Kramers' law: Validity, derivations and generalisations

TL;DR

This paper reviews the mathematical foundations, derivations, and generalizations of Kramers' law, which predicts transition times for particles in potential landscapes, highlighting its validity and limitations across physics and mathematics.

Contribution

It provides a comprehensive review of rigorous proofs, extensions, and cases where Kramers' law does not hold, bridging concepts from physics and mathematics.

Findings

Rigorous proofs of Kramers' law are summarized.

Generalizations of the law are discussed.

Situations where Kramers' law fails are identified.

Abstract

Kramers' law describes the mean transition time of an overdamped Brownian particle between local minima in a potential landscape. We review different approaches that have been followed to obtain a mathematically rigorous proof of this formula. We also discuss some generalisations, and a case in which Kramers' law is not valid. This review is written for both mathematicians and theoretical physicists, and endeavours to link concepts and terminology from both fields.