On the nonextensivity of the long range X-Y model

It will be given analytical and numerical evidence supporting that the X-Y model yields an extensive, i.e., proportional to the number of degrees of freedom N, internal energy U for any value of the interaction range. The X-Y is a well known model of Statistical Mechanics. It is defined by a set of conjugate pairs of variables (h i , p i) i=1,..., N , N being the number of degrees of freedom, with a Hamiltonian (kinetic, K, plus potential, V) function H= 1 2 C N i=1 p 2 i + o 2 C N i, j=1 1 − cos(h i − h j) r a ij — K+V (1) In this expression r ij is the distance between the sites (i, j) of a regular lattice of dimensionality d and lattice spacing equal to one. For example, in the linear case that will be analyzed in detail later, d=1, it is r ij =|i − j|. Periodic boundary conditions are assumed, and it is understood the convention of minimal distance between sites i and j. The constant o sets the interaction strength. The definition is such that, for ferromagnetic coupling o \ 0 (the only one considered in this paper) the minimum value of the energy is equal to zero. The parameter a sets the interaction range and its variation allows to recover some limits of interest. For example, the case a=., for which