  A fast spectral method (FSM) is developed to solve the Uehling-Uhlenbeck
equation for quantum gas mixtures with generalized differential cross-sections.
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.
