On the blow up and condensation of supercritical solution of the Nordheim equation for bosons

TL;DR

This paper proves that solutions to the isotropic Nordheim equation for bosons blow up in finite time when initial conditions lead to equilibrium states with a Dirac mass, indicating condensation phenomena.

Contribution

It establishes finite-time blow-up and condensation for solutions of the Nordheim equation under specific initial conditions, advancing understanding of bosonic quantum gases.

Findings

Solutions blow up in finite time in certain conditions.

A Dirac measure forms at the origin in finite time.

Condensation phenomena are rigorously demonstrated.

Abstract

In this paper we prove that the solutions of the isotropic, spatially homogeneous Nordheim equation for bosons, with bounded initial, data blow up in finite time in the $L^\infty$ norm if the values of the energy and particle density are in the range of values where the corresponding equilibria contains a Dirac mass. We also prove that, in the weak solutions, whose initial data are measures with values of particle and energy densities satisfying the previous condition, a Dirac measure at the origin forms in finite time.