\end{pmatrix} \mathcal{W} \begin{pmatrix} f \\ f^* \end{pmatrix}}{(\bar{n}+1)^2 |\beta|^4 - \bar{n}^2 |\alpha|^4},\\
\Phi &= \gamma J \int {\,\mathrm{d}}\bm{c} \frac{e^{-\bm{c}^\dagger \bm{c}}}{\pi^2}  \frac{f(V) \alpha \beta^* + f(V)^* \alpha^* \beta}{(\bar{n}+1)|\beta|^2 + \bar{n} |\alpha|^2 }.
\end{aligned}$$

We can also express these formulas in terms of angular variables. First, for the entropy flux, we use Eq. (\[fQ\_spin\]) and integrate over $\mathcal{I}$ to get $$\label{Phi_damping_angular}
\Phi = \frac{(2J+1)}{4\pi} \gamma J\int{\,\mathrm{d}}\Omega \sin\theta \bigg\{\frac{2 J \mathcal{Q} \sin\theta}{(2\bar{n}+1)-\cos\theta} - \partial_\theta \mathcal{Q}\bigg\}.$$ Similarly, for the entropy production, we get $$\label{eq:wehrl_prod_ad_def}
\Pi = \frac{\gamma(2J+1)}{8\pi}\int\frac{{\,\mathrm{d}}\Omega}{\mathcal{Q}} \frac{\begin{pmatrix} f(\mathcal{Q})^* & f(\mathcal{Q}) \end{pmatrix} \tilde{\mathcal{W}} \begin{pmatrix} f(\mathcal{Q}) \\ f(\mathcal{Q})^* \end{pmatrix}    }{(\bar{n}+1)^2 \sin^4 (\theta/2) - \bar{n}^2 \cos^4(\theta/2)},
$$ where $\tilde{\mathcal{W}} = \mathcal{W}/\mathcal{I}^2$. As $V$ is a homogeneous function \[Eq. (\[Euler\])\], one may relate the currents $f(V)$ and $\mathcal{J}_z(V)$ according to $$\label{Jz_f}
\mathcal{J}_z(V) = \frac{f(V)^* \alpha^* \beta-f(V) \alpha \beta^*}{(\bar{n}+1) |\beta|^2 - \bar{n} |\alpha|^2}.$$ Using this result and expanding the matrix $\tilde{\mathcal{W}}$, we then arrive at $$\label{Pid}
\Pi = \frac{\gamma}{2} \frac{(2J+1)}{4\pi} \int \frac{{\,\mathrm{d}}\Omega}{\mathcal{Q}} 
\Bigg\{ 
\frac{[
2 J \mathcal{Q} \sin\theta + (\cos\theta - (2\bar{n}+1)) \partial_\theta \mathcal{Q}
]^2}{(2\bar{n}+1)-\cos\theta}
+ 
 |\mathcal{J}_z(\mathcal{Q})|^2 \bigg[(2\bar{n}+1)\cos\theta-1\bigg] \frac{\cos\theta}{\sin^2\theta}
 \Bigg\}.$$

The most striking feature about this result is the appearance of the dephasing current $|\mathcal{J}_z(\mathcal{Q})|^2$ \[Eq. (\[Pi\_dephasing\])\] as a part of the entropy production. This means that the Wehrl 