On the Rate of Convergence to Equilibrium of the Andersen Thermostat in Molecular Dynamics

摘要

It has been shown in E and Li (Comm. Pure. Appl. Math., 2007, in press) that the Andersen dynamics is uniformly ergodic. Exponential convergence to the invariant measure is established with an error bound of the form $$mathit{const}cdotexp{(-mathit{const}cdotkappa(nu)nu^{2N}t)},$$ where N is the number of particles, ν is the collision frequency and κ(ν)→const as ν→0. In this article we study the dependence on ν of the rate of convergence to equilibrium. In the one dimension and one particle case, we improve the error bound to be $$mathit{const}cdotexp{(-mathit{const}cdotkappa(nu)nu t)}.$$ In the d-dimension N-particle free-streaming case, it is proved that the optimal error bound is $$mathit{const}cdotexp{biggl(-mathit{const}cdotfrac{nu}{N}tbiggr)}.$$ It is also shown that as ν→∞, on the diffusive time scale, the Andersen dynamics converges to a Smoluchowski equation.