Superstatistical equilibrium distributions

III. The Fluctuation-Dissipation Theorem for the Superstatistical Equilibrium Distributions... 

In this chapter we will extend the fluctuation-dissipation theorem for general equilibrium distributions. We will consider two typical equilibrium distributions. One is the superstatistical equilibrium distribution [13]. The other is the microcanonical equilibrium distribution. 

When a system has long-time correlation, for which we expect fractional power scaling of excess heat, our assumption of the Boltzmann equilibrium distribution may always not be valid. Actually some power distributions such as the Tsallis distribution [14] have been reported at the edge of chaos [15]. A superstatistical equilibrium distribution is written as a superposition of Boltzmann distributions with different temperatures. Beck and Cohen [13] considered many types of distributions for the inverse of temperature. For example, they chose Gaussian, uniform, gamma, log-normal, and others. In particular, the Tsallis distribution is realized for gamma distribution. We will show that excess heat can be written as a superposition of correlation functions... 

III. THE FLUCTUATION-DISSIPATION THEOREM FOR THE SUPERSTATISTICAL EQUILIBRIUM DISTRIBUTIONS... 

A superstatistical equilibrium distribution is written as a superposition of Boltzmann distributions with different temperatures. We showed that the excess heat could be written as a superposition of correlation functions with different temperatures using the generalized fluctuation-dissipation theorem. When a relaxation time depends on a temperature, we can expect various behaviors for the area of the hysteresis loop from the fluctuation-dissipation theorem. 

In this subsection we will consider generalized equilibrium distributions in superstatistics,... 

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