Physica A 416 (2014) 231–241 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Semi-classical expansion of distribution function using modified Hermite polynomials for quantum gas Ryosuke Yano ∗ Department of Advanced Energy, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan highlights • Semi-classical expansion of quantum distribution function. • Grad’s 13 moment equations for the Uehling–Uhlenbeck equation. • Moment equations for quantum gas. article info Article history: Received 15 July 2014 Available online 8 September 2014 Keywords: Quantum gas Kinetic theory for quantum gas Moment equations for quantum gas abstract The author proposes the semi-classical expansion of the distribution function using modi- fied Hermite polynomials to calculate moment equations for quantum gas. The complete- ness of the semi-classical expansion of the distribution function is not satisfied, whereas we can conjecture that moment equations obtained using the semi-classical expansion coin- cides with those obtained using Uehling–Uhlenbeck equation. Actually, Grad’s 13 moment equations, which are calculated using correct Grad’s 13 moment equation, coincide with those, which are calculated using the semi-classical expansion of the distribution function, when the collisional term of the Uehling–Uhlenbeck equation is replaced with the quantum Bhatnagar–Gross–Krook model. © 2014 Elsevier B.V. All rights reserved. 1. Introduction The quantum gas dynamics is significant for understanding of fluids such as Bose–Einstein condensation (BEC) [1], in which the quantum (spin) effect is markedly significant. In particular, the dilute quantum gas is described using the quantum Boltzmann equation, namely, Uehling–Uhlenbeck equation [2]. Meanwhile, the kinetic theory of the quantum gas on the basis of the Uehling–Uhlenbeck equation is not established in comparison with the kinetic theory of the classical gas on the basis of the classical Boltzmann equation, adequately. The one reason for such an inadequate discussion of the kinetic theory of the quantum gas depends on the complex structure of the collisional term of the Uehling–Uhlenbeck equation. Meanwhile, the author considers that the main reason for such an inadequate discussion of the kinetic theory of the quantum gas depends on the immaturity of the derivation of the hydrodynamic equation from the Uehling–Uhlenbeck equation. Actually, nobody has calculated Grad’s 13 moment equations for the quantum gas. Meanwhile, the dilute quantum gas must be discussed beyond the Navier–Stokes–Fourier (NSF) equation. In this paper, we consider the semi-classical expansion of the distribution function using the modified Hermite polyno- mials to calculate moment equations in an easy way. The coefficients of the modified Hermite polynomials are determined ∗ Tel.: +81 4 7136 3858. E-mail addresses: Zepheyrus25@aol.com, yano@k.u-tokyo.ac.jp. http://dx.doi.org/10.1016/j.physa.2014.08.067 0378-4371/© 2014 Elsevier B.V. All rights reserved.