  We introduce a class of random mechanical systems called random billiards to
study the problem of quantifying the irreversibility of nonequilibrium
macroscopic systems. In a random billiard model, a point particle evolves by
free motion through the interior of a spatial domain, and reflects according to
a reflection operator, specified in the model by a Markov transition kernel,
upon collision with the boundary of the domain. We derive a formula for entropy
production rate that applies to a general class of random billiard systems.
This formula establishes a relation between the purely mathematical concept of
entropy production rate and textbook thermodynamic entropy, recovering in
particular Clausius' formulation of the second law of thermodynamics. We also
study an explicit class of examples whose reflection operator, referred to as
the Maxwell-Smolukowski thermostat, models systems with boundary thermostats
kept at possibly different temperatures. We prove that, under certain mild
regularity conditions, the class of models are uniformly ergodic Markov chains
and derive formulas for the stationary distribution and entropy production rate
in terms of geometric and thermodynamic parameters.
