Weakly nonextensive thermostatistics and the Ising model with long–range interactions

We introduce a new nonextensive entropic measure Sχ that grows like N , where N is the size of the system under consideration. This kind of nonextensivity arises in a natural way in some N-body systems endowed with long-range interactions described by r−α interparticle potentials. The power law (weakly nonextensive) behavior exhibited by Sχ is intermediate between (1) the linear (extensive) regime characterizing the standard Boltzmann-Gibbs entropy and (2) the exponential law (strongly nonextensive) behavior associated with the Tsallis generalized q-entropies. The functional Sχ is parametrized by the real number χ ∈ [1, 2] in such a way that the standard logarithmic entropy is recovered when χ = 1. We study the mathematical properties of the new entropy, showing that the basic requirements for a well behaved entropy functional are verified, i.e., Sχ possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms except the one related to additivity since Sχ is nonextensive. For 1 < χ < 2, the entropy Sχ becomes superadditive in the thermodynamic limit. The present formalism is illustrated by a numerical study of the thermodynamic scaling laws of a ferromagnetic Ising model with long-range interactions. PACS. 05.20.-y Classical statistical mechanics – 05.20.Gg Classical ensemble theory – 02.70.Lq Monte Carlo and statistical methods 75.10.Hk Classical spin models