Fractional statistical mechanics.

The Liouville and first Bogoliubov hierarchy equations with derivatives of noninteger order are derived. The fractional Liouville equation is obtained from the conservation of probability to find a system in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Liouville and Bogoliubov equations with fractional coordinate and momenta derivatives are considered as a basis to derive fractional kinetic equations. The Fokker-Planck-Zaslavsky equation that has fractional phase-space derivatives is obtained from the fractional Bogoliubov equation. The linear fractional kinetic equation for distribution of the charged particles is considered.

[1]  Vasily E. Tarasov,et al.  Dynamics with low-level fractionality , 2005, physics/0511138.

[2]  V. E. Tarasov.  Continuous Medium Model for Fractal Media , 2005, cond-mat/0506137.

[3]  L. Zelenyi,et al.  Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics , 2004 .

[4]  Vasily E. Tarasov,et al.  Magnetohydrodynamics of fractal media , 2006, 0711.0305.

[5]  G. Zaslavsky,et al.  Directional fractional kinetics. , 2001, Chaos.

[6]  Transport Equations from Liouville Equations for Fractional Systems , 2006, cond-mat/0604058.

[7]  G. Ecker.  Theory of Fully Ionized Plasmas , 1972 .

[8]  George M. Zaslavsky.  Hamiltonian Chaos and Fractional Dynamics , 2005 .

[9]  G. Uhlenbeck,et al.  Studies in statistical mechanics , 1962 .

[10]  V. E. Tarasov.  Fractional Generalization of Gradient Systems , 2005, nlin/0604007.

[11]  Vasily E Tarasov.  Fractional systems and fractional Bogoliubov hierarchy equations. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[13]  Fractional Differential Forms II , 2003, math-ph/0301016.

[14]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[15]  Vasily E. Tarasov.  Possible experimental test of continuous medium model for fractal media , 2005 .

[16]  Vasily E. Tarasov.  MULTIPOLE MOMENTS OF FRACTAL DISTRIBUTION OF CHARGES , 2005 .

[17]  Vasily E Tarasov.  Fractional generalization of Liouville equations. , 2004, Chaos.

[18]  V. E. Tarasov.  Fractional generalization of gradient and hamiltonian systems , 2005, math/0602208.

[19]  A. Vlasov,et al.  The vibrational properties of an electron gas , 1967, Uspekhi Fizicheskih Nauk.

[20]  N. Laskin,et al.  Fractional quantum mechanics , 2008, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  J. Gibbs.  Elementary Principles in Statistical Mechanics , 1902 .

[22]  Vasily E. Tarasov,et al.  Coupled oscillators with power-law interaction and their fractional dynamics analogues , 2007 .

[23]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[24]  N. Laskin.  Fractional Schrödinger equation. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  G. A. Martynov.  Classical Statistical Mechanics , 1997 .

[26]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[27]  V. E. Tarasov,et al.  Fractional Fokker-Planck equation for fractal media. , 2005, Chaos.

[28]  M. Naber,et al.  Fractional differential forms , 2001, math-ph/0301013.

[29]  Stationary Solutions of Liouville Equations for Non-Hamiltonian Systems , 2005, cond-mat/0602409.

[30]  N. A. Krall,et al.  Principles of Plasma Physics , 1973 .

[31]  V. E. Tarasov.  Continuous limit of discrete systems with long-range interaction , 2006, 0711.0826.

[32]  A. Vlasov,et al.  Many-particle theory and its application to plasma , 1961 .

[33]  M. Naber.  Time fractional Schrödinger equation , 2004, math-ph/0410028.

[34]  N. Laskin.  Fractional quantum mechanics and Lévy path integrals , 1999, hep-ph/9910419.

[35]  Nick Laskin,et al.  Fractals and quantum mechanics. , 2000, Chaos.

[36]  G. Zaslavsky,et al.  Nonlinear fractional dynamics on a lattice with long range interactions , 2005, nlin/0512010.

[37]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[38]  CLASSICAL CANONICAL DISTRIBUTION FOR DISSIPATIVE SYSTEMS , 2003, cond-mat/0311536.

[39]  A. M. Mathai,et al.  On fractional kinetic equations , 2002 .

[40]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[41]  Vladimir V. Uchaikin,et al.  Anomalous diffusion and fractional stable distributions , 2003 .

[42]  V. E. Tarasov.  Map of discrete system into continuous , 2006, 0711.2612.

[43]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[44]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[45]  Kharkov,et al.  Fractional kinetics for relaxation and superdiffusion in a magnetic field , 2002 .

[46]  V. E. Tarasov.  Fractional variations for dynamical systems: Hamilton and Lagrange approaches , 2006, math-ph/0606048.

[47]  Vladimir V. Uchaikin,et al.  REVIEWS OF TOPICAL PROBLEMS: Self-similar anomalous diffusion and Levy-stable laws , 2003 .

[48]  Alexander I. Saichev,et al.  Fractional kinetic equations: solutions and applications. , 1997, Chaos.

[49]  V. E. Tarasov.  FOKKER PLANCK EQUATION FOR FRACTIONAL SYSTEMS , 2007, 0710.2053.

[50]  Vasily E. Tarasov.  Fractional Liouville and BBGKI Equations , 2005 .

[51]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[52]  D. Haar,et al.  Statistical Physics , 1971, Nature.

[53]  Pierre Resibois,et al.  Classical kinetic theory of fluids , 1977 .

[54]  V. E. Tarasov.  Liouville and Bogoliubov Equations with Fractional Derivatives , 2007, 0711.0859.

[55]  Vickie E. Lynch,et al.  Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model , 2001 .

[56]  G. Zaslavsky,et al.  Fractional dynamics of coupled oscillators with long-range interaction. , 2005, Chaos.

[57]  Danilo Rastovic.  Fractional Fokker–Planck Equations and Artificial Neural Networks for Stochastic Control of Tokamak , 2008 .

[58]  Vasily E. Tarasov,et al.  Electromagnetic field of fractal distribution of charged particles , 2005, physics/0610010.

[59]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .

[60]  V. E. Tarasov.  Fractional hydrodynamic equations for fractal media , 2005, physics/0602096.

[61]  金子 尚武.  N.A.Krall and A.W.Trivelpiece, Principles of Plasma Physics, McGraw-Hill, New and St. Louis, 1973, xiii+674ページ, 23.5×16cm, 9,000円(International Series in Pure and Applied Physics). , 1973 .

[62]  A. Vlasov,et al.  On the kinetic theory of an assembly of particles with collective interaction , 1945 .

[63]  Francesco Mainardi,et al.  On Mittag-Leffler-type functions in fractional evolution processes , 2000 .

[64]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[65]  Raoul R. Nigmatullin,et al.  ‘Fractional’ kinetic equations and ‘universal’ decoupling of a memory function in mesoscale region , 2006 .

[66]  G. Zaslavsky,et al.  Fractional dynamics of systems with long-range interaction , 2006, 1107.5436.

[67]  George M. Zaslavsky,et al.  Fractional kinetic equation for Hamiltonian chaos , 1994 .

[68]  M. Caputo.  Linear models of dissipation whose Q is almost frequency independent , 1966 .

[69]  George M. Zaslavsky,et al.  Fractional kinetics: from pseudochaotic dynamics to Maxwell’s Demon , 2004 .