Dynamical systems theory of irreversibilhty

Recent work on the connections between dynamical systems theory and nonequilibrium statistical mechanics is reviewed with emphasis on results which are compatible with Lionville's theorem. Starting from a general discussion of time-reversal symmetry in the Newtonian scheme, it is shown that the Liouvillian eigenstates associated with the Pollicott-Ruelle resonances spontaneously break the time-reversal symmetry. We explain that such a feature is compatible with the time reversibility of Newton's equations because of a selection of trajectories which are not time-reversal symmetric. The Pollicott-Ruelle resonances and their associated eigenstates can be constructed not only for decay processes but also for transport processes such as diffusion or viscosity, as well as for reaction-diffusion processes. The Pollicott-Ruelle resonances thus describe the relaxation toward the thermodynamic equilibrium. The entropy production of these relaxation processes can be calculated and shown to take the value expected from nonequilibrium thermodynamics. In nonequilibrium steady states, an identity is obtained which shows that the entropy production directly characterizes the breaking of time-reversal symmetry by nonequilibrium boundary conditions. The extension to quantum systems is also discussed.