Toshihico Arimitsu

Expansion Formulas in Nonequilibrium Statistical Mechanics

Damping theoretical expansion formulas are given for basic equations of non-equilibrium systems: They are expressed in terms of “partial” and “ordered” cumulant functions. Validity and applicability of the method are examined with the use of a solvable model. The time-convolutionless formula is shown to be useful and systematic in obtaining reduced description of the systems.

The world according to Rényi: thermodynamics of multifractal systems

We discuss basic statistical properties of systems with multifractal structure. This is possible by extending the notion of the usual Gibbs–Shannon entropy into more general framework—R enyi s information entropy. We address the renormalization issue for R enyi s entropy on (multi)fractal sets and consequently show how R enyi s parameter is connected with multifractal singularity spectrum. The maximal entropy approach then provides a passage between R enyi s information entropy and thermodynamics of multifractals. Important issues such as R enyi s entropy versus Tsallis–Havrda–Charvat entropy and PDF reconstruction theorem are also studied. Finally, some further speculations on a possible relevance of our approach to cosmology are discussed. 2004 Elsevier Inc. All rights reserved. PACS: 65.40.Gr; 47.53.+n; 05.90.+m

Tsallis statistics and fully developed turbulence

An analysis of fully developed turbulence is developed based on the assumption that the underlying statistics of the system is that of the Tsallis ensemble. The multifractal spectrum fT(α) corresponding to the Tsallis-type distribution function is determined self-consistently in the sense that all parameters can be obtained through the observed value of the intermittency exponent. It is shown that the scaling exponents ζm of the velocity structure function derived with the help of the multifractal spectrum fit very well with experimental data. It is revealed that the asymptotic expression of ζm for m>>1 has a log term. The present self-consistent approach narrowed down the value of intermittency exponent µ for the fully developed turbulence to µ = 0.235±0.015.

PDF of velocity fluctuation in turbulence by a statistics based on generalized entropy

An analytical formula for the probability density function (PDF) of the velocity fluctuation in fully-developed turbulence is derived, non-perturbatively, by assuming that its underlying statistics is the one based on the generalized measures of entropy, the Renyi entropy or the Tsallis entropy. The parameters appearing in the PDF, including the index q in the generalized measures, are determined self-consistently with the help of the observed value μ of the intermittency exponent. The derived PDF explains quite well the experimentally observed density functions.

Analysis of turbulence by statistics based on generalized entropies

An analytical formula of the scaling exponents of velocity structure function for fully developed turbulence is derived, non-perturbatively, by assuming that its underlying statistics is the one based on the generalized measures of entropy, the Renyi entropy or the Havrda–Charvat–Tsallis (HCT) entropy. It is revealed by a self-consistent analysis for the observed value μ=0.220(±1%) that the formula explains experimental data very well with single value, q=0.343, of the index which appears in the measures of the Renyi entropy or of the HCT entropy. The probability density functions of the velocity fluctuation and of the velocity gradient are also presented.

Canonical formalism of dissipative fields in thermo field dynamics

For the stationary case the canonical formalism of thermally dissipative fields with both positive‐ and negative‐frequency parts is constructed. This formulation enables one to follow the self‐consistent renormalization scheme which creates the dissipation spontaneously. The self‐interacting φ3 model is examined as an example of the spontaneous creation of dissipation. The parameter α appearing in the thermal state conditions as well as observables independent of the choice of α are discussed.

Analysis of PDFs for energy transfer rates from 40963 DNS – verification of the scaling relation within MPDFT

The probability density functions (PDFs) for energy transfer rates, extracted by Kaneda and Ishihara from their 40963 direct numerical simulation for fully developed turbulence, are analyzed in high accuracy by means of the multifractal probability density function theory (MPDFT), in which a new scaling relation has been proposed. MPDFT is a statistical mechanical ensemble theory for the intermittent phenomena providing fat-tail PDFs. With the proposed scaling relation, MPDFT has been improved to deal with intermittency through any series of PDFs with arbitrary magnification δ (>1). As the value of δ can be determined freely by observers, the choice of δ should not affect observables. The validity of the generalized MPDFT is successfully verified through the precise analyses of several series of PDFs with different values of δ. In addition to the verification, it is revealed that the system of fully developed turbulence has much wider area representing scaling behaviors than the inertial range. With the help of MPDFT, it has become possible to separate the coherent turbulent motion from fluctuations, which may benefit the wavelet analysis of turbulence.

Observability of Renyi's entropy

Despite recent claims we argue that Rényis entropy is an observable quantity. It is shown that, contrary to popular belief, the reported domain of instability for Rényi entropies has zero measure (Bhattacharyya measure). In addition, we show that the instabilities can be easily emended by introducing a coarse graining into an actual measurement. We also clear up any doubts regarding the observability of Rényis entropy in (multi)fractal systems and in systems with absolutely continuous probability density functions.

General Formulation for Open Systems : Mirror Operation

General formulation for open systems is given in terms of the time-convolution-less formulation of the damping theory. A diagramatic construction rule for the “kinetic coefficient” is given by introducing the mirror operation of the time evolution operator. The response function generalized to the open systems is derived by using the general formulation. Formally, the response function can be written in the same form as was given by Kubo by using the mirror commutator instead of the conventional one.

Theory of Exchange Dephasing. I. General Formulation

Theory of exchange dephasing in molecules is developed from a quantum statistical point of view. Use has been made of a time-convolution-less projection operator formalism presented by us earlier. Reservoir variables and fast exchanging mode are eliminated consistently giving a basic equation valid for any time scale and temperatures.