Eyring--Kramers law for the hyperbolic $\phi^4$ model

TL;DR

This paper analyzes the transition rates between metastable states in a stochastic wave equation with a double-well potential, connecting quantum field theory and stochastic dynamics, and computes key components of the transition frequency.

Contribution

It provides a novel computation of the invariant measure component using variational stochastic quantization and extends analysis to specific cases with random data.

Findings

Computed the invariant measure component for $d=2,3$ using variational methods.

Derived the transmission coefficient for the 2D equation with random data.

Connected quantum field theory with metastable transition analysis in stochastic wave equations.

Abstract

We study the expected transition frequency between the two metastable states of a stochastic wave equation with double-well potential. By transition state theory, the frequency factorizes into two components: one depends only on the invariant measure, given by the $\phi^4_d$ quantum field theory, and the other takes the dynamics into account. We compute the first component with the variational approach to stochastic quantization when $d = 2, 3$. For the two-dimensional equation with random data but no stochastic forcing, we also compute the transmission coefficient.