Dynamical foundations of nonextensive statistical mechanics.pdfVIP

a r X i v : c o n d - m a t / 0 1 0 5 3 7 4 v 1 [ c o n d - m a t .s t a t - m e c h ] 1 8 M a y 2 0 0 1 Dynamical foundations of nonextensive statistical mechanics Christian Beck1 Isaac Newton Institute for Mathematical Sciences, University of Cambridge, 20 Clarkson Road, Cambridge CB3 0EH, UK Abstract We construct classes of stochastic differential equations with fluc- tuating 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 un- derly 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 Eu- lerian 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 nonex- tensive 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 construc- tion, 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 temper- ature 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 veloc- ity differences in high-Reynolds number turb