The thermodynamic limit in the non-extensive thermostatistics

Thermodynamic limit is more than just: ‘the limit of the system when size tends to infinity’. This rough definition can be sufficient in the simplest cases, but precise definition of the thermodynamic limit involves subtle constraints in order to compare systems of various sizes N but in the similar thermodynamic state. We discuss in this paper how these constraints must be handled in the non-extensive thermostatistics, within the canonical ensemble. Role of the entropy index q on the proper definition of the thermodynamic limit, is emphasized. In particular, the only choice to get non-trivial thermodynamic limit is: N(q−1)=constant. It can lead to a steady state different from the Boltzmann–Gibbs thermodynamic limit. We show that the hypothesis of non-extensivity of the entropy is generally not so crucial for the infinite systems, with the notable exception of the critical systems.