[Paper Review] Metastability for general dynamics with rare transitions: escape time and critical configurations

Emilio N. M. Cirillo, Francesca R. Nardi+1 more|arXiv (Cornell University)|Dec 26, 2014

Cellular Automata and Applications39 references45 citations

TL;DR

This paper extends the pathwise approach to metastability beyond Metropolis dynamics by introducing a generalized energy landscape based on virtual energy and transition rate costs. It establishes a variational framework for computing escape times and critical configurations in general rare transitions Markov chains, enabling rigorous analysis of metastability in non-Gibbsian systems like Probabilistic Cellular Automata.

ABSTRACT

Metastability is a physical phenomenon ubiquitous in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions Markov chains. For Metropolis chains associated with Statistical Mechanics systems, this phenomenon has been described in an elegant way in terms of the energy landscape associated to the Hamiltonian of the system. In this paper, we provide a similar description in the general rare transitions setup. Beside their theoretical content, we believe that our results are a useful tool to approach metastability for non--Metropolis systems such as Probabilistic Cellular Automata.

Motivation & Objective

• To extend the pathwise approach to metastability beyond Metropolis dynamics to general Markov chains with rare transitions.
• To develop a rigorous mathematical framework for analyzing metastable escape times and critical configurations in systems without a Gibbs invariant measure.
• To provide a systematic tool for studying metastability in non-Metropolis systems such as Probabilistic Cellular Automata.
• To generalize the energy landscape description of metastability by introducing a virtual energy function derived from transition rates and invariant measure.
• To establish equivalence between cycle-based and path-based approaches in the general Freidlin–Wentzell setup using a novel path height definition.

Proposed method

• Introduces a generalized path height based on the virtual energy function and exponential transition costs, replacing the Hamiltonian used in Metropolis dynamics.
• Applies the general cycle theory from [8, 9] to analyze metastable behavior in non-reversible, non-Gibbsian Markov chains.
• Defines cycles as subsets where the process typically visits all states before exiting, with stability determined by the minimal path height to exit.
• Uses a variational problem over the induced landscape of path heights to identify critical configurations and compute escape times.
• Employs large deviation estimates and exponential moment bounds to control hitting time distributions in the zero-temperature limit.
• Develops a constructive method to compute the virtual energy step-by-step from transition rates using a reference state and path integrals of rate differences.

Experimental results

Research questions

• RQ1 How can the pathwise approach to metastability be generalized to non-Metropolis, non-reversible Markov chains with non-Gibbsian invariant measures?
• RQ2 What is the appropriate generalization of the energy landscape concept when no Hamiltonian is available?
• RQ3 How can the escape time from a metastable state be characterized in terms of transition rates and invariant measure in the low-temperature regime?
• RQ4 What defines the critical configurations that the system must pass through during metastable escape in general rare transitions?
• RQ5 Can the cycle-based and path-based approaches to metastability be reconciled in the general Freidlin–Wentzell framework?

Key findings

• The escape time from a metastable state is exponentially large in the inverse temperature β, with the exponent given by the minimal path height in the virtual energy landscape.
• The critical configurations are those lying at the top of the most convenient path (minimal height path) from the metastable state to the stable state.
• The virtual energy function H(x) is explicitly constructible from transition rates via path integrals of rate differences, even when no Hamiltonian exists.
• The path height is defined as the sum of virtual energy differences along a path, enabling a variational characterization of metastable escape.
• The cycle theory framework is extended to general chains by proving equivalence between the probabilistic cycle definition and the energy-based cycle definition using virtual energy.
• For Probabilistic Cellular Automata, the virtual energy can be computed explicitly, making the full metastability analysis tractable via the proposed framework.