Kramers rate for Brownian particles in excitable media with deformable double-well substrates

TL;DR

This paper derives analytical Kramers escape rates for Brownian particles in deformable double-well potentials, revealing a critical deformability parameter where non-Gaussian effects and a quantum tunneling transition emerge.

Contribution

It introduces three models of deformable double-well potentials and provides explicit formulas for escape rates, highlighting a critical point for non-Gaussian corrections and quantum transition.

Findings

Analytical escape rate formulas for three deformable double-well models.

Identification of a critical deformability parameter affecting escape dynamics.

Prediction of a first-order quantum tunneling transition at high deformability.

Abstract

We address the Kramers escape problem for Brownian particles in bistable substrates with deformable double-well shapes. The shape deformability is considered of three distinct forms: in one, the positions of the two degenerate minima can be shifted continuously without affecting the barrier height. In the second the minima positions are kept fix while the barrier height is continuously shifted. In the third the minima positions and barrier height can be tuned simultaneously by changing a single real parameter that we refer to as deformability parameter. We obtain the analytical expression of the Kramers escape rate for the three different double-well models, and identify a critical value of the deformability parameter above which non-Gaussian corrections become relevant. Remarkable enough in the latter region, statistical mechanics predicts a first-order transition in quantum tunneling using the functional action associated with the family of deformable double-well potentials.