A dynamical-systems interpretation of the dissipation function, T-mixing and their relation to thermodynamic relaxation

We review the notions of the dissipation function and T-mixing for non-invariant measures, recently introduced for nonequilibrium molecular dynamics models. We provide a dynamical-systems interpretation for the dissipation function and related results, providing new perspectives into results such as the second-law inequality. We then consider the problem of relaxation within this framework-the convergence of time averages along single phase-space trajectories, as opposed to the convergence of ensemble averages. As a first step in this direction, we observe that T-mixing implies convergence to a unique asymptotic ensemble, independent on the initial ensemble. In particular, the initial ensemble can be concentrated arbitrarily closely to any point in phase-space.