Superstatistics and Gravitation

Abstract

We suggest to consider the spacetime as a non-equilibrium system with a long-term stationary state that possess as a spatio-temporally fluctuating quantity β. These systems can be described by a superposition of several statistics, "superstatistics". We propose a Gamma distribution for f (β) that depends on a parameter p l . By means of it the corresponding entropy is calculated, p l is identified with the probability corresponding to this model. A generalized Newton's law of gravitation is then obtained following the entropic force formulation. We discuss some of the difficulties to try to get an associated theory of gravity.

Generalized Entropies Depending Only on the Probability and Their Quantum Statistics

“…As already mentioned in [7,8] nonextensive statistical mechanics entropies have been proposed, that depend only on the probability distribution, and not on a parameter, in the framework of superstatistics. It is based on a Γ or (χ 2 ) distribution that depends on β and also on p l .…”

“…In this section we will discuss the extension to the nonextensive statistical mechanics whose entropies depend only on the probability [7,8]; corresponding to Bose-Einstein (BEO) and Fermi-Dirac (FDO) distributions. These entropies are: .…”

“…In [7,8] it is proposed a nonextensive statistical mechanics entropy that depends only on the probability distribution and not on a parameter in the framework of superstatistics; it is based on a Γ or χ 2 distribution that depends on β and also on p l . The probabilities were calculated from the Boltzmann factor and show that it is possible to obtain the generalized entropy…”

“…All these depend on one or several parameters. By means of a modification to Superstatistics [6], one of the authors [7] has proposed generalized entropies that depend only on the probability [7,8]. There are three entropies: S I = k ∑…”

“…It is interesting to notice that the expansion in series of these entropies having as a first term S = −kΣ Ω l=1 p l ln p l in the parameter x l ≡ p l ln p l ≤ 1 cover, up to the first terms, any other expansion of any other possible function in x l , one would want to propose as another entropy. The three proposed entropies in [7,8] are then the only possible generalizations of the Boltzmann-Gibbs (BG) or Shannon entropies that depend only of the probability. One obtains a superposition of two statistics (that of β and that of p l ), hence the name superstatistics.…”