Quantum BGK model near a global Fermi-Dirac distribution

Abstract

In this paper, we consider the existence and asymptotic behavior of the fermionic quantum BGK model, which is a relaxation model of the quantum Boltzmann equation for fermions. More precisely, we establish the existence of unique classical solutions and their exponentially fast stabilization when the initial data starts sufficiently close to a global Fermi-Dirac distribution. A key difficulty unobserved in the study of the classical BGK model is that we must verify that the equilibrium parameters is uniquely determined through a set of nonlinear equations in each iteration step.

Stationary Quantum BGK Model for Bosons and Fermions in a Bounded Interval

“…For this we recall the following lemma: Lemma 3.1. [7,32] The function β B and β F defined in (1.6) satisfy the following properties.…”

“…We then introduce the equilibrium parameter a and c defined by ( + sign is for fermion and − sign is for boson, see [7,39]):…”

“…Note that c is determined implicitly. For the later convenience, we define The relations (1.4) and (1.5) arise from the requirement that F , B must share the same mass, momentum and energy with f (See [7]): Now we are ready to define the local quantum equilibriums. [7,15,32,39,52] • Bose-Einstein distribution: The local equilibrium for bosons is defined as follows:…”

“…In [9,10], Braukhoff obtained analytic solutions of quantum BGK type model arising in the study of ultracold fermionic clouds. The global existence and asymptotic behavior of fermionic quantum BGK model near a global Fermi-Dirac distribution were studied by the authors in [7]. Presently, authors are not aware of any further analytical results on the quantum BGK models.…”