Fractional Statistical Mechanics

TL;DR

This paper develops a fractional generalization of statistical mechanics by deriving fractional Liouville and Bogoliubov equations, leading to fractional kinetic equations and the fractional Fokker-Planck equation, expanding the theoretical framework for complex systems.

Contribution

It introduces fractional derivatives into fundamental equations of statistical mechanics, providing a new theoretical basis for analyzing systems with anomalous dynamics.

Findings

Derived fractional Liouville and Bogoliubov equations with noninteger derivatives.

Obtained fractional kinetic equations including the fractional Fokker-Planck equation.

Applied fractional equations to describe charged particle distributions.

Abstract

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 fractional Bogoliubov equation. The linear fractional kinetic equation for distribution of the charged particles is considered.