entropy production rate is able to capture the contribution of the amplitude damping to the loss of coherence. Thus, not only do we get a microscopic picture of the irreversible currents responsible for the entropy production, but we are also able to distinguish the different contributions related to the amplitude damping current $f$ and the dephasing current $\mathcal{J}_z$. Moreover, compared to Eq. (\[Pi\_dephasing\]), we also see a temperature-dependent prefactor multiplying $|\mathcal{J}_z(\mathcal{Q})|^2$. This term introduces an angular dependence of the dephasing and is a consequence of the fact that for the amplitude damping, decoherence is not homogeneous over the Bloch sphere.

Spin-1/2 case
-------------

We now consider the case of spin $1/2$, where all formulas become much simpler. We express all results in terms of the bath-induced magnetization $\bar{\tau}_z = -1/(2\bar{n}+1)$. The energy flux (\[PhiE\_damp\]) simplifies to $$\Phi_E = \frac{\gamma \omega}{2\bar{\tau}_z} (\bar{\tau}_z - \tau_z ).$$ The entropy flux (\[Phi\_damping\_angular\]), on the other hand, simplifies to $$\label{Phi_damp_W}
\Phi  = \frac{\nicefrac{\gamma}{2}}{\bar{\tau}_z^3}
\bigg(
\bar{\tau}_z+(\bar{\tau}_z^2-1)\tanh^{-1}(\bar{\tau}_z)
\bigg)(\tau_z-\bar{\tau}_z),$$ They are related by $$\Phi = \bigg[\frac{(1-\bar{\tau}_z^2)\tanh^{-1}(\bar{\tau}_z)-\bar{\tau}_z}{\bar{\tau}_z^2\; \omega}\bigg]\Phi_E.$$ In the limit $T\gg\omega$, this becomes approximately $$\Phi\simeq\frac{1}{3}\frac{\Phi_E}{T},$$ which is a particular case of Eq. (\[Phi\_Phi\_E\_high\_T\]).

Finally, we present the result for the entropy production rate, given by Eq. (\[Pid\]): $$\label{Pi_damp_spin12}
