  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.
