Underresolved Simulations of Heat Baths

Some recent numerical and theoretical studies indicate that it is possible to accurately simulate the macroscopic motion of a particle in a heat bath, comprising coupled oscillators, without accurately resolving the fast frequencies in the heat bath itself. Here we study this issue further by performing numerical experiments on a wide variety of mechanical heat bath models, all generalizations of the Ford–Kac oscillator model. The results indicate that the nature of the particle-bath damping in the macroscopic limit crucially affects the ability of underresolved simulations to correctly predict macroscopic behaviour. In particular, problems for which the damping is local in time pose more severe problems for approximation. The root cause is that local damping typically arises from the degeneration of a memory kernel to a delta singularity in the macroscopic limit. The approximation of such singularities is a more delicate issue than the approximation of smoother memory kernels.