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

We introduce a new nonextensive entropic measure S-chi that grows like N-ch i, where N is the size of the system under consideration. This kind of none xtensivity arises in a natural way in some N-body systems endowed with long -range interactions described by r(-alpha) interparticle potentials. The po wer law (weakly nonextensive) behavior exhibited by S-chi is intermediate b etween (1) the linear (extensive) regime characterizing the standard Boltzm ann-Gibbs entropy and (2) the exponential law (strongly nonextensive) behav ior associated with the Tsallis generalized q-entropies. The functional S-c hi is parametrized by the real number chi epsilon [1,2] in such a way that the standard logarithmic entropy is recovered when chi = 1. We study the ma thematical properties of the new entropy, showing that the basic requiremen ts for a well behaved entropy functional are verified, i.e., S-chi possesse s the usual properties of positivity, equiprobability, concavity and irreve rsibility and verifies Khinchin axioms except the one related to additivity since S-chi is nonextensive. For 1 < <chi> < 2, the entropy S-<chi> become s superadditive in the thermodynamic limit. The present formalism is illust rated by a numerical study of the thermodynamic scaling laws of a ferromagn etic Ising model with long-range interactions.