This article reviews our present knowledge of universality classes in nonequilibrium systems defined on regular lattices. The first section presents the most important critical exponents and relations, as well as the field-theoretical formalism used in the text. The second section briefly addresses the question of scaling behavior at first-order phase transitions. In Sec. III the author looks at dynamical extensions of basic static classes, showing the effects of mixing dynamics and of percolation. The main body of the review begins in Sec. IV, where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. Section V considers such nonequilibrium classes in coupled, multicomponent systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion-type systems. However, by mapping they can be related to the universal behavior of interface growth models, which are treated in Sec. VI. The review ends with a summary of the classes of absorbing-state and mean-field systems and discusses some possible directions for future research.

Universality classes in nonequilibrium lattice systems

Publisher Summary This chapter discusses the systems coupled to two reservoirs of energy in such a way that there is a steady energy flow through the system. An example is a resistor in steady state, gaining energy from a battery and losing it to the atmosphere. Even for this restricted class there is no equivalent of Gibbs' framework and, typically, distributions cannot be expressed solely in terms of the internal energies of the system. Thus, in addition to the “technical difficulties” associated with computing averages in a many-body system, one must first solve the “more fundamental” problem of finding the stationary distribution. For systems, which are only weakly perturbed so that they remain “close to equilibrium,” much is known at the level of linear response. The chapter focuses on steady states “far from equilibrium” where such schemes break down. Against this backdrop of a vast theoretical terra incognita, a reasonable approach consists in investigating systems which, while retaining the essence of the difficulties of “far from equilibrium” states, are as simple as possible. In this very spirit, Lenz suggested the Ising model in an attempt to understand the nature of ferromagnetic phase transitions. This philosophy provides one of the main motivations behind the introduction of a simple non-equilibrium system, which is referred to as the “standard model.”

Statistical mechanics of driven diffusive systems

Various thermal equilibrium and nonequilibrium phase transitions exist where the correlation lengths in different lattice directions diverge with different exponentsv
‖,v
⊥: uniaxial Lifshitz points, the Kawasaki spin exchange model driven by an electric field, etc. An extension of finite-size scaling concepts to such anisotropic situations is proposed, including a discussion of (generalized) rectangular geometries, with linear dimensionL
‖ in the special direction and linear dimensionsL
⊥ in all other directions. The related shape effects forL
‖≠L

 ⊥
 but isotropic critical points are also discussed. Particular attention is paid to the case where the generalized hyperscaling relationv
‖+(d−1)v
⊥=γ+2β does not hold. As a test of these ideas, a Monte Carlo simulation study for shape effects at isotropic critical point in the two-dimensional Ising model is presented, considering subsystems of a 1024x1024 square lattice at criticality.

Finite-size effects at critical points with anisotropic correlations: Phenomenological scaling theory and Monte Carlo simulations

Driven diffusive systems. An introduction and recent developments

We review recent progress in the understanding of phase transitions and critical phenomena, obtained by means of numerical simulations of the early dynamic evolution of systems prepared at well-specified initial conditions. This field has seen exhaustive scientific research during the last decade, when the renormalization-group (RG) theoretical results obtained at the end of the 1980s were applied to the interpretation of dynamic Monte Carlo simulation results. While the original RG theory is restricted to critical phenomena under equilibrium conditions, numerical simulations have been applied to the study of far-from-equilibrium systems and irreversible phase transitions, the investigation of the behaviour of spinodal points close to first-order phase transitions and the understanding of the early-time evolution of self-organized criticality (SOC) systems when released far form the SOC regime. The present review intends to provide a comprehensive overview of recent applications in those fields, which can give the flavour of the main ideas, methods and results, and to discuss the directions for further studies. All of these numerical results pose new and interesting theoretical challenges that remain as open questions to be addressed by new research in the coming years.

https://iopscience.iop.org/article/10.1088/0034-4885/74/2/026501/pdf

Study of phase transitions from short-time non-equilibrium behaviour

Two-dimensional lattice-gas models with attractive interactions and particleconserving happing dynamics under the influence of a very large external electric field along a principal axis are studied in the case of a critical density. A finite-size scaling analysis allows the evaluation of critical indexes for the infinite system asβ=0.230±0.003,v=0.55±0.2, and α 0. We also describe some qualitative features of the system evolution and the existence of certain anisotropic order even well above the critical temperature in the case of finite lattices.

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Nonequilibrium second-order phase transitions in stochastic lattice systems: A finite-size scaling analysis in two dimensions

Two-dimensional lattice-gas models with attractive interactions and particle-conserving hopping dynamics under the influence of a very large external electric field ⇀E along a principal axis are studied in the case of different ratiosγ between the jump rates in the field direction and perpendicular to it using different transition probabilities. We investigate the dependence of the non-equilibrium steady-state properties on the transition mechanism. We find self-similarity with respect to (T, γ) and a coexistence curve critical exponent which, for smallγ, seems independent ofγ. There is some evidence that this exponent might be halfway between the equilibrium mean field and Onsager's values. A crossover toward mean-field behavior for largeγ seems also identified.

Nonequilibrium phase transitions in stochastic lattice systems: Influence of the hopping rates

Two-dimensional lattice-gas models with attractive interactions and particle-conserving hopping dynamics under the influence of a very large external electric field along a principal axis are studied in the case of off-critical densities. We describe the corresponding nonequilibrium first-order phase transitions, evaluate coexistence and spinodal lines, and make some comparisons with experimental observations on fast ionic conductors.

J. L. Vallés

Papers

Nonequilibrium second-order phase transitions in stochastic lattice systems: A finite-size scaling analysis in two dimensions

Nonequilibrium phase transitions in stochastic lattice systems: Influence of the hopping rates

Nonequilibrium discontinuous phase transitions in a fast ionic conductor model: coexistence and spinodal lines