Power-law Spatial Dispersion from Fractional Liouville Equation

TL;DR

This paper introduces a fractional kinetic model using the Liouville equation with Caputo derivatives to describe power-law spatial dispersion of permittivity, deriving fractional differential equations for electrostatic potential.

Contribution

It presents a novel microscopic fractional kinetic model and derives fractional differential equations for media with power-law spatial dispersion.

Findings

Power-law dependence of permittivity on wave vector derived

Fractional differential equations for electrostatic potential established

Solutions for point charge potential in such media analyzed

Abstract

A microscopic model in the framework of fractional kinetics to describe spatial dispersion of power-law type is suggested. The Liouville equation with the Caputo fractional derivatives is used to obtain the power-law dependence of the absolute permittivity on the wave vector. The fractional differential equations for electrostatic potential in the media with power-law spatial dispersion are derived. The particular solutions of these equations for the electric potential of point charge in this media are considered.