The Behavior of Processes with Statistical Mechanical Properties

Abstract

Ever since Spitzer’s famous paper in 1970, there has been interest in a class of Markov processes which have as time-reversible stationary measures certain special distributions from the theory of statistical mechanics. The state space for these processes is $\Xi = {\{ - 1, + 1\}^{{{Z^d}}}}$, which is the space of conf igurations of + and — spins on the sites of the lattice Z ^d. Transitions occur when there is a “flip” at a site x ∈ Z ^d, or in other words, a change of sign in the spin at x. The probability that a flip occurs at x in a short time interval (t, t + h], given the history of the process up to time t, is c_x(ξ_t)h + o(h), where ξ_t is the state of the process at time t, and c_x is a nonnegative function defined on =, called the flip rate at x. Simultaneous flips at two different sites do not occur. A system of Markov processes with this description, me process for each possible initial state, is often called a “spin-flip system” with rates {c_x}. Spitzer pointed out that for certain kinds of interaction potentials commonly used in statistical mechanics, one can always find a set of rates {c_x} such that the corresponding spin-flip system has as time-reversible equilibria the Gibbs states that correspond to the interaction potential. (Spitzer’s results required a certain uniqueness hypothesis that was later verified for a large class of systems by Liggett (1972).)