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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonequilibrium critical behavior from the master equation

B. Yu, D. A. Browne

Published Apr 1, 1994• 2 citations
Publisher

Abstract

Kinetic models for nonequilibrium phase transitions have been extensively studied via mean-field theory, series expansions, and Monte Carlo simulation. In this paper we present a different approach based on studying the normal-mode solutions of the master equation for the system to extract information on the static and dynamic behavior of the system in the vicinity of the phase transition. This method provides insight into the physics of the transition and offers an alternative technique for analyzing kinetic phase transitions. We demonstrate the method by extracting the critical behavior of a simple model from finite-size scaling.

DOI
10.1103/physreve.49.3496

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Kinetic models for nonequilibrium phase transitions have been extensively studied via mean-field theory, series expansions, and Monte Carlo simulation. In this paper we present a different approach based on studying the normal-mode solutions of the master equation for the system to extract information on the static and dynamic behavior of the system in the vicinity of the phase transition. This method provides insight into the physics of the transition and offers an alternative technique for analyzing kinetic phase transitions. We demonstrate the method by extracting the critical behavior of a simple model from finite-size scaling.
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Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent zzz of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for zzz; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate zzz. Even in this case we find that zzz depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.
We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.
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Abstract Recent developments have led to a good understanding of universality; why phase transitions in systems as diverse as magnets, fluids, liquid crystals, and superconductors can be brought under the same theoretical umbrella and well described by simple models. This book describes the physics underlying universality and then lays out the theoretical approaches now available for studying phase transitions. Traditional techniques, mean-field theory, series expansions, and the transfer matrix, are described; the Monte Carlo method is covered, and two chapters are devoted to the renormalization group, which led to a break-through in the field. The book will be useful as a textbook for a course in `Phase Transitions', as an introduction for graduate students undertaking research in related fields, and as an overview for scientists in other disciplines who work with phase transitions but who are not aware of the current tools in the armoury of the theoretical physicist.