Ergodicity and the Numerical Simulation of Hamiltonian Systems

We discuss the long-time numerical simulation of Hamiltonian systems of ordinary differential equations. Our goal is to explain the ability of symplectic integration schemes such as Störmer-Verlet to compute accurate long-time averages for these systems in the context of molecular dynamics. This paper introduces a weakened version of ergodicity that allows us to study this problem. First, we demonstrate the utility of the weakened ergodicity definition by showing that it is a property of Hamiltonian systems robust to perturbations. Second, we study what the weakened ergodicity of a Hamiltonian system implies about numerical simulations of the system. In the case where a numerical method is volume-conserving and approximately energyconserving, we show that long-time averages are approximated well for sufficiently small step lengths.