Stretched exponentials from superstatistics

Abstract

Distributions exhibiting fat tails occur frequently in many different areas of science. A dynamical reason for fat tails can be a socalled superstatistics, where one has a superposition of local Gaussians whose variance fluctuates on a rather large spatio-temporal scale. After briefly reviewing this concept, we explore in more detail a class of superstatistics that hasn't been subject of many investigations so far, namely superstatistics for which a suitable power β η of the local inverse temperature β is χ 2 -distributed. We show that η > 0 leads to power law distributions, while η < 0 leads to stretched exponentials. The special case η = 1 corresponds to Tsallis statistics and the special case η = −1 to exponential statistics of the square root of energy. Possible applications for granular media and hydrodynamic turbulence are discussed.

Variational Principle for the Pareto Power Law

“…in non-extensive generalizations of the Boltzmann entropy [4], or modifying the basic form of the Gibbs distribution [5]. Power laws are also obtained within the framework of superstatistics [6], if some parameter, typically the temperature, is considered as a random variable.…”

Pathway model, superstatistics, Tsallis statistics, and a generalized measure of entropy

“…which is a generalized type-2 beta model for real x. Beck and Cohen's superstatistics belong to this case (2) (Beck and Cohen, 2003;Beck, 2006). For γ = 1, a = 1, δ = 1 we have Tsallis statistics for α > 1 from (2).…”

“…For general values of µ and α > 1 such that 1 α−1 − µ ν > 0 equation (38) corresponds to the pathway model of Mathai (2005) as well as the superstatistics considered by Beck and Cohen (2003) and Beck (2006). Now, consider an equation parallel to equation (17) in Mathai, Saxena, and Haubold (2005), namely,…”

“…Remark 1. Superstatistics considerations can produce only the generalized type-2 beta form of (35) with the appropriate normalization constant because Beck and Cohen (2003) and Beck (2006) consider the situation of a conditional density belonging to the generalized gamma family where a scaling parameter is having a marginal distribution belonging to a generalized gamma family. Then the unconditional density can only produce a form of the type f (x) = c 1 x α (1 + β 1 x δ ) −γ 1 , c 1 > 0, β 1 > 0, γ 1 > 0, x > 0…”

On superstatistical multiplicative-noise processes

“…In addition, superstatistical systems present a parameter,β, that fluctuates on a large scale, T , and follows a time-independent distribution, p(β). The superstatistical framework has successfully been applied on a widespread of problems like: interactions between hadrons from cosmic rays [4], fluid turbulence [3,5,6], granular material [7], electronics [8], economics [9][10][11][12], among many others [13]. Furthermore, it has been regarded as a possible foundation for non-extensive statistical mechanics [3] based on Tsallis entropy [14] as we show later on.…”