13-Moment System with Global Hyperbolicity for Quantum Gas

TL;DR

This paper addresses the lack of global hyperbolicity in the quantum Grad's 13-moment system by proposing a regularized model that ensures hyperbolicity across all thermodynamic states while maintaining accuracy.

Contribution

The authors introduce a novel regularization of the quantum 13-moment system that guarantees hyperbolicity for all admissible states, improving upon previous models.

Findings

The original quantum 13-moment system is not globally hyperbolic.

The proposed regularization achieves hyperbolicity for all thermodynamic states.

The new model preserves the approximate accuracy of the original system.

Abstract

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].