Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover

Simulations (e.g. Zhou et al., Phys. Rev. B 79, 115201 (2009)) show nonlocal effects of the ballistic/diffusive crossover. The local temperature has nonlinear spatial variation not contained in the local Fourier law $\vec{j}(\vec{r})=-\kappa\vec{\nabla}T(\vec{r})$. The heat current $\vec{j}(\vec{r})$ depends not just on the local temperature gradient $\vec{\nabla}T(\vec{r})$, but also on temperatures at points $\vec{r}^{ \ \prime}$ within phonon mean free paths, which can be micrometers long. This paper uses the Peierls-Boltzmann transport theory in non-local form to analyze the spatial variation $\Delta T(\vec{r})$. The relaxation-time approximation (RTA) is used because full solution is very challenging. Improved methods of extrapolation to obtain the bulk thermal conductivity $\kappa$ are proposed. Callaway invented an approximate method of correcting RTA for the $\vec{q}$ (phonon wavevector or crystal momentum) conservation of N (normal as opposed to Umklapp) anharmonic collisions This method is generalized to the non-local case where $\kappa(\vec{k})$ depends on wavevector of the current $\vec{j}(\vec{k})$ and temperature gradient $i\vec{k}\Delta T(\vec{k})$.