Weakly Non-Extensive Thermostatistics and the Ising Model with Long--range Interactions

We introduce a nonextensive entropic measure $S_{chi}$ that grows like $N^{chi}$,
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 $r^{-alpha}$ interparticle potentials. The power law (weakly nonextensive)
behavior exhibited by $S_{chi}$ 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_{chi} $ is parametrized by the real number $chi in[1,2]$ in such a
way that the standard logarithmic entropy is recovered when $chi=1$ . We study the
mathematical properties of the new entropy, showing that the basic requirements for a well
behaved entropy functional are verified, i.e., $S_{chi}$ possesses the usual properties
of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms
except the one related to additivity since $S_{chi}$ is nonextensive. For $1entropy $S_{chi}$ 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.