A fast spectral method for the Uehling-Uhlenbeck equation for quantum gas mixtures: Homogeneous relaxation and transport coefficients

Abstract

A fast spectral method (FSM) is developed to solve the Uehling-Uhlenbeck equation for quantum gas mixtures with generalized differential cross-sections. The computational cost of the proposed FSM is O(M dv−1 N dv +1 log N), where d v is the dimension of the problem, M dv −1 is the number of discrete solid angles, and N is the number of frequency nodes in each direction. Spatially-homogeneous relaxation problems are used to demonstrate that the FSM conserves mass and momentum/energy to the machine and spectral accuracy, respectively. Based on the variational principle, transport coefficients such as the shear viscosity, thermal conductivity, and diffusion are calculated by the FSM, which compare well with analytical solutions. Then, we apply the FSM to find the accurate transport coefficients through an iterative scheme for the linearized quantum Boltzmann equation. The shear viscosity and thermal conductivity of the three-dimensional quantum Fermi and Bose gases interacting through hard-sphere potential are calculated. For Fermi gas, the relative difference between the accurate and variational transport coefficients increases with the fugacity; for Bose gas, the relative difference in the thermal conductivity has similar behavior as the gas moves from the classical to the degenerate limits, but that in the shear viscosity decreases. Finally, the shear viscosity and diffusion coefficient have also been calculated for a two-dimensional equal-mole mixture of Fermi gases. When the molecular mass of the two components are the same, our numerical results agree well with the variational solution. However, when the molecular mass ratio is not one, large discrepancies between the accurate and variational results are observed; our results are reliable because (i) the method relies on no assumption and (ii) the ratio between shear viscosity and entropy density satisfies the minimum bound predicted by the string theory.