Journal of Statistical Physics, Vol. 28, No. 3, 1982. p. 589-602.
The Hilbert expansion to the Boltzmann equation is carried out for steady flow. It is shown that the first term in the Hilbert series for the distribution function is a local Maxwellian leading to the steady Euler equations. The steady field equations that follow from the solution of the second term in the series are derived. The formulas for thermal conductivity and for viscosity of Hilbert that appear in the steady field equations of the second approximation are shown to be precisely the same as those obtained by Chapman and Enskog. The procedure to obtain higher approximations by Hilbert's method is summarized.