only for convenience. In the TSS representation, the Wehrl entropy may be written as $$\label{S_main}
S = - \int {\,\mathrm{d}}^2\alpha {\,\mathrm{d}}^2\beta \; Q(\alpha,\beta) \ln Q(\alpha,\beta).$$ where both integrals are over the entire complex plane. The definitions (\[entropy\_spin\]) and (\[S\_main\]) are not identical, but differ by an additive constant. However, in view of Eq. (\[dSdt\]), we will only be interested in the general rate of change of the entropy so we shall not differentiate between the two definitions.

Unlike the von Neumann entropy, the Wehrl entropy can be affected by unitary transformations. This is related to the coarse-graining aspect of the Husimi function. Hamiltonians which are linear in $J_i$ do not affect $S$, but in general nonlinear Hamiltonians do [@Zyczkowski2001]. Which classes of Hamiltonians affect the unitary part is still an open question [@Wehrl1979]. Here we shall not consider this unitary contribution, as it simply adds a new term to ${\,\mathrm{d}}S/{\,\mathrm{d}}t$, but rather concentrate on the dissipative contribution to ${\,\mathrm{d}}S/{\,\mathrm{d}}t$, which from Eq. (\[FP\]) is found to be $$\label{contribution_D}
\frac{{\,\mathrm{d}}S}{{\,\mathrm{d}}t}\Bigg|_\text{diss} = - \frac{(2J+1)}{4\pi}  \int {\,\mathrm{d}}\Omega \; \mathcal{D}(\mathcal{Q}) \ln \mathcal{Q}.$$ The goal is to separate this in the form of Eq. (\[dSdt\]), i.e., to identify terms that can be interpreted as an 