  Extended Lagrangian molecular dynamics (XLMD) is a general method for
performing molecular dynamics simulations using quantum and classical many-body
potentials. Recently several new XLMD schemes have been proposed and tested on
several classes of many-body polarization models such as induced dipoles or
Drude charges, by creating an auxiliary set of these same degrees of freedom
that are reversibly integrated through time. This gives rise to a singularly
perturbed Hamiltonian system that provides a good approximation to the time
evolution of the real mutual polarization field. To further improve upon the
accuracy of the XLMD dynamics, and to potentially extend it to other many-body
potentials, we introduce a stochastic modification which leads to a set of
singularly perturbed Langevin equations with degenerate noise. We prove that
the resulting Stochastic-XLMD converges to the accurate dynamics, and the
convergence rate is both optimal and is independent of the accuracy of the
initial polarization field. We carefully study the scaling of the damping
factor and numerical noise for efficient numerical simulation for
Stochastic-XLMD, and we demonstrate the effectiveness of the method for model
polarizable force field systems.
