Tsallis ensemble as an exact orthode

We show that Tsallis ensemble of power-law distributions provides a mechanical model of equilibrium thermodynamics for possibly low dimensional interacting Hamiltonian systems, i.e., using Boltzmann's original nomenclature, we prove that it is an exact orthode. This means that the heat differential admits the inverse average kinetic energy as an integrating factor. One immediate consequence is that the logarithm of the normalization function can be identified with the entropy, instead of the q-deformed logarithm. It has been noted that such entropy coincides with Rényi entropy rather than Tsallis entropy, it is mechanically nonadditive, tends to the standard canonical entropy as the power index tends to infinity and is consistent with the free energy formula proposed in [S. Abe, et al., Phys. Lett. A 281 (2001) 126]. It is also shown that the heat differential admits the Lagrange multiplier used in nonextensive thermodynamics as an integrating factor too, and that the associated entropy is given by ordinary nonextensive entropy. The mechanical approach proposed in this work is fully consistent with an information-theoretic approach based on the maximization of Rényi entropy.