Thermodynamics of trajectories of the one-dimensional Ising model

We present a numerical study of the dynamics of the one-dimensional Ising model by applying the large-deviation method to describe ensembles of dynamical trajectories. In this approach trajectories are classified according to a dynamical order parameter and the structure of ensembles of trajectories can be understood from the properties of large-deviation functions, which play the role of dynamical free-energies. We consider both Glauber and Kawasaki dynamics, and also the presence of a magnetic field. For Glauber dynamics in the absence of a field we confirm the analytic predictions of Jack and Sollich about the existence of critical dynamical, or space–time, phase transitions at critical values of the 'counting' field s. In the presence of a magnetic field the dynamical phase diagram also displays first order transition surfaces. We discuss how these non-equilibrium transitions in the 1d Ising model relate to the equilibrium ones of the 2d Ising model. For Kawasaki dynamics we find a much simpler dynamical phase structure, with transitions reminiscent of those seen in kinetically constrained models