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

Conservation of Fractional Mean Energy in Dissipative Gases