= Kinetic thermal-conductivity approximation
{title2=$\kappa\simeq C\langle v\rangle\ell/3$}
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 $C\langle v\rangle\ell/3$ 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.
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