scholarly journals FOKKER–PLANCK EQUATION FOR FRACTIONAL SYSTEMS

2007 ◽  
Vol 21 (06) ◽  
pp. 955-967 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Phase Space ◽  
Power Systems ◽  
Fractional Power ◽  
Planck Equation ◽  
Distribution Functions ◽  
Fractional Integrals ◽  
Fractional Systems ◽  
Fokker Planck Equation ◽  
Bogoliubov Equations ◽  
Fokker Planck

The normalization condition, average values, and reduced distribution functions can be generalized by fractional integrals. The interpretation of the fractional analog of phase space as a space with noninteger dimension is discussed. A fractional (power) system is described by the fractional powers of coordinates and momenta. These systems can be considered as non-Hamiltonian systems in the usual phase space. The generalizations of the Bogoliubov equations are derived from the Liouville equation for fractional (power) systems. Using these equations, the corresponding Fokker–Planck equation is obtained.

Comparison Between Theoretical Values and Simulation Results of Transport Coefficients for the Dissipative Particle Dynamics Method

2004 ◽  
Author(s):  
Akira Satoh
Keyword(s):  
Transport Coefficients ◽  
Dissipative Particle Dynamics ◽  
Planck Equation ◽  
Equation Of Motion ◽  
Particle Dynamics ◽  
Momentum Conservation ◽  
Distribution Functions ◽  
Fokker Planck Equation ◽  
Dynamics Method ◽  
Fokker Planck

In the present study, we have derived an expression for transport coefficients such as viscosity, from the equation of motion of dissipative particles. In the concrete, we have shown the Fokker-Planck equation in phase space, and macroscopic conservation equations such as the equation of continuity and the equation of momentum conservation. The basic equations of the single-particle and pair distribution functions have been derived using the Fokker-Planck equation. The solutions of these distribution functions have approximately been solved by the perturbation method under the assumption of molecular chaos. The expression of the viscosity due to dissipative forces has been obtained using the approximate solutions of the distribution functions. Also, we have conducted non-equilibrium dynamics simulations to investigate the influence of the parameters, which have appeared in defining the equation of motion in the dissipative particle dynamics method.

scholarly journals TRANSPORT EQUATIONS FROM LIOUVILLE EQUATIONS FOR FRACTIONAL SYSTEMS

2006 ◽  
Vol 20 (03) ◽  
pp. 341-353 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Transport Equation ◽  
Fractional Power ◽  
Distribution Functions ◽  
Fractional Dimension ◽  
Fractional Systems ◽  
Dimension Space ◽  
Bogoliubov Equations ◽  
Reduced Distribution Functions ◽  
Gasdynamic Equations ◽  
Generalized Transport Equation

We consider dynamical systems that are described by fractional power of coordinates and momenta. The fractional powers can be considered as a convenient way to describe systems in the fractional dimension space. For the usual space the fractional systems are non-Hamiltonian. Generalized transport equation is derived from Liouville and Bogoliubov equations for fractional systems. Fractional generalization of average values and reduced distribution functions are defined. Gasdynamic equations for fractional systems are derived from the generalized transport equation.

scholarly journals THE FRACTIONAL CHAPMAN–KOLMOGOROV EQUATION

2007 ◽  
Vol 21 (04) ◽  
pp. 163-174 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fractional Order ◽  
Planck Equation ◽  
Kolmogorov Equation ◽  
Fractional Integrals ◽  
Fractal Media ◽  
Fokker Planck Equation ◽  
Fokker Planck

The Chapman–Kolmogorov equation with fractional integrals is derived. An integral of fractional order is considered as an approximation of the integral on fractal. Fractional integrals can be used to describe the fractal media. Using fractional integrals, the fractional generalization of the Chapman–Kolmogorov equation is obtained. From the fractional Chapman–Kolmogorov equation, the Fokker–Planck equation is derived.

THE ASYMPTOTICS OF COLLISION OPERATORS FOR TWO SPECIES OF PARTICLES OF DISPARATE MASSES

1996 ◽  
Vol 06 (03) ◽  
pp. 405-436 ◽  
Author(s):  
PIERRE DEGOND ◽  
BRIGITTE LUCQUIN-DESREUX
Keyword(s):  
Time Scales ◽  
Planck Equation ◽  
Distribution Functions ◽  
Relaxation Phenomenon ◽  
Mass Ratio ◽  
Homogeneous Case ◽  
Fokker Planck Equation ◽  
Disparate Mass ◽  
Fokker Planck

We analyze the dynamics of a disparate mass binary gas or of a plasma in the homogeneous case, at various time scales, in the framework of the Boltzmann or Fokker–Planck equation. We intend to provide a rigorous foundation to the epochal relaxation phenomenon first pointed out by Grad. From general basic physical hypotheses, we derive the scaling of the equations as a function of the mass ratio of the particles, and we expand the collision operators in powers of this mass ratio. Then, Hilbert or Chapman–Enskog type expansions of the distribution functions allow us to investigate the dynamics of the mixture at various time scales, and we verify that the behavior of the obtained models is coherent with Grad's hypothesis.

scholarly journals Clustering and Uncertainty in Perfect Chaos Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Sergey A. Kamenshchikov
Keyword(s):  
Phase Space ◽  
Transport Properties ◽  
Chaotic Systems ◽  
Dynamic Properties ◽  
Planck Equation ◽  
Uncertainty Relations ◽  
Nonlinear Dispersion ◽  
Fokker Planck Equation ◽  
Mutual Relations ◽  
Fokker Planck

The goal of this investigation was to derive strictly new properties of chaotic systems and their mutual relations. The generalized Fokker-Planck equation with a nonstationary diffusion has been derived and used for chaos analysis. An anomalous transport turned out to be natural property of this equation. A nonlinear dispersion of the considered motion allowed us to find a principal consequence: a chaotic system with uniform dynamic properties tends to instable clustering. Small fluctuations of particles density increase by time and form attractors and stochastic islands even if the initial transport properties have uniform distribution. It was shown that an instability of phase trajectories leads to the nonlinear dispersion law and consequently to a space instability. A fixed boundary system was considered, using a standard Fokker-Planck equation. We have derived that such a type of dynamic systems has a discrete diffusive and energy spectra. It was shown that phase space diffusion is the only parameter that defines a dynamic accuracy in this case. The uncertainty relations have been obtained for conjugate phase space variables with account of transport properties. Given results can be used in the area of chaotic systems modelling and turbulence investigation.

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