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 considered 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-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, Tel, 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 E-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 of Tel, Vollmer, and Breymann, and those of Ruelle, on entropy production in various systems described by Anosov or Anosov-like maps.