  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 + infinite, 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 Renyi entropy.
