  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.
