Generalisation of the Eyring-Kramers transition rate formula to irreversible diffusion processes

TL;DR

This paper extends the classical Eyring-Kramers transition rate formula to irreversible diffusion processes by incorporating the quasipotential and non-Gibbsianness effects, providing a more general understanding of metastable transition times.

Contribution

It generalizes the Eyring-Kramers formula to irreversible diffusions using quasipotential and correction terms for non-Gibbsianness, based on WKB and probabilistic analysis.

Findings

Derived a formula for transition rate prefactors in irreversible diffusions.

Identified the role of quasipotential and non-Gibbsianness in transition rates.

Provided a probabilistic and WKB analysis framework for metastable transitions.

Abstract

In the small noise regime, the average transition time between metastable states of a reversible diffusion process is described at the logarithmic scale by Arrhenius' law. The Eyring-Kramers formula classically provides a subexponential prefactor to this large deviation estimate. For irreversible diffusion processes, the equivalent of Arrhenius' law is given by the Freidlin-Wentzell theory. In this paper, we compute the associated prefactor and thereby generalise the Eyring-Kramers formula to irreversible diffusion processes. In our formula, the role of the potential is played by Freidlin-Wentzell's quasipotential, and a correction depending on the non-Gibbsianness of the system along the instanton is highlighted. Our analysis relies on a WKB analysis of the quasistationary distribution of the process in metastable regions, and on a probabilistic study of the process in the neighbourhood of saddle-points of the quasipotential.