The Boltzmann equation for Bose-Einstein particles: condensation in finite time
The paper under review deals with the isotropic and spatially homogeneous Boltzmann equation for Bose-Einstein particles in the case of the hard sphere model, and tackles the problem of Bose-Einstein condensation in finite time. The main result states that if the initial energy distribution of particles has no mass at 0 but is singular enough near the origin, then condensation occurs at all positive times. In the first section, the author introduces the equation of interest, defines the notion of a distributional solution, recalls several results such as the existence of an equilibrium distributional solution and a critical temperature, as well as a result on the convergence to equilibrium, and finally raises the question of condensation in finite time. The latter is partially answered by the main theorem in Section 2, whereas Section 3 contains the proof. Implications of the result, in particular, the link between condensation and conservation of mass, are discussed in Section 4. In the appendix, it is proved that the definition of a distributional solution given in [\textit{X. Lu}, J. Stat. Phys. 116, No. 5--6, 1597--1649 (2004; Zbl 1097.82023)] is equivalent to the one in the present paper.
scientific article; zbMATH DE number 6164077
scientific article