Geometric Characterization of the Eyring-Kramers Formula
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 minima as a ratio of two geometric quantities, i.e., the minimal cut and the geodesic distance.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- November 2023
- DOI:
- 10.1007/s00220-023-04845-z
- arXiv:
- arXiv:2206.13206
- Bibcode:
- 2023CMaPh.404..401A
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Probability