Entropy production: From open volume-preserving to dissipative systems

We generalize Gaspard's method for computing the epsilon-entropy production rate in Hamiltonian systems to dissipative systems with attractors conside red earlier by Tel, 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-gr aining sets which cover the attractor. One can also study the entropy and e ntropy production as functions of the degree of resolution of the coarse-gr aining process, and examine the limit as the coarse-graining size approache s zero. We show that this definition of the Gibbs entropy leads to a positi ve rate of irreversible entropy production for reversible dissipative syste ms. We apply the method to the case of a two-dimensional map, based upon a model considered by Vollmer, Tel, and Breymann, that is a deterministic ver sion of a biased-random walk. We treat both volume-preserving and dissipati ve versions of the basic map, and make a comparison between the two cases. We discuss the E-entropy production rate as a function of the size of the c oarse-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 of Tel, Vollmer, and Breymann, and those of Ruelle, on entropy pr oduction in various systems described by Anosov or Anosov-like maps.