Enhanced Long-term Simulation of Hamiltonian Systems Containing a Strong Non-convex Potential

Many physical systems exhibit rapid motion coupled to a slowly varying motion. Often the rapid motion is associated with a stii contribution in the potential energy function. In this context, the situation typically considered in the literature is the one with a strictly convex potential. Under some technical assumptions, one can then show that the slow motion is reproduced by a properly constrained system. In this paper we are concerned with a diierent situation: Often diierent timescales can be found because of many local minima and barrier crossing between these minima. We suggest here to replace the detailed motion in the minima and the local barrier crossings by a statistical model which is then coupled to the slow equations of motion over long periods of time. This leads to Langevin type equations of motion subject to an appropriate time transformation.