Path Entropy Changes in Adiabatic Approximation

By applying adiabatic theorem to a Markovian system, we calculate the adiabatic and diabatic entropy changes along a path. As well known, the total path entropy change is separated into two parts, system and environment entropy changes, $\Delta S_{tot} = \Delta S_{sys} + \Delta S_{env}$. The environment entropy change, $\Delta S_{env}$, is divided again into two parts, an adiabatic contribution due to work, $\Delta S_{\mathcal{W}}$, and a diabatic contributions due to heat, $\Delta S_{\mathcal{Q}}$. In an adiabatic process, total path entropy change is same with the adiabatic path entropy change, $\Delta S_{A}$, which is given by sum of system entropy change and adiabatic contribution, $\Delta S_{A} = \Delta S_{sys} + \Delta S_{\mathcal{W}}$. Mathematical form of $\Delta S_{A}$ is a type of excess heat entropy change, but $\Delta S_{A}$ is due to work. By which, it is shown that the terms adiabatic and non-adiabatic contributions of $\Delta S_{na}$ and $\Delta S_{a}$ in [Phys. Rev. Lett. {\bf 104}, 090601 (2010)] should be completely switched, $i.e.$ $\Delta S_{na} \rightarrow \Delta S_{A}$ and $\Delta S_{a} \rightarrow \Delta S_{\mathcal{Q}}$ in fact.