On thermostats and entropy production

The connection between the rate of entropy production and the rate of phase space contraction for thermostatted systems in nonequilibrium steady states is discussed for a simple model of heat flow in a Lorentz gas, previously described by Spohn and Lebowitz. It is easy to show that for the model discussed here the two rates are not connected, since the rate of entropy production is nonzero and positive, while the overall rate of phase space contraction is zero. This is consistent with conclusions reached by other workers. Fractal structures appear in the phase space for this model and their properties are discussed. We conclude with a discussion of the implications of this and related work for understanding the role of chaotic dynamics and special initial conditions for an explanation of the Second Law of Thermodynamics.