Continuous Limits of Classical Repeated Interaction Systems

Abstract: We consider the physical model of a classical mechanical system (called "small system") undergoing repeated interactions with a chain of identical small pieces (called "environment"). This physical setup constitutes an advantageous way of implementing dissipation for classical systems, it is at the same time Hamiltonian and Markovian. This kind of model has already been studied in the context of quantum mechanical systems, where it was shown to give rise to quantum Langevin equations in the limit of continuous time interactions ([2]), but it has never been considered for classical mechanical systems yet. The aim of this article is to compute the continuous limit of repeated interactions for classical systems and to prove that they give rise to particular stochastic differential equations in the limit. In particular we recover the usual Langevin equations associated to the action of heat baths. In order to obtain these results, we consider the discrete-time dynamical system induced by Hamilton's equations and the repeated interactions. We embed it into a continuous-time dynamical system and we compute the limit when the time step goes to 0. This way we obtain a discrete-time approximation of stochastic differential equation, considered as a deterministic dynamical system on the Wiener space, which is not exactly of the usual Euler scheme type. We prove the Lp and almost sure convergence of this scheme. We end up with applications to concrete physical exemples such as a charged particule in an uniform electric field or a harmonic interaction. We obtain the usual Langevin equation for the action of a heat bath when considering a damped harmonic oscillator as the small system.