On ergodic relaxation time in the three-dimensional Ising model

Abstract

We have studied the dynamical decay of the autocorrelation function of the 3D Ising model for different sizes L = 20–52 of spin cluster-cubes. The behaviour of the longest, ergodic relaxation time, τ_e, of a finite domain below the phase transition temperature T_c was mostly considered for two types of phase transition dynamics. A study of the scaling properties of τ_e demonstrates a negligible difference between the types of dynamics used, but a considerable difference for different boundary conditions. In contrast to the known result for periodic boundary conditions (τ_e ~ L^z exp [const(Lє^ν)²], where z and ν are the dynamical and correlation length exponents, respectively, and є = 1 – T/T_c), the ergodic relaxation time for open boundary conditions is proportional to L^z exp [const(Lє^ν)^2k] with coeffcient k for lattices explored in this work slightly decreasing with L in between 1.65 and 1.58. This result implies that only the lattices of sizes close to or exceeding L = 300 with open boundary conditions might have ergodic relaxation times similar to those with perodic boundary conditions.