Abstract
We point out that the quantum Grad’s 13-moment system (Yano in Physica A 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 hyperbolic regularization theory developed in Cai et al. (SIAM J Appl Math 75(5):2001–2023, 2015) and Fan et al. (J Stat Phys 162(2):457–486, 2016). 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 hyperbolic regularization theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Maple is a trademark of Waterloo Maple Inc.
References
Cai, Z., Fan, Y., Li, R.: Globally hyperbolic regularization of Grad’s moment system in one dimensional space. Commun. Math. Sci. 11(2), 547–571 (2013)
Cai, Z., Fan, Y., Li, R.: Globally hyperbolic regularization of Grad’s moment system. Commun. Pure Appl. Math. 67(3), 464–518 (2014)
Cai, Z., Fan, Y., Li, R.: On hyperbolicity of 13-moment system. Kinet. Relat. Models 7(3), 415–432 (2014)
Cai, Z., Fan, Y., Li, R.: A framework on moment model reduction for kinetic equation. SIAM J. Appl. Math. 75(5), 2001–2023 (2015)
Chapman, S.: On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas. Philos. Trans. R. Soc. A 216(538–548), 279–348 (1916)
Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S.: Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71(3), 463 (1999)
Fan, Y., Koellermeier, J., Li, J., Li, R., Torrilhon, M.: Model reduction of kinetic equations by operator projection. J. Stat. Phys. 162(2), 457–486 (2016)
Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331–407 (1949)
Ikenberry, E., Truesdell, C.: On the pressures and the flux of energy in a gas according to Maxwell’s kinetic theory. I. J. Ration. Mech. Anal. 5(1), 1–54 (1956)
Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer Tracts in Natural Philosophy, vol. 37, 2nd edn. Springer, New York (1998)
Nouri, A.: An existence result for a quantum BGK model. Math. Comput. Model. 47(3), 515–529 (2008)
Uehling, E.A., Uhlenbeck, G.E.: Transport phenomena in Einstein–Bose and Fermi–Dirac gases. I. Phys. Rev. 43(7), 552 (1933)
Yano, R.: Semi-classical expansion of distribution function using modified Hermite polynomials for quantum gas. Physica A 416, 231–241 (2014)
Yano, R.: On quantum Fokker–Planck equation. J. Stat. Phys. 158(1), 231–247 (2015)
Acknowledgements
Y. Di was supported by the National Natural Science Foundation of China (Grant No. 11271358). Fan and Li were supported in part by the National Natural Science Foundation of China (Grant No. 91330205, 11421110001, 11421101 and 11325102).
Author information
Authors and Affiliations
Corresponding author
Appendix 1: Coefficients of (22) and (25)
Appendix 1: Coefficients of (22) and (25)
The coefficients in (22) and (25) can be directly obtained by calculating the determinant of the matrix \(\mathbf{{A}}_1\). Here we list the coefficients in (22) and (25) as following.
Rights and permissions
About this article
Cite this article
Di, Y., Fan, Y. & Li, R. 13-Moment System with Global Hyperbolicity for Quantum Gas. J Stat Phys 167, 1280–1302 (2017). https://doi.org/10.1007/s10955-017-1768-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-017-1768-0