Generalized statistical mechanics for superstatistical systems

Mesoscopic systems in a slowly fluctuating environment are often well described by superstatistical models. We develop a generalized statistical mechanics formalism for superstatistical systems, by mapping the superstatistical complex system onto a system of ordinary statistical mechanics with modified energy levels. We also briefly review recent examples of applications of the superstatistics concept for three very different subject areas, namely train delay statistics, turbulent tracer dynamics, and cancer survival statistics.Comment: 9 pages, 5 figure

Superstatistical Brownian motion

As a main example for the superstatistics approach, we study a Brownian particle moving in a d-dimensional inhomogeneous environment with macroscopic temperature fluctuations. We discuss the average occupation time of the particle in spatial cells with a given temperature. The Fokker-Planck equation for this problem becomes a stochastic partial differential equation. We illustrate our results using experimentally measured time series from hydrodynamic turbulence.Comment: 11 pages, 2 figures. To appear in the proceedings of the international workshop `Complexity and Nonextensivity', Kyoto, 14-18 March 2005 (Progr. Theor. Phys. Suppl.

Superstatistics: Theory and Applications

Superstatistics is a superposition of two different statistics relevant for driven nonequilibrium systems with a stationary state and intensive parameter fluctuations. It contains Tsallis statistics as a special case. After briefly summarizing some of the theoretical aspects, we describe recent applications of this concept to three different physical problems, namely a) fully developed hydrodynamic turbulence b) pattern formation in thermal convection states and c) the statistics of cosmic rays.Comment: 16 pages, 5 figures. Submitted to Continuum Mechanics and Thermodynamics as a contribution to the topical issue on nonextensive statistical mechanic

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.Comment: 7 pages, 1 figur

Complexity of chaotic fields and standard model parameters

In order to understand the parameters of the standard model of electroweak and strong interactions (coupling constants, masses, mixing angles) one needs to embed the standard model into some larger theory that accounts for the observed values. This means some additional sector is needed that fixes and stabilizes the values of the fundamental constants of nature. In these lecture notes we describe in nontechnical terms how such a sector can be constructed. Our additional sector is based on rapidly fluctuating scalar fields that, although completely deterministic, evolve in the strongest possible chaotic way and exhibit complex behaviour. These chaotic fields generate potentials for moduli fields, which ultimately fix the fundamental parameters. The chaotic dynamics can be physically interpreted in terms of vacuum fluctuations. These vacuum fluctuations are different from those of QED or QCD but coupled with the same moduli fields as QED and QCD are. The vacuum energy generated by the chaotic fields underlies the currently observed dark energy of the universe. Our theory correctly predicts the numerical values of the electroweak and strong coupling constants using a simple principle, the minimization of vacuum energy. Implementing some additional discrete symmetry assumptions one also obtains predictions for fermion masses, as well as a Higgs mass prediction of 154 GeV.Comment: 27 pages, 7 figures. Invited lectures given at the Erice summer school `The Logic of Nature, Complexity and New Physics: From Quark-Gluon Plasma to Superstrings, Quantum Gravity and Beyond' (Erice, 29 Aug.-7. Sept. 2006