of $\sigma_x$ with eigenvalue $+1$. To illustrate the physics of the problem, in Fig. \[fig:bloch\_damping\] we show the dynamics of the spin using the Bloch sphere.

![\[fig:bloch\_damping\] Evolution of a system under a time-dependent magnetic field \[Eq. (\[H\_example\])\] and subject to the amplitude damping dissipator \[Eq. (\[damp\_diss\])\]. Left: evolution of the spin in the Bloch sphere, starting in the $x$ direction. Right: Wehrl and von Neumann entropy production rates. In this example, we are considering $b_0/\gamma=5$, $b_1/\gamma=10$, $\omega/\gamma=5$, and $\bar{\tau}_z=-1/3$. ](bloch_damp.pdf){width="100.00000%"}

![\[fig:bloch\_damping\] Evolution of a system under a time-dependent magnetic field \[Eq. (\[H\_example\])\] and subject to the amplitude damping dissipator \[Eq. (\[damp\_diss\])\]. Left: evolution of the spin in the Bloch sphere, starting in the $x$ direction. Right: Wehrl and von Neumann entropy production rates. In this example, we are considering $b_0/\gamma=5$, $b_1/\gamma=10$, $\omega/\gamma=5$, and $\bar{\tau}_z=-1/3$. ](plot9.pdf){width="100.00000%"}

In the long-time limit, the system tends to its stationary state where ${\,\mathrm{d}}S/{\,\mathrm{d}}t= 0$ so $\Pi= \Phi$. For the von Neumann case, we get, in the steady state, $$\Pi_\text{vN} = \Phi_{\text{vN}}=   -\frac{ 2\gamma b_1^2\bar{\tau}_z^2  \tanh^{-1}(\bar{\tau}_z)}{\gamma^2+2\bar{\tau}_z^2(b_1^2+2(b_0+\omega)^2)}.$$ For the Wehr entropy, we have instead $$\Pi = \Phi= - \frac{\gamma b_1^2 [\bar{\tau}_z+(\bar{\tau}_z^2-1)\tanh^{-1}(\bar{\tau}_z)]  }{\gamma^2+2\bar{\tau}_z^2(b_1^2+2(b_0+\omega)^2)}.$$ In the limit $T\rightarrow 0$ the former diverges, whereas the latter tends to $$\Pi 