Abstract
We collect recent results on deriving useful response relations also for nonequilibrium systems. The approach is based on dynamical ensembles, determined by an action on trajectory space. (Anti)Symmetry under time-reversal separates two complementary contributions in the response, one entropic the other frenetic. Under time-reversal invariance of the unperturbed reference process, only the entropic term is present in the response, giving the standard fluctuation-dissipation relations in equilibrium. For nonequilibrium reference ensembles, the frenetic term contributes essentially and is responsible for new phenomena. We discuss modifications in the Sutherland-Einstein relation, the occurence of negative differential mobilities and the saturation of response. We also indicate how the Einstein relation between noise and friction gets violated for probes coupled to a nonequilibrium environment. We end with some discussion on the situation for quantum phenomena, but the bulk of the text concerns classical mesoscopic (open) systems. The choice of many simple examples is trying to make the notes pedagogical, to introduce an important area of research in nonequilibrium statistical mechanics.
Highlights
To know a system operationally, is to be able to predict its response to a stimulus
The relation with the fluctuation–dissipation relation (FDR) of the first kind derives from the fact that the motion of a probe in a thermal bath can be considered as a stimulus there
The frenetic contribution to the response is separately measurable. It shows the experimental feasibility of the entropic–frenetic dichotomy at least for non-equilibrium micron-sized systems with a small number of degrees of freedom immersed in simple fluids
Summary
To know a system operationally, is to be able to predict its response to a stimulus. we learn about a system by observing its response. A more general framework has emerged, to begin with linear response theory around equilibrium It is the context of so called fluctuation–dissipation relations. (We ignore for the moment the issue of parity and generalized Casimir-Onsager reciprocity.) The intervention of the entropy flux, defined from a balance equation, was in essence the start of much of irreversible thermodynamics [8] Another line of response theory started with the PhD work of Pierre Curie (1896) on the magnetic susceptibility of paramagnets. A further line of relations, following from response theory and called fluctuation–dissipation relations of the second type, has been opened by the Johnson-Nyquist formula It gives an expression for the noise arising from the thermal agitations of the electrons in a resistor. Terminology is not always helpful here, as such terms as fluctuation– dissipation relations, Einstein relation, response relation etc. are used in multiple meanings throughout the literature
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