On the long-time behaviour of ensembles in a model of deterministic diffusion

We describe a large model system, based on the baker transformation, where deterministic diffusion occurs. The model is similar to one recently considered by Gaspard, and by Hasegawa and Driebe. We point out the close relationship between this system and a simple random walk, and analyse the evolution with time of ensembles in the system using the resolvent-based version of the 'subdynamics' formalism developed by Prigogine and his collaborators. We obtain an exact and rigorous description of the long-time behaviour of ensembles, including the irreversible approach to equilibrium, for the case where the system has finite size. We also consider the 'thermodynamic' limit where the size of the system becomes infinite, and derive a description of the long-time behaviour in this case, where correlations decay non-exponentially with time.