Superstatistics, thermodynamics, and fluctuations

Generalized statistical mechanics for superstatistical systems

“…We should remark that the above idea of mapping superstatistics onto the statistical mechanics of an exotic Hamiltonian is completely new and different from previous attempts of developing a generalized statistical mechanics for superstatistical systems [5,27,40,41].…”

“…If there is locally Gaussian behaviour, then the theory of superstatistics is clearly related to the theory of Gaussian-scale mixtures. More generally, one can prove a superstatistical generalization of fluctuation theorems [4], develop a variational principle for the large-energy asymptotics of general superstatistics [3], proceed to generalized entropies for general superstatistics [5,40,43], let the q-values in equation (2.1) fluctuate as well [7] and prove superstatistical versions of a central limit theorem [8]. There are also relations with fractional reaction equations [45], random matrix theory [20,21,35], networks [36] and path integrals [9].…”

“…There are interesting applications in hydrodynamic turbulence [2,19,29,30], defect turbulence [17], cosmic rays [25] and other scattering processes in high-energy physics [31,32], solar flares [18], share price fluctuations [26,27,33,34], random matrix theory [20,21,35], random networks [36], multiplicative-noise stochastic processes [37], wind velocity fluctuations [23,24], hydroclimatic fluctuations [22], the statistics of train departure delays [38] and survival statistics of cancer patients [39]. Maximum entropy principles can be generalized in a suitable way to yield the relevant probability distributions that characterize the various important universality classes in superstatistics [5,[40][41][42][43].…”

Analysis of global terrorism dynamics by means of entropy and state space portrait

“…These new entropic formulations are often expressed in terms of a set of parameters and, usually, do not obey the fourth Khinchin axiom [27]. Examples are the Tsallis [63], Rényi [55], Ubriaco [64], Kaniadakis [26], Sharma-Mittal [58] and Gamma [5] entropies, among others [3,8,11,24,66]. For particular values of the parameters, the classical entropic formulations are recovered.…”

On superstatistical multiplicative-noise processes

“…A particular case to consider is when these two time scales are clearly separated, specifically, when the time needed for the system to reach stationarity (considering a predeterminedβ) is much smaller than the scale at which that parameter changes. In the long-term, the non-equilibrium system is described by the superposition of different local dynamics at different time intervals that was coined by BECK and COHEN as superstatistics or "statistics of statistics" [1,2]. Frequently, systems that are characterised as "superstatistical" exhibit non-Gaussian distributions with kurtosis excess, or distributions with non-exponential decay.…”