2025/01/27 by Noé Blassel, Tony Lelièvre, Blassel, Noé +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Spectral Theory in Mathematical Physics #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2501.16082
openalex publication_date 2025/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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.