Two-temperature Ising model at an exact limit

Abstract

We analyze the order-disorder transition for a two dimensional Ising model. We consider a ferromagnetic exchange interaction between the nearest neighbor Ising spins. The spin exchanges are introduced in two different temperatures, at infinite and finite temperatures. The model is first proposed by Præstgaard, Schmittmann, and Zia [1]. In this thesis, we look at a limit of the system where the spin exchange at infinite temperature proceeds at a very fast rate in one of the lattice direction (the “y−direction”). In the other direction (the “x−direction”), the spin exchange at a finite temperature is driven by one of several possible exchange dynamics such as Metropolis, Glauber, and exponential rates. We investigate an exact nonequilibrium stationary state solution of the model far from equilibrium. We apply basic stochastic formalisms such as the Master equation and the Fokker-Planck equation. Our main interest is to analyze the possibility of various types of phase transitions. Using the magnetization as a phase order parameter, we observe two kinds of phase transitions: transverse segregation and longitudinal segregation with respect to the direction x. We find analytically the transition temperature and the nonequilibrium stationary state for small magnetizations at an exact limit. We show that depending on the type of microscopic interaction (such as Metropolis, Glauber, exponential spin exchange rates) the transition temperature and the phase boundary vary. For some exchange rates, we observe no transverse segregation.