content_image/405264.jpg content_image/405265.jpg content_image/405266.jpg content_image/405267.jpg
Putting these three equations together, we finally obtain 

<img src='content_image/405264.jpg'>

Again, α and β are only independent in the isotropic case; in the fully anisotropic case we must have α = β . 

## 6. The heating and cooling rates of the different phonon modes 

Substituting Eqs. (A.20) and (A.29) into Eq. (A.18), and collecting terms acting on the same phonon mode by relabelling − p → p where necessary, we obtain for the phonon steady state 

<img src='content_image/405265.jpg'>

As this does not contain atomic operators, we can cancel the atomic part | ˜ 0 〉〈 ˜ 0 | , which replaces ρ (0) ss by ρ (0) $_{phn}$. It is also quadratic in phonon operators and diagonal in phonon wavevector. For p ̸ = 0 , the coefficients M$_{h}$$_{αβ}$ ( p ) and M$_{c}$$_{αβ}$ ( p ) are given by 

<img src='content_image/405266.jpg'>

For p = 0 , there appear to be additional terms 

<img src='content_image/405267.jpg'>

However, these extra terms (the last line of each equation) are pure imaginary, symmetric in α, β and of opposite signs in M$_{h}$$_{αβ}$ and M$_{c}$$_{αβ}$ . As b$_{p}$$_{α}$b † p β − b † p $_{β}$b$_{p}$$_{α}$ = δ$_{αβ}$ is a number so commutes with ρ , such terms cancel out in Eq. (A.30), so Eq. (A.31) can also be used for p = 0 . 

