The Convergence Failure

The Boltzmann Transport Equation is the standard tool for computing thermal conductivity in crystals. It works well for most three-dimensional materials. For two-dimensional systems, it universally fails — the predicted thermal conductivity diverges or fails to converge, regardless of the specific 2D material.

Di Lucente, Marzari, and Simoncelli trace the failure to a missing ingredient. The standard BTE uses Fermi’s Golden Rule for scattering rates — sharp, energy-conserving collision events. Real phonons aren’t sharp. They have finite lifetimes, which means they have finite energy widths — collisional broadening. In 3D, this broadening is a small correction. In 2D, where flexural (out-of-plane) phonons have anomalous dispersion, the broadening is essential. Without it, the theory breaks.

The through-claim: the Boltzmann equation fails in 2D not because the physics is exotic but because the approximation drops a term that happens to be negligible in 3D. The same omission produces a universal failure mode — all 2D systems, not just pathological ones. The fix is to derive the transport equation more carefully from the Kadanoff-Baym equations, keeping the self-consistent broadening that Fermi’s Golden Rule discards.

The corrected theory — a linearized generalized BTE — converges for 2D materials and resolves long-standing discrepancies in diamond’s thermal conductivity. The old theory wasn’t wrong about the mechanism. It was wrong about what you could drop.

The approximation was invisible in 3D because it was small. It was catastrophic in 2D because it wasn’t.