A test problem for molecular dynamics integrators

We derive a test problem for evaluating the ability of time-stepping methods to preserve the statistical properties of systems in molecular dynamics. We consider a family of deterministic systems consisting of a finite number of particles interacting on a compact interval. The particles are given random initial conditions and interact through instantaneous energy- and momentum-conserving collisions. As the number of particles, the particle density, and the mean particle speed go to infinity, the trajectory of a tracer particle is shown to converge to a stationary Gaussian stochastic process. We approximate this system by one described by a system of ordinary differential equations and provide numerical evidence that it converges to the same stochastic process. We simulate the latter system with a variety of numerical integrators, including the symplectic Euler method, a fourth-order Runge-Kutta method, and an energyconserving step-and-project method. We assess the methods' ability to recapture the system's limiting statistics and observe that symplectic Euler performs significantly better than the others for comparable computational expense.