Stochastic evolution in ising models

Abstract

Stochastic evolution models are often used to study timedependent properties of many-body systems. Some examples are the Brownian motion of particles, tracks of nuclear particles through dense media, kinetics of nucleation in super-heated liquids etc. These models provide the basic framework for studying time-dependent phenomena in statistical mechanics, such as the approach to thermal equilibrium from an arbitrarily prepared initial state, or of non-equilibrium steady states in dissipative systems, or the appearance of co-operative long-time correlations in the neighbourhood of second-order phasetransitions. In the following, the relaxation properties of some kinetic Ising models are briefly discussed. The plan of these lectures is as follows: In section II, we discuss briefly ho% the probabilistic description of evolution of manyparticle systems c~a be reconciled with the deterministic (microscopic) mechanical evolution. In section III, the rate-equation for the Markovian evolution, and the condition of detailed balance are described. ' In section IV, we introduce the single-spin-flip kinetic Ising model and general crystal-growth model (of which the kinetic Ising model is a special case). The dynamical scaling hypothesis, and some of its consequences are discussed in section V. Sections VI and VII contain brief discussions of long-time relaxation in a disordered Ising model in one and higher dimensions respectively. It is shown that in the disordered Ising model with broken bonds, the relaxation of magnetization to the equilibrium-value is slower than exponential for all temperatures below the critical temperature of the model without disorder.