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Hyperbolicity in extended thermodynamics of Fermi and Bose gases. (English) Zbl 1114.82018
Summary: We consider the balance system of Extended Thermodynamics with 13 moments in the case of Fermi and Bose gases, for processes not far from equilibrium. In this case, the hyperbolicity of the differential system holds only in a neighborhood of the equilibrium state. The main aim of the paper is to evaluate the hyperbolicity region of the differential system. The knowledge of this region in the state variables is mandatory to check the admissibility of the solutions and the corresponding boundary and Cauchy data in the limit of the approximation considered. The results are obtained through numerical evaluations of the Fermi and Bose integral functions \(I^\pm_n(\alpha)\) that appear in the characteristic polynomial. Particular attention is devoted to the completely degenerate case when Fermi gas reaches the \(0 K\) and when the Bose gas is in proximity of the transition temperature \(T_{c}\). In these limiting cases, the hyperbolicity requirement is lost according to previous results. In the last section we make use of the Maxwellian iteration in order to evaluate the heat conductivity and the viscosity for the degenerate Fermi and Bose gas.

MSC:
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C03 Foundations of time-dependent statistical mechanics
35L40 First-order hyperbolic systems
80A05 Foundations of thermodynamics and heat transfer
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