On Typicality in Nonequilibrium Steady States

From the statistical mechanical viewpoint, relaxation of macroscopic systems and response theory rest on a notion of typicality, according to which the behavior of single macroscopic objects is given by appropriate ensembles: ensemble averages of observable quantities represent the measurements performed on single objects, because “almost all” objects share the same fate. In the case of non-dissipative dynamics and relaxation toward equilibrium states, “almost all” is referred to invariant probability distributions that are absolutely continuous with respect to the Lebesgue measure. In other words, the collection of initial micro-states (single systems) that do not follow the ensemble is supposed to constitute a set of vanishing, phase space volume. This approach is problematic in the case of dissipative dynamics and relaxation to nonequilibrium steady states, because the relevant invariant distributions attribute probability 1 to sets of zero volume, while evolution commonly begins in equilibrium states, i.e., in sets of full phase space volume. We consider the relaxation of classical, thermostatted particle systems to nonequilibrium steady states. We show that the dynamical condition known as ΩT-mixing is necessary and sufficient for relaxation of ensemble averages to steady state values. Moreover, we find that the condition known as weak T-mixing applied to smooth observables is sufficient for ensemble relaxation to be independent of the initial ensemble. Lastly, we show that weak T-mixing provides a notion of typicality for dissipative dynamics that is based on the (non-invariant) Lebesgue measure, and that we call physical ergodicity.