An ergodic sampling scheme for constrained Hamiltonian systems with applications to molecular dynamics

This article addresses the problem of computing the Gibbs distribution of a Hamiltonian system that is subject to holonomic constraints. In doing so, we extend recent ideas of Cances et al. (M2AN 41(2), 351-389, 2007) who could prove a Law of Large Numbers for unconstrained molecular systems with a separable Hamiltonian employing a discrete version of Hamilton's principle. Studying ergodicity for constrained Hamiltonian systems, we specifically focus on the numerical discretization error: even if the continuous system is perfectly ergodic this property is typically not preserved by the numerical discretization. The discretization error is taken care of by means of a hybrid Monte-Carlo algorithm that allows for sampling bias-free expectation values with respect to the Gibbs measure independently of the (stable) step-size. We give a demonstration of the sampling algorithm by calculating the free energy profile of a small peptide.