Geometric Characterization of the Eyring-Kramers Formula

TL;DR

This paper extends the Eyring-Kramers law to degenerate minima and saddles, providing a geometric characterization of transition times for over-damped Brownian particles in complex potential landscapes.

Contribution

It introduces a new sharp geometric characterization of capacity, extending the Eyring-Kramers law to degenerate cases with multiple saddles at the same height.

Findings

Provides an upper bound on mean transition time in degenerate cases.

Characterizes capacity as a ratio of geometric quantities.

Extends Eyring-Kramers law to flat minima and saddles.

Abstract

In this paper we consider the mean transition time of an over-damped Brownian particle between local minima of a smooth potential. When the minima and saddles are non-degenerate this is in the low noise regime exactly characterized by the so called Eyring-Kramers law and gives the mean transition time as a quantity depending on the curvature of the minima and the saddle. In this paper we find an extension of the Eyring-Kramers law giving an upper bound on the mean transition time when both the minima/saddles are degenerate (flat) while at the same time covering multiple saddles at the same height. Our main contribution is a new sharp characterization of the capacity of two local minimas as a ratio of two geometric quantities, i.e., the smallest separating surface and the geodesic distance.