The Eyring-Kramers law for potentials with nonquadratic saddles

TL;DR

This paper extends the Eyring-Kramers law to potentials with nonquadratic saddles, providing accurate transition time estimates near bifurcation points where principal curvatures vanish, thus broadening its applicability.

Contribution

It derives the correct transition time prefactor for nonquadratic saddles, including bifurcations with vanishing eigenvalues, beyond the quadratic case analyzed previously.

Findings

Derived the prefactor for nonquadratic saddles in the weak-noise limit.

Analyzed transition times near bifurcation points with vanishing eigenvalues.

Expressed the prefactor in terms of modified Bessel functions for symmetric pitchfork bifurcations.

Abstract

The Eyring-Kramers law describes the mean transition time of an overdamped Brownian particle between local minima in a potential landscape. In the weak-noise limit, the transition time is to leading order exponential in the potential difference to overcome. This exponential is corrected by a prefactor which depends on the principal curvatures of the potential at the starting minimum and at the highest saddle crossed by an optimal transition path. The Eyring-Kramers law, however, does not hold whenever one of these principal curvatures vanishes, since it would predict a vanishing or infinite transition time. We derive the correct prefactor up to multiplicative errors that tend to one in the zero-noise limit. As an illustration, we discuss the case of a symmetric pitchfork bifurcation, in which the prefactor can be expressed in terms of modified Bessel functions, as well as bifurcations with two vanishing eigenvalues. The corresponding transition times are studied in a full neighbourhood of the bifurcation point. These results extend work by Bovier, Eckhoff, Gayrard and Klein, who rigorously analysed the case of quadratic saddles, using methods from potential theory.