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Nonequilibrium response, relaxation and the t-mixing condition

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Mathematics for the Fluid Earth

Twenty years ago a relation was proposed by Evans, Cohen and Morriss, which has been since known as Fluctuation Relation. This relation was initially intended to quantify the relative probability of positive and negative energy dissipation rates in non-equilibrium steady states. Later, it developed in many steady state as well transient relations. The search for the minimal ingredients which allow such steady state fluctuation relations to hold led to the formulation of the t-mixing condition. This condition, similarly to the standard mixing condition, requires a certain decay of correlations, but different from that concerning standard mixing. We illustrate how the notion of t-mixing arose in the context of fluctuation relations, and we compare it with standard mixing properties. Further, we note that besides yielding the fluctuation relations for dissipative, time reversal invariant deterministic systems, without need to explicitly invoke special properties for the phase space invariant measures, t-mixing leads to a number of other results. We will discuss, in particular, those concerning relaxation to equilibrium or non-equilibrium stationary states and those concerning a general Green-Kubo-like response formula, which may be applied arbitrarily far from equilibrium to large as well as small systems (with correspondingly different meanings).

This talk is part of the Isaac Newton Institute Seminar Series series.

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