Analysis of the backward-euler/langevin method for molecular dynamics

This paper develops the theory of a recently introduced computational method for molecular dynamics. The method in question uses the backward-Euler method to solve the classical Langevin equations of a molecular system. Parameters are chosen to produce a cutoff frequency ω_c, which may be set equal to kT/h to simulate quantum-mechanical effects. In the present paper, an ensemble of identical Hamiltonian systems modeled by the backward-Euler/Langevin method is considered, an integral equation for the equilibrium phase-space density is derived, and an asymptotic analysis of that integral equation in the limit Δt → 0 is performed. The result of this asymptotic analysis is a second-order partial differential equation for the equilibrium phase-space density expressed as a function of the constants of the motion. This equation is solved in two special cases: a system of coupled harmonic oscillators and a diatomic molecule with a stiff bond.