Quantitative low-temperature spectral asymptotics for reversible diffusions in temperature-dependent domains

TL;DR

This paper develops new low-temperature spectral asymptotics for reversible diffusions in temperature-dependent domains, extending classical formulas and providing insights into the spectral gap's sensitivity to domain shape, with applications in molecular dynamics.

Contribution

It introduces novel asymptotic estimates for the spectrum of Langevin dynamics with temperature-dependent boundaries, extending the Eyring-Kramers formula to this setting.

Findings

Precise estimates of spectral gap and principal eigenvalue.

Extension of Eyring-Kramers formula to temperature-dependent domains.

Insights into the influence of domain shape near critical points.

Abstract

We derive novel low-temperature asymptotics for the spectrum of the infinitesimal generator of the overdamped Langevin dynamics. The novelty is that this operator is endowed with homogeneous Dirichlet conditions at the boundary of a domain which depends on the temperature. From the point of view of stochastic processes, this gives information on the long-time behavior of the diffusion conditioned on non-absorption at the boundary, in the so-called quasistationary regime. Our results provide precise estimates of the spectral gap and principal eigenvalue, extending the Eyring-Kramers formula. The phenomenology is richer than in the case of a fixed boundary and gives new insight into the sensitivity of the spectrum with respect to the shape of the domain near critical points of the energy function. Our work is motivated by--and is relevant to--the problem of finding optimal hyperparameters for accelerated molecular dynamics algorithms.