  We introduce a fast Fourier spectral method for the multi-species Boltzmann
collision operator. The method retains the riveting properties of the
single-species fast spectral method (Gamba et al. SIAM J. Sci. Comput., 39 pp.
B658--B674 2017) including: (a) spectral accuracy, (b) reduced computational
complexity compared to direct spectral method, (c) reduced memory requirement
in the precomputation, and (d) applicability to general collision kernels. The
fast collision algorithm is then coupled with discontinuous Galerkin
discretization in the physical space (Jaiswal et al. J. Comp. Phys., 378 pp.
178--208 2019) to result in a highly accurate deterministic method (DGFS) for
the full Boltzmann equation of gas mixtures. A series of numerical tests is
performed to illustrate the efficiency and accuracy of the proposed method.
Various benchmarks highlighting different collision kernels, different mass
ratios, momentum transfer, heat transfer, and in particular the diffusive
transport have been studied. The results are directly compared with the direct
simulation Monte Carlo (DSMC) method.
