Time-Dependent Statistics of the Ising Model

TLDR

The Ising model assumes spins randomly change states under an external agency, with their dynamics constrained by detailed balance of the equilibrium state. Coupling is introduced by making spin transition probabilities depend on neighboring spins, and the resulting Markov process is analyzed for a closed N‑member chain. Explicit expressions for time‑dependent spin expectations and pair correlations are derived, the weak‑field frequency‑dependent magnetic susceptibility is obtained, and fluctuation‑dissipation theorems link this susceptibility to the Fourier transform of the equilibrium magnetization correlation function.

Abstract

The individual spins of the Ising model are assumed to interact with an external agency (e.g., a heat reservoir) which causes them to change their states randomly with time. Coupling between the spins is introduced through the assumption that the transition probabilities for any one spin depend on the values of the neighboring spins. This dependence is determined, in part, by the detailed balancing condition obeyed by the equilibrium state of the model. The Markoff process which describes the spin functions is analyzed in detail for the case of a closed N-member chain. The expectation values of the individual spins and of the products of pairs of spins, each of the pair evaluated at a different time, are found explicitly. The influence of a uniform, time-varying magnetic field upon the model is discussed, and the frequency-dependent magnetic susceptibility is found in the weak-field limit. Some fluctuation-dissipation theorems are derived which relate the susceptibility to the Fourier transform of the time-dependent correlation function of the magnetization at equilibrium.