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: Recent developments and applications

We review some recent developments which make use of the concept of `superstatistics', an effective description for nonequilibrium systems with a varying intensive parameter such as the inverse temperature. We describe how the asymptotic decay of stationary probability densities can be determined using a variational principle, and present some new results on the typical behaviour of correlation functions in dynamical superstatistical models. We briefly describe some recent applications of the superstatistics concept in hydrodynamics, astrophysics, and finance.Comment: 13 pages. To appear in 'Complexity, Metastability and Nonextensivity', eds. C. Beck, G. Benedek, A. Rapisarda and C. Tsallis (World Scientific, 2005

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

Recent developments in superstatistics

We provide an overview on superstatistical techniques applied to complex systems with time scale separation. Three examples of recent applications are dealt with in somewhat more detail: the statistics of small-scale velocity differences in Lagrangian turbulence experiments, train delay statistics on the British rail network, and survival statistics of cancer patients once diagnosed with cancer. These examples correspond to three different universality classes: Lognormal superstatistics, chi-square superstatistics and inverse chi-square superstatistics.Comment: 7 pages, 7 figures. Based on invited lectures given at the Sigma-Phi 08 conference, Kolympari, Crete (July 2008) and at the NEXT2008 conference, Foz do Iguasso, Brazil (October 2008). List of references extende

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

Lagrangian quantum turbulence model based on alternating superfluid/ normal fluid stochastic dynamics

Inpired by recent measurements of the velocity and acceleration statistics of Lagrangian tracer particles embedded in a turbulent quantum liquid we propose a new superstatistical model for the dynamics of tracer particles in quantum turbulence. Our model consists of random sequences S/N/S/... where the particle spends some time in the superfluid (S) and some time in the normal fluid (N). This model leads to a superposition of power law distributions generated in the superfluid and Gaussian distributions in the normal liquid, in excellent agreement with experimental measurements. We include memory effects into our analysis and present analytic predictions for probability densities and correlation functions.Comment: 6 pages, 4 figure

Extreme Value Laws for Superstatistics

We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fr\'echet and Weibull distribution. On the other hand, superstatistics contains three important universality classes, namely $\chi^2$-superstatistics, inverse $\chi^2$-superstatistics, and lognormal superstatistics, all maximizing different effective entropy measures. We investigate how the three classes of extreme value theory are related to the three classes of superstatistics. We show that for any superstatistical process whose local equilibrium distribution does not live on a finite support, the Weibull distribution cannot occur. Under the above mild asymptotic independence assumptions, we also show that $\chi^2$-superstatistics generally leads an extreme value statistics described by a Fr\'echet distribution, whereas inverse $\chi^2$-superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution.Comment: To appear in Entrop