BGK Model of the Multi-species Uehling-Uhlenbeck Equation

We propose a BGK model of the quantum Boltzmann equation for gas mixtures. We also provide a suffcient condition that guarantees the existence of equilibrium coeffcients 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 coeffcients 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 coeffcients under the condition, leading to the well-definedness of our model.