  Despite its conceptual and practical importance, the rigorous derivation of
the steady incompressible Navier-Stokes-Fourier system from the Boltzmann
theory has been {an} outstanding {open problem} for general domains in 3D. We
settle this open question in {the} affirmative, in the presence of a small
external field and a small boundary temperature variation for the diffuse
boundary condition. We employ a recent quantitative $L^{2}-L^{\infty }$
approach with new $L^{{6}}$ estimates for the hydrodynamic part $\mathbf{P}f$
of the distribution function. Our results also imply the validity of Fourier
law in the hydrodynamical limit, and our method {leads to {asymptotical}
stability of steady Boltzmann solutions as well as the derivation of the
{unsteady} Navier-Stokes Fourier system}.
