Langevin dynamics with constraints and computation of free energy differences

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)

Reliable Approximation of Long Relaxation Timescales in Molecular Dynamics

“…The initial states x 0 are sampled from the probability measure µ z ; this can be achieved by using the numerical schemes proposed in [15,34,35], which simulate the original dynamics (1) and then project the state onto the submanifold ξ −1 (z).…”

Investigating rare events with nonequilibrium work measurements. II. Transition and reaction rates

“…Note that the discrete version of Langevin equation is typically associated with so-called discretization errors. 54,55 The formalism presented here was based on continuous equations of motion. However, in practice, the particular discretization algorithm along with the particular timestep used to play a role in determining the associated error.…”

Ambient space formulations and statistical mechanics of holonomically constrained Langevin systems

“…On the other hand it remains unclear whether or not the impetus-striction formulation can lead to particularly efficient numerical implementations. To our knowledge very few of the existing numerical schemes for Langevin equations can handle holonomic constraints [31,18,21] or general (possibly degenerate) Hamiltonians [27].…”

“…To our knowledge the Langevin theory, in which the stochastic motion of both translational and rotational degrees of freedom can be strongly coupled, has only recently been fully described [32]; cf. also [11,20,21].…”