Kinetic thermal-conductivity approximation ($\kappa \simeq C⟨v⟩\ell /3$)

In an isotropic near-equilibrium quasiparticle transport picture, an angular average of squared velocity components supplies the factor1/3. Mode-resolved heat capacities times $v_g^2\tau$ then sum to thermal conductivity. Replacing them by $C⟨v⟩\ell /3$ is a characteristic-scale approximation, not an exact product of separately chosen averages. A solid can use this kinetic structure because mobile phonon and electron excitations transport energy even when the constituent atoms remain localized.