The issue of relaxation has been addressed in terms of ergodic theory in the
past. However, the application of that theory to models of physical interest is
problematic, especially when dealing with relaxation to nonequilibrium steady
states. Here, we consider the relaxation of classical, thermostatted particle
systems to equilibrium as well as to nonequilibrium steady states, using
dynamical notions including decay of correlations. We show that the condition
known as {\Omega}T-mixing is necessary and sufficient to prove relaxation of
ensemble averages to steady state values. We then observe that the condition
known as weak T-mixing applied to smooth observables is sufficient for
relaxation to be independent of the initial ensemble. Lastly, weak T-mixing for
integrable functions makes relaxation independent of the ensemble member, apart
from a negligible set of members enabling the result to be applied to
observations from a single physical experiment. The results also allow us to
give a microscopic derivation of Prigogine's principle of minimum entropy
production in the linear response regime. The key to deriving these results
lies in shifting the discussion from characteristics of dynamical systems, such
as those related to metric transitivity, to physical measurements and to the
behaviour of observables. This naturally leads to the notion of physical
ergodicity.Comment: 44 pages, 1 figur

Evans, Denis J.

Rondoni, Lamberto

Searles, Debra J.

Williams, Stephen R.

English

arXiv

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

Evans, Denis J

RONDONI, Lamberto

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

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 towhich the behavior of singlemacroscopic
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 otherwords, the collection of initialmicro-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  T-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.We would also like to thank the Australian Research Council for support of this research. LR
thanks the European Research Council, for funding under the European Community’s Seventh Framework
Programme (FP7/2007-2013)/ERC Grant Agreement No. 202680

The Australian National University

Journal of Statistical Physics

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 Ω T-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

University of Queensland eSpace

PORTO Publications Open Repository TOrino