Clausius' theorem and the Second law in the process of isoenergetic thermalization

Isoenergetic thermalization amongst $n$ bodies is a well-known irreversible process, bringing the bodies to a common temperature $T_F$ and leading to a rise in the total entropy of the bodies. We express this change in entropy using the Clausius formula over a reversible path connecting $T_F$ with $T_f$ which corresponds to the entropy-preserving temperature of the initial nonequilibrium state. Under the assumption of positive heat capacities of the bodies, the Second law inequality simply follows from the fact that $T_F > T_f$. We extend this approach to the continuum case of an unequally heated rod, illustrating with the special case of the rod with constant heat capacity and a linear temperature profile. An interpolating profile between the discrete and the continuum models is studied whereby $T_f$, given by the geometric mean temperature over $n$ elements, is shown to approach the identric mean of the lowest and the highest temperatures as $n$ becomes large. We also discuss the case of negative heat capacity in a two-body set up where isoenergetic thermalization may be forbidden by the Second law. However, the alternate scheme in which first work is performed reversibly on the system and then an equivalent amount of energy is extracted in the form of heat, brings the system to an energy-preserving common temperature.