Fluctuation–dissipation relations. Achievements and misunderstandings

Abstract

We discuss the "generalized fluctuation-dissipation relations (theorems)" for the first time suggested by us in 1977-1984 as statistical-thermodynamical consequences of time symmetry (reversibility) of microscopic dynamics. It is shown, in particular, that our old results in essence contain, as alternative formulations or special cases, various similar relations (including the "fluctuation theorems") what appeared in 1997 and later. PACS 05.20.-y, 05.40.-a, 05.70.-a

Nonequilibrium fluctuation theorems from equilibrium fluctuations

“…Theoretical derivations for thermostated and stochastic systems soon followed [3][4][5][6][7][8], together with related predictions for the work performed on systems driven away from the equilibrium by an external agent [9,10]. It was later appreciated that these results are closely related to an earlier body of research by Bochkov and Kuzovlev [11][12][13][14]; see [15][16][17][18][19] for detailed discussions.…”

“…where φ(x) is defined by equation (19). Hence, β −1 N α φ(x α ) plays the role of a potential of mean force for the order parameter x α .…”

Work statistics in stochastically driven systems

“…It has been experimentally validated thanks to remarkable progress in experimental techniques, making it possible to measure and manipulate systems at the sub-micron level. Among these achievements are the nonequilibrium versions of the fluctuation-dissipation theorem [6][7][8][9][10][11][12][13][14][15], the connection between thermodynamics and information theory [16][17][18][19][20][21][22][23][24], and-perhaps most important-the formulation and verification of the so-called fluctuation theorem (and its variants) [25][26][27][28][29][30][31][32][33][34]. According to this theorem, stochastic positive entropy production is exponentially more likely to be observed than the corresponding negative entropy production.…”

Time-reversal symmetric Crooks and Gallavotti–Cohen fluctuation relations in driven classical Markovian systems

“…We complete the introductory part of the paper by considering briefly the stochastic entropy productions. Stochastic entropy production of the environment, ∆s m , generalizes the concept of the heat for the systems violating the LDB condition (2). Indeed, like heat Q (5), this quantity sums the jumps ∆s m = i ∆s m,n j−1 →n j (t…”