Metastability in Markov processes

TL;DR

This paper develops a formalism for describing metastable states in finite Markov chains with detailed balance, linking phase space partitioning, thermodynamic descriptions, and numerical validation in the kinetic Ising model.

Contribution

It introduces a new formalism for identifying and characterizing metastable states in Markov processes, including thermodynamic descriptions and numerical validation.

Findings

Metastable states can be characterized by restricted Markov chains.

The probability distribution of metastable states is proportional to the equilibrium distribution.

Numerical tests confirm the theoretical proportionality in the kinetic Ising model.

Abstract

We present a formalism to describe slowly decaying systems in the context of finite Markov chains obeying detailed balance. We show that phase space can be partitioned into approximately decoupled regions, in which one may introduce restricted Markov chains which are close to the original process but do not leave these regions. Within this context, we identify the conditions under which the decaying system can be considered to be in a metastable state. Furthermore, we show that such metastable states can be described in thermodynamic terms and define their free energy. This is accomplished showing that the probability distribution describing the metastable state is indeed proportional to the equilibrium distribution, as is commonly assumed. We test the formalism numerically in the case of the two-dimensional kinetic Ising model, using the Wang--Landau algorithm to show this proportionality explicitly, and confirm that the proportionality constant is as derived in the theory. Finally, we extend the formalism to situations in which a system can have several metastable states.