Glaubers Ising chain between two thermostats

Abstract

We consider a one-dimensional Ising model with N spins, each in contact with two thermostats of distinct temperatures, T1 and T2. Under Glauber dynamics the stationary state happens to coincide with the equilibrium state at an effective intermediate temperature . The system nevertheless carries a nontrivial energy current between the thermostats. By means of the fermionization technique, for a chain initially in equilibrium at an arbitrary temperature T0 we calculate the Fourier transform of the probability for the time-integrated energy current during a finite time interval τ. In the long time limit we determine the corresponding generating function for the cumulants per site and unit of time, , and explicitly give those with n = 1, 2, 3, 4. We exhibit various phenomena in specific regimes: kinetic mean-field effects when one thermostat flips any spin less often than the other one, as well as dissipation towards a thermostat at zero temperature. Moreover, when the system size N goes to infinity while the effective temperature T vanishes, the cumulants of per unit of time grow linearly with N and are equal to those of a random walk process. In two adequate scaling regimes involving T and N we exhibit the dependence of the first correction upon the ratio of the spin–spin correlation length and the size N.