Multifractal Analysis of Various PDF in Turbulence based on Generalized Statistics: A Way to Tangles in Superfluid He

By means of the multifractal analysis (MFA), the expressions of the probability density functions (PDFs) are unified in a compact analytical formula which is valid for various quantities in turbulence. It is shown that the formula can explain precisely the experimentally observed PDFs both on log and linear scales. The PDF consists of two parts, i.e., the {\it tail} part and the {\it center} part. The structure of the tail part of the PDFs, determined mostly by the intermittency exponent, represents the intermittent large deviations that is a manifestation of the multifractal distribution of singularities in physical space due to the scale invariance of the Navier-Stokes equation for large Reynolds number. On the other hand, the structure of the center part represents small deviations violating the scale invariance due to thermal fluctuations and/or observation error.Comment: 10 pages and 5 figure

量子性と散逸 : いくつかの新しい視点(第3回『非平衡系の統計物理』シンポジウム(その1),研究会報告)

この論文は国立情報学研究所の電子図書館事業により電子化されました。非平衡状態にある量子系を扱う正準理論の体系(非平衡Thermo Field Dynamics [NETFD]と呼ばれている)を解説する。この体系は,非平衡系の統計物理学で重要な4つの視点(Boltzmann方程式,Fokker-Planck方程式,Langevin方程式,確率的Liouville方程式によりそれぞれ記述される視点)を,一貫したひとつの土俵で議論することを可能にした。NETFD建設の背景と動機,それと「量子コヒーレンスと散逸」に関わる問題を扱う際に有効となる新しい観点に重点をおいて紹介する。また,外場を導入する際に2つの方法があることを示す。NETFDの体系に内在するこの特性は,微視的な視点と巨視的な視点の橋渡しに従来にない観点を与えるであろう

The world according to Renyi: thermodynamics of fractal systems

We discuss a basic thermodynamic properties of systems with multifractal structure. This is possible by extending the notion of Gibbs-Shannon's entropy into more general framework - Renyi's information entropy. We show a connection of Renyi's parameter q with the multifractal singularity spectrum f(\alpha) and clarify a relationship with the Tsallis-Havrda-Charvat entropy. Finally, we generalize Hagedorn's statistical theory and apply it to high-energy particle collisions.Comment: 8 LaTeX pages, Proc. of the International Symposium on Non-equilibrium and Nonlinear Dynamics in Nuclear and Other Finite System

Analyses of turbulence in a wind tunnel by a multifractal theory for probability density functions

The probability density functions (PDFs) for energy dissipation rates, created from time-series data of grid turbulence in a wind tunnel, are analyzed in a high precision by the theoretical formulae for PDFs within multifractal PDF theory which is constructed under the assumption that there are two main elements constituting fully developed turbulence, i.e., coherent and incoherent elements. The tail part of PDF, representing intermittent coherent motion, is determined by Tsallis-type PDF for singularity exponents essentially with one parameter with the help of new scaling relation whose validity is checked for the case of the grid turbulence. For the central part PDF representing both contributions from the coherent motion and the fluctuating incoherent motion surrounding the former, we introduced a trial function specified by three adjustable parameters which amazingly represent scaling behaviors in much wider area not restricted to the inertial range. From the investigation of the difference between two difference formulae approximating velocity time-derivative, it is revealed that the connection point between the central and tail parts of PDF extracted by theoretical analyses of PDFs is actually the boundary of the two kinds of instabilities associated respectively with coherent and incoherent elements.Comment: 17 pages, 4 figures, 3 tables. Contents are based on the talk at BIFD 2001 held at Barcelona in Spai

Harmonious Representation of PDF's reflecting Large Deviations

The framework of multifractal analysis (MFA) is distilled to the most sophisticated one. Within this transparent framework, it is shown that the harmonious representation of MFA utilizing two distinct Tsallis distribution functions, one for the tail part of probability density function (PDF) and the other for its center part, explains the recently observed PDF's of turbulence in the highest accuracy superior to the analyses based on other models such as the log-normal model and the $p$ model.Comment: 11 pages, 2 figure