  We generalize Gaspard's method for computing the \epsilon-entropy production
rate in Hamiltonian systems to dissipative systems with attractors considered
earlier by T\'el, Vollmer, and Breymann. This approach leads to a natural
definition of a coarse grained Gibbs entropy which is extensive, and which can
be expressed in terms of the SRB measures and volumes of the coarse graining
sets which cover the attractor. One can also study the entropy and entropy
production as functions of the degree of resolution of the coarse graining
process, and examine the limit as the coarse graining size approaches zero. We
show that this definition of the Gibbs entropy leads to a positive rate of
irreversible entropy production for reversible dissipative systems. We apply
the method to the case of a two dimensional map, based upon a model considered
by Vollmer, T\'el and Breymann, that is a deterministic version of a
biased-random walk. We treat both volume preserving and dissipative versions of
the basic map, and make a comparison between the two cases. We discuss the
\epsilon-entropy production rate as a function of the size of the coarse
graining cells for these biased-random walks and, for an open system with flux
boundary conditions, show regions of exponential growth and decay of the rate
of entropy production as the size of the cells decreases. This work describes
in some detail the relation between the results of Gaspard, those of T\'el,
Vollmer and Breymann, and those of Ruelle, on entropy production in various
systems described by Anosov or Anosov-like maps.
