  This paper presents an {\it ab initio} derivation of the expression given by
irreversible thermodynamics for the rate of entropy production for different
classes of diffusive processes. The first class are Lorentz gases, where
non-interacting particles move on a spatially periodic lattice, and collide
elastically with fixed scatterers. The second class are periodic systems where
$N$ particles interact with each other, and one of them is a tracer particle
which diffuses among the cells of the lattice. We assume that, in either case,
the dynamics of the system is deterministic and hyperbolic, with positive
Lyapunov exponents. This work extends methods originally developed for a
chaotic two-dimensional model of diffusion, the multi-baker map, to higher
dimensional, continuous time dynamical systems appropriate for systems with one
or more moving particles. Here we express the rate of entropy production in
terms of hydrodynamic measures that are determined by the fractal properties of
microscopic hydrodynamic modes that describe the slowest decay of the system to
an equilibrium state.
