Mean First Passage Times and Eyring-Kramers formula for Fluctuating Hydrodynamics

TL;DR

This paper extends the Eyring-Kramers formula to systems with conserved quantities, including fluctuating hydrodynamics, providing a rigorous way to estimate transition times in complex physical and socio-economic systems.

Contribution

It derives a generalized Eyring-Kramers formula applicable to systems with zeromodes, broadening its applicability to fluctuating hydrodynamics and related fields.

Findings

Derived a generalized Eyring-Kramers formula for systems with conserved quantities.

Applied the formula to stochastic PDEs in fluctuating hydrodynamics.

Provided estimates for transition times in nanofilm rupture and social segregation.

Abstract

Thermally activated phenomena in physics and chemistry, such as conformational changes in biomolecules, liquid film rupture, or ferromagnetic field reversal, are often associated with exponentially long transition times described by Arrhenius' law. The associated subexponential prefactor, given by the Eyring-Kramers formula, has recently been rigorously derived for systems in detailed balance, resulting in a sharp limiting estimate for transition times and reaction rates. Unfortunately, this formula does not trivially apply to systems with conserved quantities, which are ubiquitous in the sciences: The associated zeromodes lead to divergences in the prefactor. We demonstrate how a generalised formula can be derived, and show its applicability to a wide range of systems, including stochastic partial differential equations from fluctuating hydrodynamics, with applications in rupture of nanofilm coatings and social segregation in socioeconomics.