  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.
