Conservation of Fractional Mean Energy in Dissipative Gases

TL;DR

This paper introduces a fractional-calculus based conserved quantity in dissipative Maxwell gases, explaining the emergence of stationary power-law energy tails and connecting to the standard mean energy in thermal limits.

Contribution

It presents a novel fractional-calculus extension of mean energy that is conserved in dissipative gases, supported by Monte-Carlo simulations.

Findings

Conservation of fractional mean energy explains stationary power-law tails.

The fractional mean energy reduces to standard mean energy in thermal limit.

Monte-Carlo simulations validate the conservation law in inelastic gases.

Abstract

I show a nontrivial functional giving a conservation quantity in the collisional energy cascade of dissipative Maxwell gases: a fractional-calculus extension of the mean energy. The conservation of this quantity directly leads the power-law energy tail that is stationary during the temporal evolution. In the thermal limit, this quantity naturally reduces to the standard mean energy. This conservation law and its extension to particles with other interactions are demonstrated with a Monte-Carlo simulation for inelastic gases.