  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.
