  The Boltzmann equation with the Bhatnagar-Gross-Krook collision operator is
considered for the Bose-Einstein and Fermi-Dirac equilibrium distribution
functions. We show that the expansion of the microscopic velocity in terms of
Hermite polynomials must be carried until the fourth order to correctly
describe the energy equation. The viscosity and thermal coefficients,
previously obtained by J.Y. Yang et al through the Uehling-Uhlenbeck approach,
are also derived here. Thus the construction of a lattice Boltzmann method for
the quantum fluid is possible provided that the Bose-Einstein and Fermi-Dirac
equilibrium distribution functions are expanded until fourth order in the
Hermite polynomials.
