Macroscopic diffusion from a Hamilton-like dynamics

We introduce and analyze a model for the transport of particles or energy in extended lattice systems. The dynamics of the model acts on a discrete phase space at discrete times but has nonetheless some of the characteristic properties of Hamiltonian dynamics in a confined phase space : it is deterministic, periodic, reversible and conservative. Randomness enters the model as a way to model ignorance about initial conditions and interactions between the components of the system. The orbits of the particles are non-intersecting random loops. We prove, by a weak law of large number, the validity of a diffusion equation for the macroscopic observables of interest for times that are arbitrary large, but small compared to the minimal recurrence time of the dynamics.