Abstract
When the temperature of a trapped Bose gas is below the Bose-Einstein transition temperature and above absolute zero, the gas is composed of two distinct components: the Bose-Einstein condensate and the cloud of thermal excitations. The dynamics of the excitations can be described by quantum Boltzmann models. We establish a connection between quantum Boltzmann models and chemical reaction networks. We prove that the discrete differential equations for these quantum Boltzmann models converge to an equilibrium point. Moreover, this point is unique for all initial conditions that satisfy the same conservation laws. In the proof, we then employ a toric dynamical system approach, similar to the one used to prove the global attractor conjecture, to study the convergence to equilibrium of quantum kinetic equations.
Highlights
Several years after the invention of the Boltzmann–Nordheim equation, which is the quantum version of the classical Boltzmann one, to describe the evolution of dilute quantum gases, a renewal in the kinetic theory of bosons has started by the pioneering work of Kirkpatrick and Dorfman [34, 35]
The most important problem in the theory of toric dynamical systems is the Global Attractor Conjecture, which says that the complex balanced equilibrium of a toric dynamical system is a globally attracting point within each linear invariant subspace
This global attractor question is strongly related to the convergence to equilibrium problem in the study of kinetic equations
Summary
Several years after the invention of the Boltzmann–Nordheim equation, which is the quantum version of the classical Boltzmann one, to describe the evolution of dilute quantum gases (cf. [40, 51]), a renewal in the kinetic theory of bosons has started by the pioneering work of Kirkpatrick and Dorfman [34, 35]. It has been pointed out in [49, 50] that when the temperature of the system is lower but closed to the Bose-Einstein condensation transition temperature, the Bogoliubov dispersion relation can be replaced by the Hatree-Fock energy (ω(p) ≈ c|p|2) In this regime, the two collision operators C12 and C22 dominate the collisional processes. The most important problem in the theory of toric dynamical systems is the Global Attractor Conjecture, which says that the complex balanced equilibrium of a toric dynamical system is a globally attracting point within each linear invariant subspace This global attractor question is strongly related to the convergence to equilibrium problem in the study of kinetic equations. By using an approach inspired by the theory of toric dynamical system, we prove in Theorem 2.2 that the solution of the discrete version of a simplified version of (1.11), that contains only C12, converges to the equilibrium exponentially in time. – In Theorem 4.1 of Section 4, we extend Theorem 3.2 to a simplified version of (1.11), that contains only C12 + C22, and the modified quantum Boltzmann model of the thermal cloud (1.11), that contains only C12 + C22 + C31
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