An introduction to phase transitions in stochastic dynamical systems

TL;DR

This paper introduces phase transitions in stochastic dynamical systems, covering both equilibrium and nonequilibrium cases, and provides a unified framework for understanding their steady-state behaviors and phase transitions.

Contribution

It offers a unified conceptual framework for phase transitions in both equilibrium and nonequilibrium stochastic systems, extending the notions of partition function and free energy.

Findings

Provides expressions for partition function and free energy in nonequilibrium systems.

Predicts macroscopic phase behavior in exactly solved nonequilibrium models.

Unifies the treatment of phase transitions across equilibrium and nonequilibrium dynamics.

Abstract

We give an introduction to phase transitions in the steady states of systems that evolve stochastically with equilibrium and nonequilibrium dynamics, the latter defined as those that do not possess a time-reversal symmetry. We try as much as possible to discuss both cases within the same conceptual framework, focussing on dynamically attractive `peaks' in state space. A quantitative characterisation of these peaks leads to expressions for the partition function and free energy that extend from equilibrium steady states to their nonequilibrium counterparts. We show that for certain classes of nonequilibrium systems that have been exactly solved, these expressions provide precise predictions of their macroscopic phase behaviour.