Fractality of the non-equilibrium stationary states of open volume-preserving systems: I. Tagged particle diffusion

Deterministic diffusive systems such as the periodic Lorentz gas, multi-baker map, as well as spatially periodic systems of interacting particles, have non-equilibrium stationary states with fractal properties when put in contact with particle reservoirs at their boundaries. We study the macroscopic limits of these systems and establish a correspondence between the thermodynamics of the macroscopic diffusion process and the fractality of the stationary states that characterize the phase-space statistics. In particular the entropy production rate is recovered from first principles using a formalism due to Gaspard [J. Stat. Phys. 88, 1215 (1997)]. This article is the first of two; the second article considers the influence of a uniform external field on such systems.