Fluctuation-dissipation theorems time-domain formulation

III. Time-Domain Formulation of the Fluctuation-Dissipation Theorem... 

The aim of this chapter is to show how the concepts of FDT violation and effective temperature can be illustrated in the framework of the above quoted system, as done experimentally in Ref. 12 and theoretically in Refs. 15-19. We do not discuss here the vast general domain of aging effects in glassy systems, which are reviewed in Refs. 2-4. Since the present contribution should be understood by beginners in the field, some relevant fundamental topics of equilibrium statistical physics—namely, on the one hand, the statistical description of a system coupled to an environment and, on the other hand, the fluctuation-dissipation theorem (in a time domain formulation)—are first recalled. Then, questions specifically related to out-of-equilibrium dynamics, such as the description of aging effects by means of an effective temperature, are taken up in the framework of the above-quoted model system. 

In Section III, we recall the fluctuation-dissipation theorem (FDT), valid for dynamical variables evolving in equilibrium. We provide its formulation in the time domain (in the whole range of temperatures). The choice of the time-domain formulation is motivated by the fact that, when out-of-equilibrium variables are concerned, the necessary modification of the equilibrium FDT can conveniently be carried out by introducing a fluctuation-dissipation ratio defined in terms of time-dependent quantities. 

III. TIME-DOMAIN FORMULATION OF THE FLUCTUATION-DISSIPATION THEOREM... 

N. Pottier and A. Mauger, Quantum fluctuation-dissipation theorem A time-domain formulation. Physica A 291, 327 (2001). 

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