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.

中文翻译：

---

多物种Uehling-Uhlenbeck方程的BGK模型

我们提出了气体混合物量子玻尔兹曼方程的BGK模型。我们还提供了一个保证平衡系数存在的充分条件，以便该模型与量子Boltzmann方程共享相同的守恒律和$ H $定理。与经典的气体混合物BGK不同，量子多物种气体的局部平衡的平衡系数是通过不可明确求解的高度非线性关系定义的。我们以统一的方式验证这种非线性关系在条件下唯一确定平衡系数，从而导致我们模型的明确性。