The Arrhenius formula in kinetic theory and Witten's spectral asymptotics

TL;DR

This paper introduces a novel proof of the Arrhenius formula in kinetic theory by linking diffusion in a potential to quantum tunneling via a Schrödinger operator, connecting it to Witten's spectral asymptotics.

Contribution

It presents a new approach to derive the Arrhenius formula using spectral analysis and Witten's method, bridging kinetic theory and quantum mechanics.

Findings

Arrhenius formula derived from Schrödinger operator analysis

Connection established between diffusion equations and quantum tunneling

Witten spectral asymptotics interpreted as low temperature limit

Abstract

A new approach to the proof of the Arrhenius formula of kinetic theory is proposed. We prove this formula starting from the equation of diffusion in a potential. We put this diffusion equation in the form of evolutionary equation generated by some Schroedinger operator. We show that the Arrhenius formula for the rate of over the barrier transitions follows from the formula for the rate of quantum tunnel transitions for the considered Schroedinger operator. Relation of the proposed approach and the Witten method of the proof of the Morse inequalities is discussed. In our approach the Witten spectral asymptotics takes the form of the low temperature limit and the Arrhenius formula is a correction to the Witten asymptotics.