Constructing a statistical mechanics for Beck-Cohen superstatistics

Abstract

The basic aspects of both Boltzmann-Gibbs (BG) and nonextensive statistical mechanics can be seen through three different stages. First, the proposal of an entropic functional (S BG = −k i p i ln p i for the BG formalism) with the appropriate constraints ( i p i = 1 and i p i E i = U for the BG canonical ensemble). Second, through optimization, the equilibrium or stationary-state distribution (p i = e −βE i /Z BG with Z BG = j e −βE j for BG). Third, the connection to thermodynamics (e.g.,Assuming temperature fluctuations, Beck and Cohen recently proposed a generalized Boltzmann factor B(E) = ∞ 0 dβf (β)e −βE . This corresponds to the second stage above described. In this letter we solve the corresponding first stage, i.e., we present an entropic functional and its associated constraints which lead precisely to B(E). We illustrate with all six admissible examples given by Beck and Cohen.

Superstatistics, thermodynamics, and fluctuations

“…This conditional entropic measure and the corresponding thermodynamics developed from it have a direct physical interpretation and it differs radically from previous work in Ref. [6], which introduces a generalized entropy. We believe that our approach is physically relevant in the sense that the thermodynamic-like relations obtained here correctly describe the physics of superstatistical nonequilibrium systems.…”

“…After the original work in Ref. [1], a lot of efforts have been made for further theoretical elaboration [5][6][7][8][9][10][11][12]. At the same time, it has also been applied successfully to a variety of systems and phenomena, including hydrodynamic turbulence [9,13,14], pattern formation [15], cosmic rays [16], solar flares [17], mathematical finance [18][19][20], random matrices [21], complex networks [22], wind velocity fluctuations [23], and hydro-climatic fluctuations [24].…”

Superstatistics

“…Then there are many different superstatistics described by different B(E), and the densities given by B(E) do not extremize the Tsallis entropies in general. Rather, as shown very recently by Tsallis and Souza [25], they extremize more general classes of entropy-like functions. The Boltzmann factor of the examples of superstatistics that we considered above, if expressed in terms of the universal parameters q and β 0 , can be written as…”

“…In other words, type-B superstatistics with f is equivalent to type-A superstatistics withf . Our formula (25) relating q and the variance of the β fluctuations is valid for both type-A and type-B superstatistics, just that all expectations ... must be formed with either f (type-A) orf (type-B). For example, if E = 1 2 u 2 , then Z(β) = 2π/β and hencef (β) ∼ β 1/2 f (β).…”

“…Since for small σE all superstatistics are the same and given by Tsallis statistics, we also have a maximum entropy principle for the normalized Boltzmann factors of all our superstatistics in terms of the Tsallis entropies [3,19] with an entropic index q given by eq. (25). For sufficiently small variance of the fluctuations they all extremize the Tsallis entropies subject to given constraints, no matter what the precise form of f (β) is.…”