Superstatistics

Abstract

Einstein repeatedly criticized Boltzmann's statistics S(E) = k B log W(E) on the grounds that the probabilities W(E) to find a system in an energy state E should in principle follow a posteriori from its dynamics and not be postulated a priori. This implied the possible existence of other classical statistics than that of Boltzmann the first of which was proposed by Tsallis. In this paper, we discuss a class of generalized Boltzmann statistics: "superstatistics", developed by Beck and myself, which can be seen as a statistics of a statistics and contains Tsallis and other statistics as well. All these superstatistics appear to occur in systems with complicated dynamics, have non-additive and non-extensive entropies and apply in general to non-equilibrium systems. Two very accurate experiments will be discussed both done in fully developed turbulent circular Couette flow: one by La Porta et al. on the fluctuations of the acceleration of a tracer particle, the other by Lewis and Swinney for the fluctuations of the longitudinal velocity.

Nuclear and quark matter at high temperature

“…Or at least investigate the effects due to un-sharp values of it -a step forward-testing some simple distributions. This so-called superstatistical approach [188][189][190][191][192][193][194] is based on a particular distribution of β values in the thermal weight, exp(−βH). The simplest such distribution, having a width of β-values and allowing only non-negative ones, is an Euler-Gamma distribution.…”

“…Regarding the superstatistics view [190][191][192], thermodynamical β-fluctuations and n-fluctuations are related by the Poisson transform. From…”

Weakest-Link Scaling and Extreme Events in Finite-Sized Systems

“…In (13), it is possible to consider correlated random variables λ i as coefficients. In this case, the random variable X becomes doubly stochastic and is described by superstatistics [18,26]. We expect that the variability of link parameters is a plausible scenario in heterogeneous physical or engineered systems.…”

Superstatistics and renewal critical events

“…Superstatistics [1][2][3] offers a general framework to describe nonequilibrium stationary states of complex systems (e.g., a Brownian particle in a inhomogeneous fluid environment). The statistical distribution associated with the stationary state is described as a superposition of Boltzmann-like factors, each one generated by a local and fast dynamical component of the system.…”