13-Moment System with Global Hyperbolicity for Quantum Gas
This work addresses a fundamental mathematical deficiency in a kinetic model for quantum gases, providing a corrected model that ensures hyperbolicity for all admissible states.
The authors identified that the quantum Grad's 13-moment system lacks global hyperbolicity and that thermodynamic equilibrium is not an interior point of the hyperbolicity region. They proposed a regularization that yields a new model which is hyperbolic for all admissible thermodynamic states while preserving the original system's accuracy.
We point out that the quantum Grad's 13-moment system [R. Yano, Physica A: Statistical Mechanics and its Applications, 416:231-241, 2014] is lack of global hyperbolicity, and even worse, the thermodynamic equilibrium is not an interior point of the hyperbolicity region of the system. To remedy this problem, by fully considering Grad's expansion, we split the expansion into the equilibrium part and the non-equilibrium part, and propose a regularization for the system with the help of the new theory developed in [Z. Cai et al., SIAM J. Appl. Math., 75(5):2001-2023, 2015, Y. Fan, J. Stat. Phys., 161(4), 2015]. This provides us a new model which is hyperbolic for all admissible thermodynamic states, and meanwhile preserves the approximate accuracy of the original system. It should be noted that this procedure is not a trivial application of the theory in [Z. Cai et al., SIAM J. Appl. Math., 75(5):2001-2023, 2015, Y. Fan, J. Stat. Phys., 161(4), 2015].