Multifractal PDF analysis for intermittent systems

The formula for probability density functions (PDFs) has been extended to include PDF for energy dissipation rates in addition to other PDFs such as for velocity fluctuations, velocity derivatives, fluid particle accelerations, energy transfer rates, etc, and it is shown that the formula actually explains various PDFs extracted from direct numerical simulations and experiments performed in a wind tunnel. It is also shown that the formula with appropriate zooming increment corresponding to experimental situation gives a new route to obtain the scaling exponents of velocity structure function, including intermittency exponent, out of PDFs of velocity fluctuations.Comment: 10 pages, 5 figure

Analysis of Velocity Derivatives in Turbulence based on Generalized Statistics

A theoretical formula for the probability density function (PDF) of velocity derivatives in a fully developed turbulent flow is derived with the multifractal aspect based on the generalized measures of entropy, i.e., the extensive Renyi entropy or the non-extensive Tsallis entropy, and is used, successfully, to analyze the PDF's observed in the direct numerical simulation (DNS) conducted by Gotoh et al.. The minimum length scale r_d/eta in the longitudinal (transverse) inertial range of the DNS is estimated to be r_d^L/eta = 1.716 (r_d^T/eta = 2.180) in the unit of the Kolmogorov scale eta.Comment: 6 pages, 1 figur

Quantum Stochastic Differential Equations in View of Non-Equlibrium Thermo Field Dynamics

Most of the mathematical approaches for quantum Langevin equation are based on the non-commutativity of the random force operators. Non-commutative random force operators are introduced in order to guarantee that the equal-time commutation relation for the stochastic annihilation and creation operators preserves in time. If it is true, it means that the origin of dissipation is of quantum mechanical. However, physically, it is hard to believe it. By making use of the unified canonical operator formalism for the system of the quantum stochastic differential equations within Non-Equilibrium Thermo Field Dynamics, it is shown that it is not true in general.Comment: 14 page

Analysis of Velocity Fluctuation in Turbulence based on Generalized Statistics

The numerical experiments of turbulence conducted by Gotoh et al. are analyzed precisely with the help of the formulae for the scaling exponents of velocity structure function and for the probability density function (PDF) of velocity fluctuations. These formulae are derived by the present authors with the multifractal aspect based on the statistics that are constructed on the generalized measures of entropy, i.e., the extensive R\'{e}nyi's or the non-extensive Tsallis' entropy. It is revealed that there exist two scaling regions separated by a crossover length, i.e., a definite length approximately of the order of the Taylor microscale. It indicates that the multifractal distribution of singularities in velocity gradient in turbulent flow is robust enough to produce scaling behaviors even for the phenomena out side the inertial range.Comment: 10 Pages, 5 figure

Quantum stochastic differential equations for boson and fermion systems -- Method of Non-Equilibrium Thermo Field Dynamics

A unified canonical operator formalism for quantum stochastic differential equations, including the quantum stochastic Liouville equation and the quantum Langevin equation both of the It\^o and the Stratonovich types, is presented within the framework of Non-Equilibrium Thermo Field Dynamics (NETFD). It is performed by introducing an appropriate martingale operator in the Schr\"odinger and the Heisenberg representations with fermionic and bosonic Brownian motions. In order to decide the double tilde conjugation rule and the thermal state conditions for fermions, a generalization of the system consisting of a vector field and Faddeev-Popov ghosts to dissipative open situations is carried out within NETFD.Comment: 69 page