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On the Planckian bound for heat diffusion in insulators

High temperature thermal transport in insulators has been conjectured to be subject to a Planckian bound on the transport lifetime $τ\gtrsim τ_\text{Pl} \equiv \hbar/(k_B T)$, despite phonon dynamics being entirely classical at these temperatures. We argue that this Planckian bound is due to a quantum mechanical bound on the sound velocity: $v_s < v_M$. The `melting velocity&#39; $v_M$ is defined in terms of the melting temperature of the crystal, the interatomic spacing and Planck&#39;s constant. We show that for several classes of insulating crystals, both simple and complex, $τ/τ_\text{Pl} \approx v_M/v_s$ at high temperatures. The velocity bound therefore implies the Planckian bound.