The grazing collision limit of Kac caricature of Bose–Einstein particles

Abstract

We discuss the grazing collision limit of certain kinetic models of Bose–Einstein particles obtained from a suitable modification of the one-dimensional Kac caricature of a Maxwellian gas without cut-off. We recover in the limit a non-linear Fokker–Planck equation which presents many similarities with the one introduced by Kaniadakis and Quarati in Phys. Rev. E 49 (1994), 5103–5110. In order to do so, we perform a study of the moments of the solution. Moreover, as is typical in Maxwell models, we make an essential use of the Fourier version of the equation.