Thermodynamics of trajectories of the one-dimensional Ising model

Abstract

We present a numerical study of the dynamics of the one-dimensional Ising modelby applying the large-deviation method to describe ensembles of dynamicaltrajectories. In this approach trajectories are classified according to a dynamicalorder parameter and the structure of ensembles of trajectories can be understoodfrom the properties of large-deviation functions, which play the role of dynamicalfree-energies. We consider both Glauber and Kawasaki dynamics, and also thepresence of a magnetic field. For Glauber dynamics in the absence of a field weconfirm the analytic predictions of Jack and Sollich about the existence of criticaldynamical, or space–time, phase transitions at critical values of the ‘counting’ fields. In the presence of a magnetic field the dynamical phase diagram also displays first ordertransition surfaces. We discuss how these non-equilibrium transitions in the 1d Ising modelrelate to the equilibrium ones of the 2d Ising model. For Kawasaki dynamics we find amuch simpler dynamical phase structure, with transitions reminiscent of those seen inkinetically constrained models.