Testing the energy diffusion approximation for the escape of a Brownian particle from a potential pocket

TL;DR

This study rigorously compares the energy diffusion approximation with exact numerical models for thermal decay of metastable states, revealing limitations of the approximation at higher damping levels.

Contribution

The paper constructs and validates a Langevin-type equation for the action, providing a detailed comparison with exact decay rates across various potentials and parameters.

Findings

Action diffusion approach agrees within 50% at low damping

Exact numerical modeling uses Langevin equations for coordinate and momentum

Limitations of the energy diffusion approximation are identified at higher damping

Abstract

For the first time, the energy diffusion approximation is confronted at the percent level with the exact numerical modeling of thermal decay of a metastable state. The latter is performed using the quasistationary decay rates resulting from the Langevin equations for the coordinate and conjugated momentum. For the energy (or action) diffusion approach, a Langevin-type equation for the action is constructed, validated, and solved numerically. The comparison of two approaches is performed for four potentials (two of which are anharmonic) in a wide range of two dimensionless scaling parameters: the governing parameter $G$ reflecting how high is the barrier with respect to the temperature and the damping parameter $\varphi$ expressing the friction strength. It turns out that the action diffusion approach produces the rate which is in 50% agreement with the exact one only at $\varphi<0.02$ contrary to $\varphi<1$ as claimed in the literature.