Langevin dynamics with constraints and computation of free energy differences

TL;DR

This paper develops and analyzes discretization schemes for constrained Langevin dynamics, enabling exact sampling on submanifolds, accurate free energy gradient estimation, and fluctuation relation proofs, with practical numerical methods.

Contribution

It introduces a simple, corrected discretization of constrained Langevin processes that samples exactly, and proves new results on free energy gradients and fluctuation relations.

Findings

Discretization scheme samples exactly the constrained canonical measure.

Long-term average of Lagrange multipliers yields free energy gradient.

Jarzynski-Crooks relation is valid for constrained Langevin processes.

Abstract

In this paper, we consider Langevin processes with mechanical constraints. The latter are a fundamental tool in molecular dynamics simulation for sampling purposes and for the computation of free energy differences. The results of this paper can be divided into three parts. (i) We propose a simple discretization of the constrained Langevin process based on a standard splitting strategy. We show how to correct the scheme so that it samples {\em exactly} the canonical measure restricted on a submanifold, using a Metropolis rule in the spirit of the Generalized Hybrid Monte Carlo (GHMC) algorithm. Moreover, we obtain, in some limiting regime, a consistent discretization of the overdamped Langevin (Brownian) dynamics on a submanifold, also sampling exactly the correct canonical measure with constraints. The corresponding numerical methods can be used to sample (without any bias) a probability measure supported by a submanifold. (ii) For free energy computation using thermodynamic integration, we rigorously prove that the longtime average of the Lagrange multipliers of the constrained Langevin dynamics yields the gradient of a rigid version of the free energy associated with the constraints. A second order time discretization using the Lagrange multipliers is proposed. (iii) The Jarzynski-Crooks fluctuation relation is proved for Langevin processes with mechanical constraints evolving in time. An original numerical discretization without time-step error is proposed. Numerical illustrations are provided for (ii) and (iii).