content_image/417779.jpg content_image/417786.jpg
<img src='content_image/417779.jpg'>

In the fully anisotropic case (three different phonon frequencies ν$_{x}$ , ν$_{y}$ , ν$_{z}$ ), Eq. (A.30) simplifies to 

<img src='content_image/417786.jpg'>

In this case each phonon mode p α evolves independently, with a heating (phonon creation) rate Re M$_{h}$$_{αα}$ ( p ), a cooling (phonon decay) rate Re M$_{c}$$_{αα}$ ( p ), and an energy dispersion Im M$_{h}$$_{αα}$ ( p ) + Im M$_{c}$$_{αα}$ ( p ). 

## 7. Consistency with diagrammatic method 

Comparing Eq. (A.31) with Eq. (29), we see that both methods give the same heating and cooling rates, i.e. Re M$_{h}$$_{αα}$ = R$_{h}$$_{α}$, Re M$_{c}$$_{αα}$ = R$_{c}$$_{α}$ . 

