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

We discuss the grazing collision limit of certain kinetic models of Bose-Einstein particles obtained from a suitable modication of the one- dimensional Kac caricature of a Maxwellian gas without cut-o. We re- cover in the limit a nonlinear Fokker-Planck equation which presents many similarities with the one introduced by Kaniadakis and Quarati. 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. The research that led to the present paper was partially supported by a grant of the group GNFM of INdAM.