Superstatistics

Einstein repeatedly criticized Boltzmann’s statistics S(E)=kB 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.