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Fractional statistical mechanics.

Vasily E Tarasov1

  • 1Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia. tarasov@theory.sinp.msu.ru

Chaos (Woodbury, N.Y.)
|October 4, 2006
PubMed
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This study derives fractional Liouville and Bogoliubov hierarchy equations, introducing fractional derivatives into statistical mechanics for Hamiltonian systems. These fractional kinetic equations, including the Fokker-Planck-Zaslavsky equation, describe particle distribution.

Area of Science:

  • Statistical Mechanics
  • Nonlinear Dynamics
  • Fractional Calculus

Background:

  • The Liouville and Bogoliubov hierarchy equations are fundamental in classical and quantum statistical mechanics.
  • Generalizations of these equations are needed to describe complex systems with anomalous dynamics.

Purpose of the Study:

  • To derive and analyze Liouville and first Bogoliubov hierarchy equations using derivatives of noninteger order.
  • To establish a framework for the statistical mechanics of fractional Hamiltonian systems.
  • To obtain fractional kinetic equations, including the Fokker-Planck-Zaslavsky equation.

Main Methods:

  • Derivation of fractional Liouville equation from probability conservation in fractional volume elements.
  • Application of fractional Liouville equation to obtain fractional Bogoliubov hierarchy and kinetic equations.

Related Experiment Videos

  • Consideration of fractional coordinate and momenta derivatives in Liouville and Bogoliubov equations.
  • Derivation of the Fokker-Planck-Zaslavsky equation from the fractional Bogoliubov equation.

    • Main Results:

    • The fractional Liouville equation was derived, forming the basis for further generalizations.
    • Fractional Bogoliubov hierarchy and kinetic equations with fractional derivatives were obtained.
    • A statistical mechanics framework for fractional Hamiltonian systems was discussed.
    • The Fokker-Planck-Zaslavsky equation with fractional phase-space derivatives was successfully derived.

    Conclusions:

    • Fractional derivatives provide a powerful tool for generalizing Liouville and Bogoliubov equations.
    • The derived fractional kinetic equations offer new approaches to modeling complex systems.
    • This work extends the applicability of statistical mechanics to systems exhibiting anomalous behavior.