Recent THz spectroscopy of the quantum paraelectric ${\mathrm{SrTiO}}_3$ (K. S. Kumar et al., arXiv:2501.15771) and a high-$T_c$ cuprate (D. Chaudhuri et al., arXiv:2503.15646) has renewed interest in energy relaxation in correlated electron systems. We consider a situation in which single-phonon scattering is forbidden by symmetry or momentum conservation, while two-phonon scattering is allowed. Solving the Boltzmann equation, we show that above the Bloch–Grüneisen temperature the energy relaxation rate from two soft transverse optical phonons exceeds the single-phonon one: while the latter scales as $1/T$, the former is linear in $T$. This dominance of two-phonon scattering invalidates the usual picture of energy diffusion arising from frequent scattering by subthermal phonons; instead, energy relaxes via rare scattering events involving thermal phonons. Below the Bloch–Grüneisen temperature, the energy relaxation rate scales as the single-particle rate, namely as $T^3$ for soft phonons. For anisotropic electron bands, an intermediate regime appears between two Bloch–Grüneisen temperatures, in which both allowed single-phonon and two-phonon processes scale as $T^2$.

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• Diffusion
• Electron relaxation
• Electron-phonon coupling
• Kinetic theory
• Optical phonons
• Boltzmann theory