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Journal of the Physical Society of Japan

Critical Relaxation of Three-Dimensional Kinetic Ising Model

Macoto Kikuchi, Yutaka Okabe

Published Apr 15, 1986• 26 citations
Publisher

Abstract

We report a Monte Carlo study on the critical relaxation of the three-dimensional kinetic Ising model from a nonequilibrium initial state. It is shown that the details of the method, namely, a choice of the sampling procedure and a functional form of the transition probability do not affect an asymptotic power-law relaxation. The finite-size effects on the critical relaxation is also studied. Cross-over from the power-law relaxation to the exponential one is found, which is discussed in view of the dynamical finite-size scaling theory.

DOI
10.1143/jpsj.55.1359

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We report a Monte Carlo study on the critical relaxation of the three-dimensional kinetic Ising model from a nonequilibrium initial state. It is shown that the details of the method, namely, a choice of the sampling procedure and a functional form of the transition probability do not affect an asymptotic power-law relaxation. The finite-size effects on the critical relaxation is also studied. Cross-over from the power-law relaxation to the exponential one is found, which is discussed in view of the dynamical finite-size scaling theory.
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We investigate nonequilibrium relaxations of Ising models at the critical point by using a cluster update. While preceding studies imply that nonequilibrium cluster-flip dynamics at the critical point are universally described by the stretched-exponential function, we find that the dynamics changes from the stretched exponential to the power function as the dimensionality is increased: The two-, three-, four-, and infinite-dimensional Ising models are numerically studied, and the four- and infinite-dimensional Ising models exhibit the power-law relaxation. We also show that the finite-size scaling analysis using the normalized correlation length is markedly effective for the analysis of relaxational processes rather than the direct use of the Monte Carlo step.
We present field-theoretic and numerical studies of finite-size effects on the exponential relaxation times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$ of the order parameter and the square of the order parameter near the critical point of three-dimensional Ising-like systems. Finite-size scaling functions of ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$ are calculated for the relaxational dynamics of the ${\ensuremath{\phi}}^{4}$ model in cubic geometry with periodic boundary conditions. At Tc{T}_{c}Tc​ we predict the universal ratio $\frac{{\ensuremath{\tau}}_{1}}{{\ensuremath{\tau}}_{2}}=5.4$. Below Tc{T}_{c}Tc​, a maximum of ${\ensuremath{\tau}}_{2}$ is predicted. New Monte Carlo data for the Ising model are presented which are in good agreement with the predictions.
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.
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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.