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

摘要

We introduce a new nonextensive entropic measure S_χ that grows like N~χ, 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 S_χ 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 S_χ 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., S_χ possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms except the one related to additivity since S_χ is nonextensive. For 1 < χ < 2, the entropy S_χ 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.