Thermodynamical limit in non-extensive and Renyi statistics

Previous results on R´enyi and Wang’s formalism of the Tsallis thermostatics are founded by using an extensive variable z connected to the entropic parameter q . It is shown that in the thermodynamical limit both thermostatics meet all the requirements of equilibrium thermodynamics. In particular, both the Tsallis and R´enyi entropies are extensive functions of state and the temperature of the system is intensive. In the thermodynamical limit Wang’s incomplete nonextensive statistics resembles the Tsallis one, but the R´enyi thermostatics is reduced to the usual Boltzmann-Gibbs one. The principle of additivity and the zeroth law of thermodynamics in the canonical ensemble for both thermostatics are demonstrated on the particular example of the classical ideal gas of identical particles.