Weakly Nonextensive Thermostatistics and the Ising Model with Long--range Interactions

Abstract

We introduce a 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 the (2) the exponential law (strongly nonextensive) behavior associated with the Tsallis generalized $q$-entropies. The functional $S_χ $ is parametrized by the real number $χ\in[1,2]$ 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.