Phonon collisional broadening and heat transport beyond the Boltzmann equation

In crystals, macroscopic technological properties such as thermal conductivity stem from microscopic phonon drift and scattering, described by the Boltzmann Transport Equation (BTE). Despite its widespread use, the general space–time nonlocal form of the BTE still lacks a rigorous derivation of its collision term based on Fermi's Golden Rule (FGR) and becomes inadequate when the energy-variation scale set by phonon dispersion steepness approaches collisional broadening. A hallmark of this issue is the poor numerical convergence of conductivity with respect to the smearing used to compute FGR rates. This is often bypassed with adaptive schemes that, however, violate detailed balance and allow unphysical negative eigenvalues in the collision operator. Here, we overcome these limitations by rigorously deriving the space–time-dependent BTE from the Kadanoff–Baym Equation (KBE), and introduce a beyond-FGR BTE that includes phonon collisional broadening and energy-nonconserving scattering. More generally, we establish a hierarchy of ansätze for Green's and spectral functions, enabling controlled extensions of the semiclassical BTE and a roadmap toward quantum KBE accuracy. Finally, first-principles calculations and analytical arguments show that this approach resolves two long-standing problems of the FGR-based BTE: (i) conductivity convergence issues, especially in heat conductors; (ii) universal failure in 2D systems due to FGR predicting unphysical overdamping of flexural modes.