Phonon mean free path ($\ell =v_g\tau$)

The phonon mean free path is the typical distance traveled by a phonon between the relevant scattering events. For heat transport, the relevant relaxation time must describe relaxation of heat current, not merely count all collisions. An isotropic kinetic estimate is $\kappa \simeq c_V{v}_s\ell /3$, where $c_V$ is heat capacity per unit volume. Boundary-limited $\ell$ and the low-temperature Debye model give $\kappa \propto T^3$.