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Generalized Langevin And Nos{é}-hoover Processes Absorbed At The Boundary Of A Metastable Domain

Arnaud Guillin, D I Lu, Boris Nectoux, Liming Wu

TL;DR

This work investigates metastable dynamics of molecular systems by proving the existence, uniqueness, and exponential convergence of quasi-stationary distributions for the generalized Langevin and Nosé-Hoover processes in highly nonregular domains and with singular potentials. The authors develop an energy-splitting framework together with carefully constructed Lyapunov functions W_δ to obtain return-from-infinity properties and to verify a set of regularity conditions (C1)–(C5) (or their extensions) that drive the QSD results. The results cover both locally Lipschitz drifts and singular interaction potentials, including the extension to non-gradient force fields (b non-gradient), and they extend existing outcomes for kinetic Langevin dynamics by weakening boundary regularity requirements. The established QSDs come with exponential convergence, a positive spectral gap, and robustness under weak regularity, providing a rigorous underpinning for accelerated dynamics methods in molecular simulation and for understanding metastable transitions across basins of attraction. Overall, the paper advances mathematical foundations for metastability in hypoelliptic, nonreversible stochastic systems and broadens applicability to realistic, singular interaction potentials and boundary geometries.

Abstract

In this paper, we prove in a very weak regularity setting existence and uniqueness of quasi-stationary distributions as well as exponential conver- gence towards the quasi-stationary distribution for the generalized Langevin and the Nos{é}-Hoover processes, two processes which are widely used in molecular dynamics. The case of singular potentials is considered. With the techniques used in this work, we are also able to greatly improve existing results on quasi-stationary distributions for the kinetic Langevin process to a weak regularity setting.

Generalized Langevin And Nos{é}-hoover Processes Absorbed At The Boundary Of A Metastable Domain

TL;DR

This work investigates metastable dynamics of molecular systems by proving the existence, uniqueness, and exponential convergence of quasi-stationary distributions for the generalized Langevin and Nosé-Hoover processes in highly nonregular domains and with singular potentials. The authors develop an energy-splitting framework together with carefully constructed Lyapunov functions W_δ to obtain return-from-infinity properties and to verify a set of regularity conditions (C1)–(C5) (or their extensions) that drive the QSD results. The results cover both locally Lipschitz drifts and singular interaction potentials, including the extension to non-gradient force fields (b non-gradient), and they extend existing outcomes for kinetic Langevin dynamics by weakening boundary regularity requirements. The established QSDs come with exponential convergence, a positive spectral gap, and robustness under weak regularity, providing a rigorous underpinning for accelerated dynamics methods in molecular simulation and for understanding metastable transitions across basins of attraction. Overall, the paper advances mathematical foundations for metastability in hypoelliptic, nonreversible stochastic systems and broadens applicability to realistic, singular interaction potentials and boundary geometries.

Abstract

In this paper, we prove in a very weak regularity setting existence and uniqueness of quasi-stationary distributions as well as exponential conver- gence towards the quasi-stationary distribution for the generalized Langevin and the Nos{é}-Hoover processes, two processes which are widely used in molecular dynamics. The case of singular potentials is considered. With the techniques used in this work, we are also able to greatly improve existing results on quasi-stationary distributions for the kinetic Langevin process to a weak regularity setting.
Paper Structure (38 sections, 32 theorems, 195 equations)

This paper contains 38 sections, 32 theorems, 195 equations.

Table of Contents

  1. Setting and main results
  2. Introduction
  3. Purpose of this work
  4. Contributions
  5. On the approach to check the regularity conditions
  6. Improving existing results on the kinetic Langevin process
  7. Related results
  8. Notation
  9. Quasi-stationary distributions for the generalized Langevin process
  10. Generalized Langevin process with locally Lipschitz drifts
  11. Quasi-stationary distribution for the Nosé-Hoover process
  12. Main result
  13. On Assumption [V sing2]
  14. Generalized Langevin process: proof of Theorem \ref{['th.1']}
  15. On the conditions (C1), (C2), and (C4)
  16. ...and 23 more sections

Key Result

Proposition 1.2

(wu1999) Assume that a sequence of random variables $(Y_n)_{n\ge 0}$ defined on a probability space $(\Omega, \mathcal{F}, \mathbf P)$ with values in a Polish space $\mathscr S$ converges in $\mathbf P$-probability to $Y$. Assume also that there exists a fixed probability measure $\mu$ such that for

Theorems & Definitions (62)

  • Definition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • proof
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • proof
  • Theorem 1.7
  • Remark 1.8
  • ...and 52 more