as mode $i$, $R^{ji}_{21}(\omega)$ is the reflection coefficient of mode $j$ at frequency $\omega$ from side 2 back into side 2 as mode $i$, and so on.

The next question is how $T^{ij}_{12}(\omega)$ is related to the other reflection and transmission coefficients. The reflection coefficients are related to the transmission coefficients by energy conservation given by $$\sum_j T^{ij}_{12}(\omega)+R^{ij}_{12}(\omega) = 1,$$ and $$\sum_j T^{ij}_{21}(\omega)+R^{ij}_{21}(\omega) = 1.$$ $T^{ji}_{21}(\omega)$ is related to $T^{ij}_{12}(\omega)$ through the principle of detailed balance, which requires that no net heat flux can transmit across the interface when both materials are at an equilibrium temperature $T$. Applying this condition to every phonon mode on each side of the interface for each polarization and frequency gives: $$\label{eq:DetailedBalance}
T^{ij}_{12} (\omega)C^{i}_{1\omega}v^{i}_{1\omega} = T^{ji}_{21} (\omega) C^{j}_{2\omega}v^{j}_{2\omega}.$$ Therefore, we need to specify $T^{ij}_{12}(\omega)$, $R^{ij}_{12}(\omega)$, and $R^{ij}_{21}(\omega)$.

Let us first consider a special case where no mode conversion is allowed ($T^{ij}_{12}(\omega)$, $T^{ij}_{21}(\omega)$, $R^{ij}_{12}(\omega)$, $R^{ij}_{21}(\omega) = 0$ for $i \neq j$). Then, the interface conditions become $$\begin{aligned}
\label{eq:IC_1_special}
\int_0^1 g_{2\omega}^{i+} v_{2\omega}\mu d\mu &=& T^{ii}_{12}(\omega)\int_0^1 g_{1\omega}^{i+} v^i_{1\omega}\mu d\mu + R^{ii}_{21}(\omega) \int_0^1 g_{2\omega}^{i-}v^{i}_{2\omega}\mu d\mu, \\
\label{eq:IC_2_special}
\int_0^1 g_{1\omega}^{i-} v_{1\omega}\mu d\mu &=& T^{ii}_{21}(\omega)\int_0^1 g_{2\omega}^{i-} v^i_{2\omega}\mu d\mu+R^{ii}_{12}(\omega)\int_0^1 g_{1\omega}^{i+} v^{i}_{1\omega}\mu d\mu,\end{aligned}$$ and the detail balance becomes $$\label{eq:DetailedBalance_special}
T^{ii}_{12} (\omega)C^{i}_{1\omega}v^{i}_{1\omega} = T^{ii}_{21} 