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Overdamped limits for Langevin dynamics with position-dependent coefficients via $L^2$-hypocoercivity

Noé Blassel

Abstract

This note provides a simple derivation of the overdamped approximation for kinetic (or underdamped) equilibrium Langevin dynamics, in cases where certain coefficients depend on the position variable. The equivalent small-mass limit of these dynamics, known as the Kramers--Smoluchowski approximation, in the case of a state-dependent friction coefficient, has been previously studied by a variety of approaches. Our new approach uses hypocoercivity estimates, which may be of interest in their own right, and lead to a very direct derivation, providing in particular a clear explanation of the ``noise-induced drift'' term in the overdamped equation in the case of a state-dependent friction term. Using the same approach, we also treat several effective kinetic dynamical models derived from a coarse-graining approximation of a high-dimensional system, as well as a class of kinetic dynamics with position-dependent mass matrices. All of these models are relevant to applications in computational chemistry. We finally identify a mistake in a related work and suggest a solution.

Overdamped limits for Langevin dynamics with position-dependent coefficients via $L^2$-hypocoercivity

Abstract

This note provides a simple derivation of the overdamped approximation for kinetic (or underdamped) equilibrium Langevin dynamics, in cases where certain coefficients depend on the position variable. The equivalent small-mass limit of these dynamics, known as the Kramers--Smoluchowski approximation, in the case of a state-dependent friction coefficient, has been previously studied by a variety of approaches. Our new approach uses hypocoercivity estimates, which may be of interest in their own right, and lead to a very direct derivation, providing in particular a clear explanation of the ``noise-induced drift'' term in the overdamped equation in the case of a state-dependent friction term. Using the same approach, we also treat several effective kinetic dynamical models derived from a coarse-graining approximation of a high-dimensional system, as well as a class of kinetic dynamics with position-dependent mass matrices. All of these models are relevant to applications in computational chemistry. We finally identify a mistake in a related work and suggest a solution.
Paper Structure (20 sections, 10 theorems, 151 equations, 1 figure)

This paper contains 20 sections, 10 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and first main result
  3. Notation.
  4. Assumptions on $U,V$ and $D$.
  5. Overdamped approximation.
  6. Proofs
  7. Proof of Theorem \ref{['thm:overdamped_limit']}: a first Itô computation.
  8. Proof of Theorem: proof of \ref{['eq:convergence_in_l2']}.
  9. Proof of Corollary \ref{['corr:pathwise']}: tightness.
  10. Proof of Corollary \ref{['corr:traj_averages']}.
  11. Overdamped limit of coarse-grained Langevin dynamics
  12. Overdamped limit of some Langevin equations with position-dependent mass matrices
  13. A canonical transformation.
  14. An Itô computation.
  15. Overdamped limit.
  16. ...and 5 more sections

Key Result

Theorem 1

Assume that $\mu_0\ll \mu$ is such that Suppose that Assumptions hyp:smoothness, hyp:hypoellipticity, hyp:elliptic and hyp:UVreg are satisfied, and let $q\in[1,\infty)$ be the Hölder-conjugate of $p$, namely $q = p/(p-1)$. Then for any $T>0$ and $0<\alpha\leq\frac{2}{q}$, there exists $C(T,\alpha,\mu_0)>0$ such that

Figures (1)

  • Figure 1: Non-commutation of the overdamped and effective/coarse-grained dynamical approximations. Horizontal arrows correspond to overdamped approximations, while vertical arrows correspond to taking effective dynamics through the collective variable $\xi$. The coarse-graining of $X$ through $\xi$ is not equal to $Z$ for general $\xi$.

Theorems & Definitions (17)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Lemma 1: Uniform hypocoercivity in $L^2(\mu)$
  • Theorem 2
  • proof : (Sketch of proof)
  • Lemma 2
  • Corollary 3
  • proof
  • proof
  • ...and 7 more