Phase Transition and Free Action of Non-equilibrium Systems

Abstract

Extending the concept of free energy to non-equilibrium systems is a central problem in non-equilibrium statistical mechanics. In this dissertation, we discuss the issue for two classes of non-equilibrium systems. In the first part, we focus on systems endowed with a stochastic dynamics that produces non-equilibrium steady states.We first show that a naive generalization of the free energy, based on replacing the canonical ensemble by the non-equilibrium steady state distribution, does not capture dynamical information at steady state. To resolve this, we introduce a new concept,

which we term the "free action". This can be viewed as free energy on path-space and it naturally captures macroscopic transition rates in the thermodynamic limit. Moreover, the path-space formulation allows us to develop an efficient numerical algorithm, based on Hamiltonian Monte-Carlo and thermodynamic integration, to calculate the free action. We illustrate our framework with two examples. The first is a minimal model of a chemical reaction network exhibiting a first order phase transition, where we use the free action to identify the transition point and transition mechanisms. The second example is based the Staver-Levin model of forest-savanna landscape formation. Here, we show that our path-based framework remains powerful even when direct methods becomes prohibitively inefficient.

In the second part of the thesis, we consider non-ergodic systems having deterministic evolution dynamics but random initial data. A prototypical model, which can be considered a zero-temperature Ising model, is discussed. We define and calculate exactly the free energy and the free action. Essentially, in these systems the role of the canonical ensemble is played by the ensemble of initial conditions. The usefulness of this approach is demonstrated by analyzing the laminar-turbulent transition in 2D Poiseuille flow. By extensive numerical computations, we show the existence of a thermodynamic limit and the free energy, action and various thermodynamic relations. Moreover, we show that in the thermodynamic framework, the laminar-turbulent transition can be regarded as a continuous phase transition. Although the set-up of such non-ergodic models are very different from those in classical statistical

physics, one can nevertheless uncover interesting thermodynamic structures.