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Langevin dynamics with constraints and computation of free energy differences


Authors: Tony Lelièvre, Mathias Rousset and Gabriel Stoltz
Journal: Math. Comp. 81 (2012), 2071-2125
MSC (2010): Primary 82B80, 65C30; Secondary 82B35
Published electronically: February 22, 2012
MathSciNet review: 2945148
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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 splitting strategy. We show how to correct the scheme so that it samples exactly the canonical measure restricted on a submanifold, using a Metropolis-Hastings correction 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. (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 discretization error is proposed, and its overdamped limit is studied. Numerical illustrations are provided for (ii) and (iii).


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Additional Information

Tony Lelièvre
Affiliation: Université Paris Est, CERMICS and INRIA, MICMAC project-team Ecole des Ponts ParisTech, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée, France
Email: lelievre@cermics.enpc.fr

Mathias Rousset
Affiliation: INRIA Lille, Nord Europe, Parc Scientifique de la Haute Borne, 40 avenue Halley, Bt. A Park Plaza, 59650 Villeneuve d’Ascq, France
Email: mathias.rousset@inria.fr

Gabriel Stoltz
Affiliation: Université Paris Est, CERMICS and INRIA, MICMAC project-team Ecole des Ponts ParisTech, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée, France
Email: stoltz@cermics.enpc.fr

DOI: https://doi.org/10.1090/S0025-5718-2012-02594-4
Keywords: Constrained stochastic differential equations, free energy computations, nonequilibrium dynamics
Received by editor(s): June 23, 2010
Received by editor(s) in revised form: April 18, 2011, and June 24, 2011
Published electronically: February 22, 2012
Additional Notes: We would like to thank the anonymous referee for a careful reading of the manuscript and useful suggestions. This work was supported by the Agence Nationale de la Recherche, under the grant ANR-09-BLAN-0216-01 (MEGAS)
Article copyright: © Copyright 2012 American Mathematical Society