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Statistical Mechanics of Fractal Phase Space Distributions

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Part of the book series: Nonlinear Physical Science ((NPS,volume 0))

Abstract

In this chapter, we consider fractal distributions of states in the phase space. We use a continuous phase space model to describe those distributions. In this model, the fractal distributions of states are described by fractional generalizations of expectation values and normalization conditions. These generalizations use integrals of non-integer order.

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References

  • J. de Boer, G.E. Uhlenbeck (Eds.), 1962, Studies in Statistical Mechanics, North-Holland, Amsterdam.

    MATH  Google Scholar 

  • N.N. Bogolyubov, 1946, Kinetic equations, Zhurnal Eksperimental’ noi i Teoreticheskoi Fiziki, 16, 691–702. In Russian; and Journal of Physics USSR, 10, 265–274.

    Google Scholar 

  • N.N. Bogolyubov, 1970, Selected Works, Vol.2, Naukova Dumka, Kiev.

    Google Scholar 

  • N.N. Bogolyubov, 2005a, Collection of Scientific Works in 12 Volumes, Vol.5, Nauka, Moscow. In Russian.

    Google Scholar 

  • N.N. Bogolyubov, 2005b, Collection of Scientific Works in 12 Volumes, Vol.6, Nauka, Moscow. In Russian.

    Google Scholar 

  • K.P. Gurov, 1966, Foundation of Kinetic Theory. Method of N.N. Bogolyubov, Nauka, Moscow. In Russian.

    Google Scholar 

  • A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.

    MATH  Google Scholar 

  • R.L. Liboff, 1998, Kinetic Theory: Classical, Quantum and Relativistic Description, 2nd ed., Wiley, New York.

    Google Scholar 

  • G.A. Martynov, 1997, Classical Statistical Mechanics, Kluwer, Dordrecht.

    MATH  Google Scholar 

  • D.Ya. Petrina, V.I. Gerasimenko, P.V. Malishev, 2002, Mathematical Foundation of Classical Statistical Mechanics 2nd ed., Taylor and Francis, London.

    Google Scholar 

  • M. Praprotnik, K. Kremer, L. Delle Site, 2007a, Fractional dimensions of phase space variables: a tool for varying the degrees of freedom of a system in a multiscale treatment, Journal of Physics A, 40, F281–F288.

    Article  ADS  MATH  Google Scholar 

  • M. Praprotnik, K. Kremer, L. Delle Site, 2007b, Adaptive molecular resolution via a continuous change of the phase space dimensionality, Physical Review E, 75, 017701.

    Article  ADS  Google Scholar 

  • S.G. Samko, A.A. Kilbas, O.I. Marichev, 1993, Integrals and Derivatives of Fractional Order and Applications, Nauka i Tehnika, Minsk, 1987, in Russian; and Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993.

    Google Scholar 

  • V.E. Tarasov, 2004, Fractional generalization of Liouville equations, Chaos, 14, 123–127.

    Article  ADS  Google Scholar 

  • V.E. Tarasov, 2005a, Fractional systems and fractional Bogoliubov hierarchy equations, Physical Review E, 71, 011102.

    Article  MathSciNet  ADS  Google Scholar 

  • V.E. Tarasov, 2005b, Fractional Liouville and BBGKI equations, Journal of Physics: Conference Series 7, 17–33.

    Article  ADS  Google Scholar 

  • V.E. Tarasov, 2006, Transport equations from Liouville equations for fractional systems, International Journal of Modern Physics B, 20, 341–353.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • V.E. Tarasov, 2007, Fokker-Planck equation for fractional systems, International Journal of Modern Physics B, 21, 955–967.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • G.E. Uhlenbeck, G.W. Ford, 1963, Lectures in Statistical Mechanics, American Mathematical Society, Providence.

    MATH  Google Scholar 

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© 2010 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg

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Tarasov, V.E. (2010). Statistical Mechanics of Fractal Phase Space Distributions. In: Fractional Dynamics. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14003-7_7

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