The question of coarse-graining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarse-grained variable ξ(x), where x describes the configuration of the system in a high-dimensional space R n , and ξ is a smooth function with value in R (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on x ∈ R n , the equilibrium properties on ξ(x) are completely determined by the free energy. On the other hand, the question of the effective dynamics on ξ(x) is much more difficult to address. Starting from an overdamped Langevin equation on x ∈ R n , we propose an effective dynamics for ξ(x) ∈ R using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics.AMS classification scheme numbers: 35B40, 82C31, 60H10 ‡ In this article, we do not address the difficult question of how to find a good reaction coordinate. See for instance [24] for some discussion on that point.

Abstract. The purpose of the present article is to compare different phase-space sampling methods, such as purely stochastic methods (Rejection method, Metropolized independence sampler, Importance Sampling), stochastically perturbed Molecular Dynamics methods (Hybrid Monte Carlo, Langevin Dynamics, Biased Random Walk), and purely deterministic methods (Nosé-Hoover chains, Nosé-Poincaré and Recursive Multiple Thermostats (RMT) methods). After recalling some theoretical convergence properties for the various methods, we provide some new convergence results for the Hybrid Monte Carlo scheme, requiring weaker (and easier to check) conditions than previously known conditions. We then turn to the numerical efficiency of the sampling schemes for a benchmark model of linear alkane molecules. In particular, the numerical distributions that are generated are compared in a systematic way, on the basis of some quantitative convergence indicators.Mathematics Subject Classification. 82B80, 37M25, 65C05, 65C40. Phase-space integrals are widely used in Statistical Physics to relate the macroscopic properties of a system to the elementary phenomena at the microscopic scale [17]. In constant temperature (NVT) molecular simulations, these integrals take the formIn the above expression, M denotes the position space (also called the configuration space), and T * M its cotangent space. Typically, M = T 3N (a torus of dimension 3N ) for simulations with periodic boundary conditions (PBC) and N atoms in the simulation cell. In this case, T * M = T 3N × R 3N . Let us note that, for biological systems currently studied, N is typically more than 100 000. A generic element of the position space M will be denoted by q = (q 1 , . . . , q N ) and a generic element of the momentum space R 3N by p = (p 1 , . . . , p N ). The so-called canonical probability measure µ appearing in (1) is given by dµ(q, p) = Z −1 exp(−βH(q, p)) dq dp,

We follow up on our previous work [C. Le Bris, F. Legoll, and A. Lozinski, Chinese Ann. Math. Ser. B, 34 (2013), pp. 113-138], where we have studied a multiscale finite element method (MsFEM) in the vein of the classical Crouzeix-Raviart FEM that is specifically adapted for highly oscillatory elliptic problems. We adapt the approach to address here a multiscale problem on a perforated domain. An additional ingredient of our approach is the enrichment of the multiscale finite element space using bubble functions. We first establish a theoretical error estimate. We next show that, for the problem we consider, the approach we propose outperforms all dedicated existing variants of the MsFEM we are aware of.

No abstract