A generalized thermodynamics for power-law statistics

We show that there exists a natural way to define a condition of generalized thermal equilibrium between systems governed by Tsallis thermostatistics, under the hypotheses that i) the coupling between the systems is weak, ii) the structure functions of the systems have a power-law dependence on the energy. It is found that the $q$ values of two such systems at equilibrium must satisfy a relationship involving the respective numbers of degrees of freedom. The physical properties of a Tsallis distribution can be conveniently characterized by a new parameter $\eta$ which can vary between $0$ and $+\infty$, these limits corresponding respectively to the two opposite situations of a microcanonical distribution and of a distribution with a predominant power-tail at high energies. We prove that the statistical expression of the thermodynamic functions is univocally determined by the requirements that a) systems at thermal equilibrium have the same temperature, b) the definitions of temperature and entropy are consistent with the second law of thermodynamics. We find that, for systems satisfying the hypotheses i) and ii) specified above, the thermodynamic entropy is given by R\'enyi entropy.