BGK model of the multi-species Uehling-Uhlenbeck equation

Abstract

We propose a BGK model of the quantum Boltzmann equation for gas mixtures. We also provide a sufficient condition that guarantees the existence of equilibrium coefficients so that the model shares the same conservation laws and H-theorem with the quantum Boltzmann equation. Unlike the classical BGK for gas mixtures, the equilibrium coefficients of the local equilibriums for quantum multi-species gases are defined through highly nonlinear relations that are not explicitly solvable. We verify in a unified way that such nonlinear relations uniquely determine the equilibrium coefficients under the condition, leading to the well-definedness of our model.

BGK model for multi-component gases near a global Maxwellian

“…Examples include the model of Gross and Krook [37], the model of Hamel [43], the model of Greene [35], the model of Garzo, Santos and Brey [33], the model of Sofonea and Sekerka [60], the model by Andries, Aoki and Perthame [1], the model of Brull, Pavan and Schneider [17], the model of Klingenberg, Pirner and Puppo [47], the model of Haack, Hauck, Murillo [42], the model of Bobylev, Bisi, Groppi, Spiga [13]. The BGK model for gas mixtures has also been extended to the ES-BGK model, polyatomic molecules, chemical reactions, or the quantum case; See for example [4,11,12,36,46,48,55,62,72]. For the applications of the mixture BGK models, we refer to [8,9,14,27,28,31,54,56].…”

A Consistent BGK Model with Velocity-Dependent Collision Frequency for Gas Mixtures

“…There are many BGK models for gas mixtures proposed in the literature [1,5,10,12,[14][15][16]23,28], many of which satisfy these basic requirements and, in addition, are able to match some prescribed relaxation rates and/or transport coefficients that come from more complicated physics models or from experiment. Many of these approaches have been extended to accommodate ellipsoid statistical (ES-BGK) models, polyatomic molecules, chemical reactions or quantum gases; see for example [3,4,13,[24][25][26][27]31].…”

A consistent BGK model with velocity-dependent collision frequency for gas mixtures

“…There are many BGK models for gas mixtures proposed in the literature [14,16,10,12,26,21,15,5,1], many of which satisfy these basic requirements and, in addition, are able to match some prescribed relaxation rates and/or transport coefficients that come from more complicated physics models or from experiment. Many of these approaches have been extended to accommodate ellipsoid statistical (ES-BGK) models, polyatomic molecules, chemical reactions or quantum gases; see for example [22,29,13,23,24,3,4,25].…”