Fluctuation-dissipation and virtual processes in interacting phonon systems

Phonon-phonon interactions are fundamental to understanding a wide range of material properties, including thermal transport and vibrational spectra. In conventional perturbative approaches, energy conservation during each microscopic phonon interaction is enforced using delta functions. We demonstrate that these delta functions stem from an incomplete treatment, that violates the fluctuation-dissipation theorem governing systems at equilibrium. By replacing delta functions with convolutions and introducing a self-consistency condition for the phonon spectral function, we provide a more accurate and physically consistent framework. For systems where phonon dynamics can be approximated as Markovian, we simplify this approach, reducing the dissipative component to a single parameter tied to phonon lifetimes. Applying this method to boron arsenide, we find that self-consistent linewidths better capture the phonon scattering processes, significantly improving agreement with experimental thermal conductivity values. These results also challenge the conventional view of four-phonon processes as dominant in BAs, demonstrating the adequacy of a three-phonon description, provided it is self-consistent. With this method we address critical limitations of perturbative approaches, offering new insights into dissipation and phonon-mediated processes, and enabling more accurate modeling of anharmonic materials.