A theorem on the equilibrium thermodynamics of Hamiltonian systems

We prove a theorem stating that, within an interesting class of stationary solutions of the Liouville equation, the Tsallis q-distributions are the only ones that satisfy a condition which guarantees the existence of an integrating factor for the heat. The functional forms of the distribution as well as that of entropy, temperature and heat capacity are derived. We then explain why the Gibbs distribution, as a limit of the Tsallis one, plays a privileged role for large size ordinary molecular systems.