Long time behavior of non-autonomous Fokker-Planck equations and the cooling of granular gases

TL;DR

This paper investigates the long-term behavior of non-autonomous Fokker-Planck equations with time-dependent coefficients, demonstrating relaxation to a Maxwellian distribution and applying findings to granular gases to show the attraction to a cooling state.

Contribution

It provides explicit conditions for relaxation to Maxwellian distributions and applies these results to granular gases, revealing algebraic convergence to the cooling state.

Findings

Relaxation towards Maxwellian distribution with time-dependent temperature

Algebraic rate of attraction to the Homogeneous Cooling State in granular gases

Explicit computable conditions for asymptotic behavior

Abstract

We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases as introduced J. J. Brey, J. Dufty and A. Santos (1999), by showing that the Homogenous Cooling State attracts any solution at an algebraic rate.