Dynamical foundations of nonextensive statistical mechanics

We construct classes of stochastic differential equations with fluctuating friction forces that generate a dynamics correctly described by Tsallis statistics and nonextensive statistical mechanics. These systems generalize the way in which ordinary Langevin equations underly ordinary statistical mechanics to the more general nonextensive case. As a main example, we construct a dynamical model of velocity fluctuations in a turbulent flow, which generates probability densities that very well fit experimentally measured probability densities in Eulerian and Lagrangian turbulence. Our approach provides a dynamical reason why many physical systems with fluctuations in temperature or energy dissipation rate are correctly described by Tsallis statistics. 1 permanent address: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS. Recently there has been considerable interest in the formalism of nonextensive statistical mechanics (NESM) as introduced by Tsallis [1] and further developed by many others (e.g. [2]–[4]). In the mean time there is growing evidence that the formalism, rather than being just a theoretical construction, is of relevance to many complex physical systems. Applications in various areas have been reported, mainly for systems with either long-range interactions [5]–[7], multifractal behaviour [8, 9], or fluctuations of temperature or energy dissipation rate [10]–[14]. A recent interesting application of the formalism is that to fully developed turbulence [9, 11, 12]. Precision measurements of probability density functions (pdfs) of longitudinal velocity differences in high-Reynolds number turbulent Couette-Taylor flows are found to agree quite perfectly with analytic formulas of pdfs as predicted by NESM [12]. Despite this apparent success of the nonextensive approach, still the question remains why in many cases (such as the above turbulent flow) NESM works so well. To answer this question, let us first go back to ordinary statistical mechanics and just consider a very simple well known example, the Brownian particle [15]. Its velocity u satisfies the linear Langevin equation u̇ = −γu + σL(t), (1) where L(t) is Gaussian white noise, γ > 0 is a friction constant, and σ describes the strength of the noise. The stationary probability density of u is Gaussian with average 0 and variance β, where where β = γ σ can be identified with the inverse temperature of ordinary statistical mechanics (we assume that the Brownian particle has mass 1). The above simple situation completely changes if one allows the parameters γ and σ in the stochastic differential equation (SDE) to fluctuate as well. To be specific, let us assume that either γ or σ or both fluctuate in such a way that β = γ/σ is χ-distributed with degree n. This means the probability density of β is given by f(β) = 1 Γ ( n 2 ) { n 2β0 } n 2 β n 2 −1 exp {