Gibbs entropy and irreversible thermodynamics

Recently, a number of approaches have been developed to connect the microscopic dynamics of particle systems to the macroscopic properties of systems in non-equilibrium stationary states, via the theory of dynamical systems. In this way a direct connection between dynamics and irreversible thermodynamics has been claimed to have been found. However, the main quantity used in these studies is a (coarse-grained) Gibbs entropy, which to us does not seem suitable, in its present form, to characterize non-equilibrium states. Various simplified models have also been devised to give explicit examples of how the coarse-grained approach may succeed in giving a full description of the irreversible thermodynamics. We analyse some of these models and point out a number of difficulties which, in our opinion, need to be overcome in order to establish a physically relevant connection between these models and irreversible thermodynamics. AMS classification scheme numbers: 82C05, 80A20, 70F25.