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On typicality in nonequilibrium steady states

Evans, Denis J.; Williams, Stephen R.; Searles, Debra J.; Rondoni, Lamberto

Description

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...[Show more]

dc.contributor.authorEvans, Denis J.
dc.contributor.authorWilliams, Stephen R.
dc.contributor.authorSearles, Debra J.
dc.contributor.authorRondoni, Lamberto
dc.date.accessioned2017-02-13T22:59:27Z
dc.date.available2017-02-13T22:59:27Z
dc.identifier.issn0022-4715
dc.identifier.urihttp://hdl.handle.net/1885/112266
dc.description.abstractFrom 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.
dc.description.sponsorshipWe 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.
dc.format16 pages
dc.format.mimetypeapplication/pdf
dc.publisherSpringer Verlag (Germany)
dc.rights© Springer Science+Business Media New York 2016
dc.sourceJournal of Statistical Physics
dc.subjectErgodicity
dc.subjectMixing
dc.subjectTransient states
dc.subjectNecessary conditions
dc.titleOn typicality in nonequilibrium steady states
dc.typeJournal article
local.identifier.citationvolume164
dcterms.dateAccepted2016-06-07
dc.date.issued2016-06-23
local.publisher.urlhttp://link.springer.com/
local.type.statusPublished Version
local.contributor.affiliationEvans, Denis J., Department of Applied Mathematics, CPMS Research School of Physics and Engineering, The Australia National University
local.contributor.affiliationWilliams, Stephen R., RSC General, CPMS Research School of Chemistry, The Australian National Univesity
local.bibliographicCitation.issue4
local.bibliographicCitation.startpage842
local.bibliographicCitation.lastpage857
local.identifier.doi10.1007/s10955-016-1563-3
dcterms.accessRightsOpen Access
CollectionsANU Research Publications

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