terms of the function $V$ since, it turns out, most differential operators act trivially on the exponential prefactor $e^{-\bm{c}^\dagger \bm{c}}$. The dissipator then becomes $$\label{contribution_Di}
\mathcal{D}(Q) = \frac{\gamma}{2}\frac{e^{-\bm{c}^\dagger \bm{c}}}{\pi^2}   \left[
\mathcal{J}_{-}(f(V)) - \mathcal{J}_{+}(f^*(V))  \right],$$ where $$\label{amp_fV}
f(V) = \left[  (\bar{n}+1) \beta \partial_\alpha - \bar{n} \alpha^* \partial_{\beta^*}\right] V.$$

Inserting these currents into Eq. (\[contribution\_D\]), integrating by parts, and writing everything in terms of $V$ quantities, we get $$\label{tmp3412}
\frac{d S}{dt}\bigg|_\text{diss} =  \frac{\gamma}{2} \int \frac{{\,\mathrm{d}}\bm{c}}{V}\; \frac{e^{-\bm{c}^\dagger \bm{c}}}{\pi^2} \;
\left[ f(V) \mathcal{J}_-(V) - f(V)^* \mathcal{J}_+(V)\right].$$ where

$$\label{current_sep}
 f(V) \mathcal{J}_-(V) - f(V)^* \mathcal{J}_+(V) = -  \frac{2 J V (f^* \alpha^* \beta+ f \alpha \beta^*)}{(\bar{n}+1) |\beta|^2 + \bar{n}|\alpha|^2} + \frac{\begin{pmatrix} f^* & f \end{pmatrix} \mathcal{W} \begin{pmatrix} f \\ f^* \end{pmatrix}}{(\bar{n}+1)^2 |\beta|^4 - \bar{n}^2 |\alpha|^4}$$

with $$\mathcal{W} = \begin{pmatrix}
(\bar{n}+1) |\beta|^4 - \bar{n} |\alpha|^4  &   (\alpha^* \beta)^2  \\[0.2cm]
(\alpha\beta^*)^2   &   (\bar{n}+1) |\beta|^4 - \bar{n} |\alpha|^4
\end{pmatrix}.$$ The first term in Eq. (\[current\_sep\]) is linear in the relevant currents, whereas the second one is quadratic. Hence, the first term should naturally be associated with an entropy flux rate and the latter with an entropy production rate. That is, we may separate Eq. (\[tmp3412\]) as $$\begin{aligned}
\Pi &= \frac{\gamma}{2} \int \frac{{\,\mathrm{d}}\bm{c}}{V} \frac{e^{-\bm{c}^\dagger \bm{c}}}{\pi^2} \; 
\frac{\begin{pmatrix} f^* & f 