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Deterministic diffusive systems such as the periodic Lorentz gas, multi-bakermap, as well as spatially periodic systems of interacting particles, havenon-equilibrium stationary states with fractal properties when put in contactwith particle reservoirs at their boundaries. We study the macroscopic limitsof these systems and establish a correspondence between the thermodynamics ofthe macroscopic diffusion process and the fractality of the stationary statesthat characterize the phase-space statistics. In particular the entropyproduction rate is recovered from first principles using a formalism due toGaspard [J. Stat. Phys. 88, 1215 (1997)]. This article is the first of two; thesecond article considers the influence of a uniform external field on suchsystems.

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