Transport Equations from Liouville Equations for Fractional Systems

TL;DR

This paper derives generalized transport and hydrodynamic equations for fractional systems, which are described by fractional powers of coordinates and momenta, extending classical equations to fractional-dimensional spaces.

Contribution

It introduces a fractional generalization of Liouville, Bogoliubov, and hydrodynamic equations for non-Hamiltonian systems in fractional spaces.

Findings

Derived a generalized transport equation for fractional systems.

Defined fractional averages and reduced distribution functions.

Established hydrodynamic equations for fractional systems.

Abstract

We consider dynamical systems that are described by fractional power of coordinates and momenta. The fractional powers can be considered as a convenient way to describe systems in the fractional dimension space. For the usual space the fractional systems are non-Hamiltonian. Generalized transport equation is derived from Liouville and Bogoliubov equations for fractional systems. Fractional generalization of average values and reduced distribution functions are defined. Hydrodynamic equations for fractional systems are derived from the generalized transport equation.