  We consider a one-dimensional persisent random walk viewed as a deterministic
process with a form of time reversal symmetry. Particle reservoirs placed at
both ends of the system induce a density current which drives the system out of
equilibrium. The phase space distribution is singular in the stationary state
and has a cumulative form expressed in terms of generalized Takagi functions.
The entropy production rate is computed using the coarse-graining formalism of
Gaspard, Gilbert and Dorfman. In the continuum limit, we show that the value of
the entropy production rate is independent of the coarse-graining and agrees
with the phenomenological entropy production rate of irreversible
thermodynamics.
