Asymptotic description of Dirac mass formation in kinetic equations for quantum particles

Abstract

In this paper, the detailed asymptotic behaviour of the solutions of a kinetic equation for quantum particles is studied. It is shown that this behaviour is sensitive not only to the total mass of the initial data but also to its precise behaviour near the origin. In some cases, solutions develop a Dirac mass at the origin for long times in a self-similar manner that is analysed in detail. r

Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

“…In this case our method breaks down because we lack the spectral gap (the essential spectrum reaches 0 and one only expects some polynomial rates of convergence to equilibrium, which is consistent with [14]). …”

The Shape of the Emerging Condensate in Effective Models of Condensation

“…We further assume that we also have (2.7), and find that γ = σ(0, y)p ∞ (dy). Note that (3.8) is weaker than the assumption σ(0, 0) > 0 that was made in the appendix of [9], where non-constant b are treated non-rigorously, using matched asymptotic expansions. Finally we need to verify (2.4).…”

“…(1.1) with some functional F . We will give three concrete examples of such systems below: Kingman's model of selection and mutation [11], an approximate model for Bose-Einstein condensation due to Buffet, de Schmedt and Pulé [1] (henceforth called the BSP-model), and a model for bosons in a heat bath investigated by Escobedo, Mischler and Velazquez [7,8,9] (referred to als EMV-model below). Another effective model of condensation is the Boltzmann-Nordheim equation [12,13,15,10], but as we will see below, it is too singular for our theory to apply.…”

“…They also find the influence of the initial condition, depending on its behavior near the condensation point, that will appear in our results below. The results rely on explicit solution formulas for all times, although in [9] a formal asymptotic expansion is used in order to cover also cases where such a solution formula is missing.…”

“…Acknowledgements: The authors would like to thank Daniel Ueltschi for many fruitful discussions and J.J.L. Velazquez for useful comments on the Boltzmann-Nordheim equation, and for pointing out reference [9].…”