First-order Chapman--Enskog velocity distribution function in a granular   gas
  A method is devised to measure the first-order Chapman-Enskog velocity distribution function associated with the heat flux in a dilute granular gas. The method is based on the application of a homogeneous, anisotropic velocity-dependent external force which produces heat flux in the absence of gradients. The form of the force is found under the condition that, in the linear response regime, the deviation of the velocity distribution function from that of the homogeneous cooling state obeys the same linear integral equation as the one derived from the conventional Chapman-Enskog expansion. The Direct Simulation Monte Carlo method is used to solve the corresponding Boltzmann equation and measure the dependence of the (modified) thermal conductivity on the coefficient of normal restitution $\alpha$. Comparison with previous simulation data obtained from the Green--Kubo relations [Brey et al., J. Phys.: Condens. Matter 17, S2489 (2005)] shows an excellent agreement, both methods consistently showing that the first Sonine approximation dramatically overestimates the thermal conductivity for high inelasticity ($\alpha\lesssim 0.7$). Since our method is tied to the Boltzmann equation, the results indicate that the failure of the first Sonine approximation is not due to velocity correlation effects absent in the Boltzmann framework. This is further confirmed by an analysis of the first-order Chapman-Enskog velocity distribution function and its three first Sonine coefficients obtained from the simulations. 
J. M. Montanero, A. Santos, and V. Garzo