= Solution
A solid's atoms do not travel freely through it, but its <phonons> and mobile electronic <quasiparticles> carry energy between scattering events. In a near-equilibrium isotropic diffusive regime, their transport can therefore use a <kinetic thermal-conductivity approximation>. Carriers moving along the local temperature gradient sample energies characteristic of their last scattering position. Linearizing that temperature difference and averaging squared directional cosines gives
$$
\boxed{\mathbf j_Q=-\kappa\nabla T,\qquad\kappa\simeq\frac13C\langle c\rangle\ell.}
$$
Here C is the contributing carrier heat capacity per unit volume, c its group speed and ell a transport mean free path. More accurately, the formula is a mode/energy-weighted sum of heat capacities times $v_g^2\tau/3$, not a product of arbitrary separately averaged values. Anisotropic crystals require a tensor, and ballistic samples or strongly nonquasiparticle regimes need a different description.
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