Thermodynamics of trajectories of the one-dimensional Ising model

TL;DR

This paper investigates the dynamical behavior of the one-dimensional Ising model using large-deviation theory, revealing phase transitions and differences between Glauber and Kawasaki dynamics, with implications for understanding non-equilibrium phenomena.

Contribution

It provides a numerical analysis of dynamical phase transitions in the 1D Ising model, confirming theoretical predictions and exploring effects of magnetic fields and different dynamics.

Findings

Confirmation of dynamical phase transitions in Glauber dynamics

Identification of first order transition surfaces with magnetic field

Simpler phase structure in Kawasaki dynamics

Abstract

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 1$d$ Ising model relate to the equilibrium ones of the 2$d$ Ising model. For Kawasaki dynamics we find a much simple dynamical phase structure, with transitions reminiscent of those seen in kinetically constrained models.