On Typicality in Nonequilibrium Steady States

Abstract

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 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 B Lamberto Rondoni 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.

Asymmetry relations and effective temperatures for biased Brownian gyrators

“…Like other investigations of fluctuation relations led to results of more general interest (e.g. novel response theories [19,27]), our asymptotic asymmetry relations produce a new form of effective temperature, expected to be measurable e.g. in settings similar to those of Refs.…”

“…Although fluctuations are not normally observable in macroscopic systems, FRs have been verified in gravitational wave detectors, that are indeed meant to reveal microscopic fluctuations in macroscopic systems [22]. The literature on FRs is abundant [11][12][13][14][15][16][17][18][19][20][21].…”

A Derivation of the Gibbs Equation and the Determination of Change in Gibbs Entropy from Calorimetry

“…see the study therein. [3] In the present proof, we consider a system as a collection of particles whose positions (q 1 , q 2 , y) and momenta (p 1 , p 2 , y) are represented by the phase space vector, G ðq 1 ; . .…”

“…This is not the case -the integrating factor for the heat (apart from a simple scaling) is unique. It cannot be replaced by any other monotonic function of the reciprocal temperature such as 1=T 3 . This is because it comes directly from the algebraic form for the canonical equilibrium distribution function (Eqn 2) and then the associated form for the corresponding Helmholtz free energy (Eqn 6).…”

Temperature and correlations in 1-dimensional systems

“…Away from equilibrium, the situation is more problematic. Equipartition is violated [26,31,32], the statistic describing the state of the system is model dependent, and the ergodic properties of the particles dynamics are only partially understood [33,34]. Hence, there is no universally accepted microscopic notion of nonequilibrium temperature [20][21][22][23][24][25][26].…”