Theoretical and Numerical Comparison of Some Sampling Methods for Molecular Dynamics

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. Received July 30, 2005. 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 form 〈A〉 = ∫ T∗M A(q, p) dμ(q, p). (1) In the above expression, M denotes the position space (also called the configuration space), and T ∗M its cotangent space. Typically, M = T (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 × R . Let us note that, for biological systems currently studied, N is typically more than 100000. A generic element of the position space M will be denoted by q = (q1, . . . , qN ) and a generic element of the momentum space R by p = (p1, . . . , pN). The so-called canonical probability measure μ appearing in (1) is given by dμ(q, p) = Z−1 exp(−βH(q, p)) dq dp, (2)