Rigorous derivation of weakly anharmonic phonon damping scaling

Derive rigorously the long-wavelength phonon damping scaling \(\Gamma_k\sim k^{5/3}(\beta T)^{4/3}\) observed in the weakly anharmonic \(\beta\)-Fermi–Pasta–Ulam–Tsingou chain.

Background

The paper studies heat transport in the weakly anharmonic $\beta$-Fermi–Pasta–Ulam–Tsingou chain by resolving the heat current into phonon-mode contributions. Numerical extrapolation indicates that the long-wavelength phonon damping rate scales as $\Gamma_k\sim k^{5/3}(\beta T)^{4/3}$, which leads to a heat-current autocorrelation decay $C_J(t)\sim t^{-3/5}+\mathcal{O}(t^{-4/5})$ and a conductivity divergence $\kappa(N)\sim N^{2/5}+\mathcal{O}(N^{1/5})$.

Although the observed damping law is consistent with the transport scaling extracted from the simulations and with prior theoretical expectations, the paper states that a rigorous analytical derivation of this specific dependence on wave number and anharmonicity remains unresolved. Establishing such a derivation would provide a mathematical foundation for the phonon-based explanation of anomalous heat transport in this model.