge from collectively behaving many-body systems at criticality, almost regardless of the precise microscopic realization of interactions in those systems. As a consequence, vastly different phenomena show striking similarities at their respective phase transitions. In this dissertation we pursue the question of whether the universal properties of critical phenomena are preserved when the system is subjected to topological perturbations. For this purpose, we perform numerical simulations of several prototypical systems of statistical physics which show a continuous phase transition. In particular, the equilibrium spin-1/2 Ising model and its generalizations represent -- among other applications -- fairly natural approaches to model magnetism in solids, whereas the non-equilibrium contact process serves as a toy model for percolation in porous media and epidemic spreading. Finally, the Manna sandpile model is strongly related to the concept of self-organized criticality, where a complex dynamic system reaches a critical state without fine-tuning of external variables. Our results reveal that the prevailing understanding of the influence of topological randomness on critical phenomena is insufficient. In particular, by considering very specific and newly developed lattice structures, we are able to show that -- contrary to the popular opinion -- spatial correlations in the number of interacting neighbours are not a key measure for predicting whether disorder ultimately alters the behaviour of a given critical system.