Critical Behavior of the Kramers Escape Rate in Asymmetric Classical Field Theories

TL;DR

This paper investigates how asymmetric classical field theories exhibit a phase transition in escape rates under weak noise, revealing critical behavior and crossover phenomena near a specific interval length.

Contribution

It introduces an asymmetric Ginzburg-Landau model and analyzes the critical behavior of escape rates, extending understanding beyond symmetric models.

Findings

Critical length l_c marks a second-order phase transition in escape behavior.

Prefactor divergence indicates a crossover from non-Arrhenius to Arrhenius regimes.

Similar phenomena are observed in more general asymmetric models.

Abstract

We introduce an asymmetric classical Ginzburg-Landau model in a bounded interval, and study its dynamical behavior when perturbed by weak spatiotemporal noise. The Kramers escape rate from a locally stable state is computed as a function of the interval length. An asymptotically sharp second-order phase transition in activation behavior, with corresponding critical behavior of the rate prefactor, occurs at a critical length l_c, similar to what is observed in symmetric models. The weak-noise exit time asymptotics, to both leading and subdominant orders, are analyzed at all interval lengthscales. The divergence of the prefactor as the critical length is approached is discussed in terms of a crossover from non-Arrhenius to Arrhenius behavior as noise intensity decreases. More general models without symmetry are observed to display similar behavior, suggesting that the presence of a ``phase transition'' in escape behavior is a robust and widespread phenomenon.