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