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Eyring-Kramers Law for the Underdamped Langevin Process

Seungwoo Lee, Mouad Ramil, Insuk Seo

TL;DR

This work derives the Eyring-Kramers law for the underdamped Langevin process in a double-well potential, establishing the precise low-temperature prefactor in addition to the Arrhenius-type exponential term. Facing irreversibility and hypoellipticity, the authors replace classical capacity-based methods with a novel observable and a perturbative, harmonic-measure framework, coupled with delicate short-time, boundary, and zero-noise analyses. The main result yields a sharp mean transition time from the local minimum to the saddle region, with a prefactor expressed in terms of Hessians at the minimum and saddle and the small negative eigenvalue of the saddle Hessian. The approach broadens metastability analysis to irreversible, degenerate diffusion and provides tools potentially applicable to other hypoelliptic settings, with implications for molecular dynamics and rare-event sampling at low temperature.

Abstract

Consider the underdamped Langevin process $(q(t),p(t))_{t\geq0}$ in $\R^d\times\R^d$. We derive the low-temperature asymptotic of its mean-transition time between basins of attraction for a double-well potential. This asymptotic is called Eyring-Kramers law and often relies in the literature on Potential theory tools which are ill-defined for hypoelliptic processes like the underdamped Langevin process. In this work, we implement a novel approach which circumvents the use of these traditional methods.

Eyring-Kramers Law for the Underdamped Langevin Process

TL;DR

This work derives the Eyring-Kramers law for the underdamped Langevin process in a double-well potential, establishing the precise low-temperature prefactor in addition to the Arrhenius-type exponential term. Facing irreversibility and hypoellipticity, the authors replace classical capacity-based methods with a novel observable and a perturbative, harmonic-measure framework, coupled with delicate short-time, boundary, and zero-noise analyses. The main result yields a sharp mean transition time from the local minimum to the saddle region, with a prefactor expressed in terms of Hessians at the minimum and saddle and the small negative eigenvalue of the saddle Hessian. The approach broadens metastability analysis to irreversible, degenerate diffusion and provides tools potentially applicable to other hypoelliptic settings, with implications for molecular dynamics and rare-event sampling at low temperature.

Abstract

Consider the underdamped Langevin process in . We derive the low-temperature asymptotic of its mean-transition time between basins of attraction for a double-well potential. This asymptotic is called Eyring-Kramers law and often relies in the literature on Potential theory tools which are ill-defined for hypoelliptic processes like the underdamped Langevin process. In this work, we implement a novel approach which circumvents the use of these traditional methods.

Paper Structure

This paper contains 35 sections, 56 theorems, 476 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Metastability of the overdamped Langevin process
  3. Metastability of the underdamped Langevin process
  4. Model and main results
  5. Underdamped Langevin process
  6. Notation
  7. Main result: Eyring-Kramers law
  8. Sketch of Proof
  9. Preliminary analysis
  10. Tightness of invariant measure
  11. Non-explosion and positive recurrence of the underdamped Langevin process
  12. Proof of Proposition \ref{['prop:harm']}
  13. Perturbed underdamped Langevin process
  14. Distance between $X^{\epsilon}(t)$ and $X^{\alpha,\,\epsilon}(t)$
  15. Approximation of $h_{\mathcal{M}_{\epsilon},\,\mathcal{S}_{\epsilon}}^{*}$
  16. ...and 20 more sections

Key Result

Theorem 2.5

We have that where $\mu^{\sigma}=\frac{-\gamma+\sqrt{\gamma^{2}+4\lambda^{\sigma}}}{2}>0$ and $-\lambda^{\sigma}$ is the unique negative eigenvalue of $\mathbb{H}_{U}^{\sigma}$.

Figures (1)

  • Figure 1.1: An illustration of a double-well potential $U$.

Theorems & Definitions (113)

  • Remark 2.3
  • Theorem 2.5: Eyring-Kramers law
  • Remark 2.6
  • Proposition 2.7: Capacity analogous
  • Remark 2.8
  • Proposition 2.9
  • Proposition 2.10
  • Proposition 2.11
  • Proposition 2.12
  • proof : Proof of Theorem \ref{['thm:ek']}
  • ...and 103 more