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.
Categories: quant-ph cond-mat.mes-hall cond-mat.stat-mech