Local, Cluster, and Transitional Monte Carlo Dynamics

TL;DR

This paper reviews local and cluster Monte Carlo methods, introduces a new transitional Monte Carlo dynamics, and provides an exact solution for the relaxation dynamics of the Ising model across dimensions.

Contribution

It introduces transitional Monte Carlo dynamics and analytically solves the relaxation master equation for the Ising model in all dimensions.

Findings

Exact solution for 1D case.

Continuum limit governed by a PDE.

Complete understanding of relaxation dynamics.

Abstract

We review the local Monte Carlo dynamics and Swendsen-Wang cluster algorithm. We introduce and analyze a new Monte Carlo dynamics known as transitional Monte Carlo. The transitional Monte Carlo algorithm samples energy probability distribution P(E) with a transition matrix obtained from single-spin-flip dynamics. We analyze the relaxation dynamics master equation, d P(E, t)/ dt = sum{E'} T(E,E') P(E',t), associated with Ising model in d dimensions. In one dimension, we obtain an exact solution. We show in all dimensions in the continuum limit the dynamics is governed by the partial differential equation d P/dt' = d^2 P / d x^2 + x dP/dx + P. where x and t' are rescaled energy deviation from the equilibrium value and rescaled time, respectively. This equation is readily solved. Thus, we have a complete understanding of the dynamics.