Kinetic modelling of the quantum gases in the normal phase

Abstract

Using the maximum entropy principle, a kinetic model equation is proposed to simplify the intricate collision term in the semi-classical Boltzmann equation for dilute quantum gases in the normal phase. The kinetic model equation keeps the main properties of the Boltzmann equation, including conservation of mass, momentum and energy, the entropy dissipation property, and rotational invariance. It also produces the correct Prandtl numbers for the Fermi gases. To validate the proposed model, the kinetic model equation is numerically solved in the hydrodynamic and kinetic flow regimes using the asymptotic preserving scheme. The results agree well with those of the quantum Euler and Boltzmann equations.

Numerical solutions of ideal quantum gas dynamical flows governed by semiclassical ellipsoidal-statistical distribution

“…In order to improve the classical Boltzmann-BGK model equation that does not produce correct Prandtl numbers, a new kinetic model called the ES model was proposed by Lowell & Holway [19]. Analogously, a semiclassical ES kinetic model for quantum gases in the normal phase has been recently developed by Wu et al [20]. This new model mimics the classical ES model of Lowell & Holway [19], and was derived by maximizing the corresponding entropy function under some constraints.…”

Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation

“…Numerical solutions of the semiclassical Boltzmann ES kinetic model equation proposed by Wu et al [23] have been obtained for two-dimensional Riemann problem in rarefied gas of particles of arbitrary statistics. The semiclassical ES equilibrium distribution is anisotropic and was derived through the maximum entropy principle and involves not only the mass, momentum and energy, but also pressure tensor quantities and differs from the standard BE or FD distribution.…”

“…The full semiclassical Boltzmann equation and its related theory can be found in [1,2]. Here we follow the development in [28] and consider the semiclassical Boltzmann ES model equation by Wu et al [23] ∂f ∂t…”

“…An H-theorem for the ES model has been proved [22]. Analogously, a semiclassical ES kinetic model for quantum gases in the normal phase has been recently developed by Wu et al [23]. This new model mimics the classical ES model of Holway [21] and was derived by maximizing the corresponding entropy function under some constraints.…”

“…Physically, in the BGK model, the conservative macroscopic variables, density, mean velocity and temperature are involved in the post-collision state, whereas in the ES model, in addition to these conservative normal variables, the local stress tensor quantities are also involved in the post-collision state. In the semiclassical Boltzmann-BGK equation, if the Bose-Einstein (BE) or Fermi-Dirac (FD) distribution is replaced by the semiclassical ES distribution, then one has the semiclassical Boltzmann-ES-BGK equation [23]. It is emphasized here that in the FD or BE statistics, only the normal solution variables such as number density, mean velocity, temperature and chemical potential (or fugacity) are involved; however, in the semiclassical ES distribution, additional pressure tensor variables are also involved.…”