pseudoenergy
What is the pseudoenergy? It is some effective "energy" (actually in units of power) that is mapped to a "local" temperature. Derivation: for scattering operator (C[f]), energy conservation enforces $$\sum_{\omega,p} \hbar \omega C[f] = 0.$$
The relaxation time replaces the scattering operator with the deviation dependent scattering lifetime $$C[f] = -\frac{f-f^{loc}}{\tau(\omega, p)}.$$
To enforce energy conservation, we require $$\sum_{\omega,p} \hbar \omega \frac{f-f^{loc}}{\tau(\omega,p)} = 0$$.
This motivates the definition of the pseudoenergy \(\tilde{E}\) as the lifetime-weighted moment of the energy $$\tilde{E} \equiv\sum_{\omega,p} \hbar \omega \frac{f}{\tau(\omega,p)}.$$
Overall, we compute the pseudotemperature by calculating the power emitted from the volume by scattering. This is found in two ways, given in eq. 8. The top one says "we have N phonons with energy \(\hbar \omega\) that scatter on avg every \(\tau\) seconds" and the bottom says "the emission of a volume at temperature \(T_{loc}\)" and T is found via equality.