Kinetic model for steady heat flow

We construct a consistent solution of the Bhatnagar-Gross-Krook (BGK) model kinetic equation describing a system in a steady state with constant pressure and nonuniform temperature. The thermal profile is not linear and depends on the interaction potential. All the moments of the distribution function are given as polynomials in the local thermal gradient. In particular, the heat flux always obeys the (linear) Fourier law.