Liouville and Bogoliubov Equations with Fractional Derivatives

TL;DR

This paper derives fractional-order Liouville, Bogoliubov, and Vlasov equations, extending classical statistical mechanics to systems with non-integer order derivatives, and discusses their implications for fractional Hamiltonian systems.

Contribution

It introduces fractional derivatives into fundamental kinetic equations, providing a new framework for analyzing systems with non-integer order dynamics.

Findings

Derived fractional Liouville equation from probability conservation.

Obtained fractional Bogoliubov hierarchy and kinetic equations.

Discussed fractional Hamiltonian systems and charged particle dynamics.

Abstract

The Liouville equation, first Bogoliubov hierarchy and Vlasov equations with derivatives of non-integer order are derived. Liouville equation with fractional derivatives is obtained from the conservation of probability 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. Fractional kinetic equation for the system of charged particles are considered.