Transport coefficients from computer experiments: A stochastic Ising model

Abstract

Transport coefficients for a two-dimensional stochastic Ising model with conserved energy and magnetization are calculated by computer simulation. Transport coefficients are derived from the current-current correlations in the hydrodynamic regime. Energy and magnetization currents are evaluated by a Monte Carlo method. By summing these currents over finite space-time regions (bins) it is possible to circumvent many of the statistical problems associated with the $(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})\ensuremath{\rightarrow}0$ limit. The limiting forms of the autocorrelations of such currents for large bin sizes are related to the transport coefficients. The coefficient of spin diffusion is also calculated from the mean-square distance that a spin travels in a given time interval. The spin diffusion $D$ and the kinetic coefficients $\frac{\ensuremath{\chi}D}{\ensuremath{\beta}}$ and $\frac{\ensuremath{\kappa}T}{\ensuremath{\beta}}$ are calculated for the case of zero net magnetization over a range of energy (temperature) values. Here $\ensuremath{\chi}$ denotes the susceptibility and $\ensuremath{\beta}$ the inverse temperature $\frac{1}{{k}_{B}T}$, ${k}_{B}$ being the Boltzmann constant. High- and low-temperature behavior is discussed. Dips are observed at temperatures where the specific heat of the corresponding finite Ising model has a maximum. It is argued that this reflects the critical behavior of transport coefficients, which has not previously been observed in the computer studies of transport coefficients. The methods developed are applicable to other, physically more-realistic systems.