Superstatistics

Abstract

We consider nonequilibrium systems with complex dynamics in stationary states with large fluctuations of intensive quantities (e.g. the temperature, chemical potential or energy dissipation) on long time scales. Depending on the statistical properties of the fluctuations, we obtain different effective statistical mechanical descriptions. Tsallis statistics follows from a χ 2 -distribution of an intensive variable, but other classes of generalized statistics are obtained as well. We show that for small variance of the fluctuations all these different statistics behave in a universal way.

Short note on the emergence of fractional kinetics

“…This latter concept can be linked with the pioneering work by Beck [12] that has led to so-called superstatistics [13]. However, the present research differs from that because it starts from a different explanatory idea and it is based on particle trajectories modeled by a CTRW rather than by the Langevin equation, even though in both cases a superposition integral is used.…”

Superstatistics, thermodynamics, and fluctuations

“…The concept of superstatistics has been introduced by two of the present authors [1] after some preliminary considerations in [2,3]. The stationary distributions of superstatistical systems typically exhibit a non-Gaussian behavior with fat tails, which can decay, e.g., as a power law, a stretched-exponential law, or in an even more complicated way [4].…”

Superstatistics and System Identification for a Class of Generalized Cauchy Processes

“…Possible generalizations with a third order nonlinearity [2] and with a long-memory [3] are given. According to Beck-Cohen superstatistics [4], the model is derived by Bayes' theorem: P s (u) = ∫ P s (u|β)g(β) dβ , under the Gausssian distribution P s (u|β) with the Gamma distribution, g(β) = ν µ Γ(µ) β µ−1 exp(−νβ) . Their theory is equivalent to considering the linear Langevin equation [4] …”

“…We need to (i) exhibit many different stochastic processes (cf. [4]), and (ii) to discriminate their features of nonlinear interactions and natures of intermittency. This paper is organized as follows.…”