Abstract
Statistical mechanics is the application of probability theory to study the dynamics of systems of arbitrary number of particles (Gibbs, 1960; Bogoliubov, 1960; Bogolyubov, 1970). Equations with derivatives of non-integer order have many applications in physical kinetics (see, for example, (Zaslavsky, 2002, 2005; Uchaikin, 2008) and (Zaslavsky, 1994; Saichev and Zaslavsky, 1997; Weitzner and Zaslavsky, 2001; Chechkin et al., 2002; Saxena et al., 2002; Zelenyi and Milovanov, 2004; Zaslavsky and Edelman, 2004; Nigmatullin, 2006; Tarasov and Zaslavsky, 2008; Rastovic, 2008)). Fractional calculus is used to describe anomalous diffusion, and transport theory (Montroll and Shlesinger, 1984; Metzler and Klafter, 2000; Zaslavsky, 2002; Uchaikin, 2003a,b; Metzler and Klafter, 2004). Application of fractional integration and differentiation in statistical mechanics was also considered in (Tarasov, 2006a, 2007a) and (Tarasov, 2004, 2005b,a, 2006b, 2007b). Fractional kinetic equations usually appear from some phenomenological models. We suggest fractional generalizations of some basic equations of statistical mechanics. To obtain these equations, the probability conservation in a fractional differential volume element of the phase space can be used (Tarasov, 2006a, 2007a). This element can be considered as a small part of the phase space set with non-integer-dimension. We derive the Liouville equation with fractional derivatives with respect to coordinates and momenta. The fractional Liouville equation (Tarasov, 2006a, 2007a) is obtained from the conservation of probability to find a system in a fractional volume element.
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References
J. de Boer, G.E. Uhlenbeck, (Eds.), 1962, Studies in Statistical Mechanics, North-Holland, Amsterdam.
N.N. Bogolyubov, 1946, Kinetic equations, Zhurnal Eksperimentaal’noi i Teoreticheskoi Fiziki, 16, 691–702. In Russian; and Journal of Physics USSR, 10, 265–274.
N.N. Bogoliubov, 1960, Problems of Dynamic Theory in Statistical Physics, Oak Ridge, Technical Information Service; and OGIZ, Moscow, 1946. In Russian.
N.N. Bogolyubov, 1970, Selected Works, Vol.2, Naukova Dumka, Kiev.
N.N. Bogoliubov, 1991, Selected Works. Part II. Quantum and Classical Statistical Mechanics, Gordon and Breach, New York.
N.N. Bogolyubov, N.N. Bogolyubov, Jr., 1982, Introduction to Quantum Statistical Mechanics, World Scientific Publishing, Singapore.
A. Campa, T. Dauxois, S. Ruffo, 2009, Statistical mechanics and dynamics of solvable models with long-range interactions, Physics Reports, 480, 57–159.
A.V. Chechkin, V.Yu. Gonchar, M. Szydlowsky, 2002, Fractional kinetics for relaxation and superdiffusion in magnetic field, Physics of Plasmas, 9, 78–88.
G. Ecker, 1972, Theory of Fully Ionized Plasmas, Academic Press, New York.
V. Feller, 1971, An introduction to Probability Theory and its Applications, Vol.2, Wiley, New York.
D. Forster, 1975, Hydrodynamics Fluctuations, Broken Symmetry, and Correlation Functions, Benjamin, London.
J.W. Gibbs, 1960, Elementary Principles in Statistical Mechanics, Dover, New York, 1960; and Yale University Press, New Haven, 1902.
K.P. Gurov, 1966, Foundation of Kinetic Theory. Method of N.N. Bogolyubov, Nauka, Moscow. In Russian.
A. Isihara, 1971, Statistical Physics, Academic Press, New York. Appendix IV, and Section 7.5.
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
N.A. Krall, A.W. Trivelpiece, 1973, Principles of Plasma Physics, McGraw-Hill, New York.
R.L. Liboff, 1998, Kinetic Theory: Classical, Quantum and Relativistic Description, 2nd ed., Wiley, New York.
G.A. Martynov, 1997, Classical Statistical Mechanics, Kluwer, Dordrecht.
R. Metzler, J. Klafter, 2000, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339, 1–77.
R. Metzler, J. Klafter, 2004, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of Physics A, 37, R161–R208.
E.W. Montroll, M.F. Shlesinger, 1984, The wonderful world of random walks, In Studies in Statistical Mechanics, Vol.11, J. Lebowitz, E. Montroll (Eds.), North-Holland, Amsterdam, 1–121.
R.R. Nigmatullin, 2006, Fractional kinetic equations and universal decoupling of a memory function in mesoscale region, Physica A, 363, 282–298.
D.Ya. Petrina, V.I. Gerasimenko, P.V. Malishev, 2002, Mathematical Foundation of Classical Statistical Mechanics, 2nd ed., Taylor and Francis, London, New York, 338p.; and Naukova Dumka, Kiev, 1985. In Russian.
D. Rastovic, 2008, Fractional Fokker-Planck equations and artificial neural networks for stochastic control of tokamak, Journal of Fusion Energy, 27, 182–187.
P. Resibois, M. De Leener, 1977, Classical Kinetic Theory of Fluids, Wiley, New York. Section IX.4.
A.I. Saichev, G.M. Zaslavsky, 1997, Fractional kinetic equations: solutions and applications, Chaos, 1, 753–764.
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.
R.K. Saxena, A.M. Mathai, H.J. Haubold, 2002, On fractional kinetic equations, Astrophysics and Space Science, 282, 281–287.
V.E. Tarasov, 2004, Fractional generalization of Liouville equations, Chaos, 14, 123–127.
V.E. Tarasov, 2005a, Fractional systems and fractional Bogoliubov hierarchy equations, Physical Review E, 71, 011102.
V.E. Tarasov, 2005b, Fractional Liouville and BBGKI equations, Journal of Physics: Conference Series 7, 17–33.
V.E. Tarasov, 2005c, Fractional generalization of gradient and Hamiltonian systems, Journal of Physics A, 38, 5929–5943.
V.E. Tarasov, 2005d, Stationary solutions of Liouville equations for non-Hamiltonian systems, Annals of Physics, 316, 393–413.
V.E. Tarasov, 2005e, Fractional generalization of gradient systems, Letters in Mathematical Physics, 73, 49–58.
V.E. Tarasov, 2005f, Fractional Fokker-Planck equation for fractal media, Chaos 15, 023102.
V.E. Tarasov, 2006a, Fractional statistical mechanics, Chaos, 16, 033108.
V.E. Tarasov, 2006b, Transport equations from Liouville equations for fractional systems, International Journal of Modern Physics B, 20, 341–353.
V.E. Tarasov, 2006c, Fractional variations for dynamical systems: Hamilton and Lagrange approaches, Journal of Physics A, 39, 8409–8425.
V.E. Tarasov, 2006d, Continuous limit of discrete systems with long-range interaction, Journal of Physics A, 39, 14895–14910.
V.E. Tarasov, 2006e, Map of discrete system into continuous, Journal of Mathematical Physics, 47, 092901.
V.E. Tarasov, 2007a, Liouville and Bogoliubov equations with fractional derivatives, Modern Physics Letters B, 21, 237–248.
V.E. Tarasov, 2007b, Fokker-Planck equation for fractional systems, International Journal of Modern Physics B, 21, 955–967.
V.E. Tarasov, 2008, Fractional vector calculus and fractional Maxwell’s equations, Annals of Physics, 323, 2756–2778.
V.E. Tarasov, G.M. Zaslavsky, 2008, Fokker-Planck equation with fractional coordinate derivatives, Physica A, 387, 6505–6512.
V.V. Uchaikin, 2003a, Self-similar anomalous diffusion and Levy-stable laws, Physics-Uspekhi, 46, 821–849.
V.V. Uchaikin, 2003b, Anomalous diffusion and fractional stable distributions, Journal of Experimental and Theoretical Physics, 97, 810–825.
V.V. Uchaikin, 2008, Method of Fractional Derivatives, Artishok, Ulyanovsk. In Russian.
A.A. Vlasov, 1938, The vibrational properties of an electron gas, Zhurnal EksperimentaVnoi i Teoreticheskoi Fiziki, 8, 291–318.
A.A. Vlasov, 1968, The vibrational properties of an electron gas, Soviet Physics Uspekhi, 10, 721–733.
A.A. Vlasov, 1945, On the kinetic theory of an assembly of particles with collective interaction, Journal of Physics USSR, 9, 25–40.
A.A. Vlasov, 1961, Many-particle Theory and its Application to Plasma, Gordon and Breach, New York.
A.A. Vlasov 1966, Statistical Distribution Functions, Nauka, Moscow. In Russian.
A.A. Vlasov, 1978, Nonlocal Statistical Mechanics, Nauka, Moscow. In Russian.
H. Weitzner, G.M. Zaslavsky, 2001, Directional fractional kinetics, Chaos, 11, 384–396.
S.W. Wheatcraft, M.M. Meerschaert, 2008, Fractional conservation of mass, Advances in Water Resources, 31, 1377–1381.
G.M. Zaslavsky, 1994, Fractional kinetic equation for Hamiltonian chaos, Physica D, 76, 110–122.
G.M. Zaslavsky, 2002, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371, 461–580.
G.M. Zaslavsky, 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford.
G.M. Zaslavsky, M.A. Edelman, 2004, Fractional kinetics: from pseudochaotic dynamics to Maxwell’s demon, Physica D, 193, 128–147.
L.M. Zelenyi, A.V. Milovanov, 2004, Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics, Physics Uspekhi, 47, 749–788.
D.N. Zubarev, M.Yu. Novikov, 1972, Generalized formulation of the boundary condition for the Liouville equation and for BBGKY hierarchy, Theoretical and Mathematical Physics, 13, 1229–1238.
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Tarasov, V.E. (2010). Fractional Statistical Mechanics. In: Fractional Dynamics. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14003-7_15
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