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

Article type: Research Article

Authors: Allemand, Thibaut | Toscani, Giuseppe

Affiliations: DMA, École Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France. E-mail: allemand@math.ens.fr | Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy

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.

Keywords: Bose gas, Kac model, grazing collisions, Fokker–Planck

DOI: 10.3233/ASY-2011-1031

Journal: Asymptotic Analysis, vol. 72, no. 3-4, pp. 201-229, 2011