Regularized Boltzmann entropy determines macroscopic adiabatic accessibility

How the thermodynamic entropy $S_{TD}$ is related to the Boltzmann entropy $S_{B}$ has been one of the central issues since the beginning of statistical mechanics. Today, it is believed that the thermodynamic entropy $S_{TD}$ is equal to a function $\tilde{S}_{B}$ that is defined by regularizing the Boltzman entropy in order to ensure extensivity. However, it is not known whether $\tilde{S}_{B}$ completely determines the possibility of a macroscopic adiabatic transformation in the same way as $S_{TD}$ does. In this paper, by formulating the possibility of a macroscopic adiabatic transformations in terms of "coarse-graining" of quantum operations, we prove that $\tilde{S}_{B}$ provides a necessary and sufficient condition for the possibility of a macroscopic adiabatic transformation.