Mean Field in Long-Range Ferromagnets and Periodic Boundary Conditions

Periodic boundary conditions are applied to a ferromagnetic spin lattice. A symmetrical lattice and its contributions all over space are being used. Results, for the Ising model with ferromagnetic interaction that decays as a $1/r^{D+\nu}$ law, are discussed in the mean field approximatio

Critical Temperature and Nonextensivity in Long-range Interacting Lennard-Jones-like Fluids

Molecular dynamic simulations for systems with $D=2,3$ Lennard-Jones-like interactions are studied. In the model, we assume that, at long distances, the two-body attractive potential decays as $r^{-\alpha}$. Thermodynamic extensivity (nonextensivity) is observed for $\alpha > D$ ($0\leq \alpha \leq D$). Particular attention is payed to the liquid-gas critical point located, in the temperature-pressure plane, at ($T_c,P_c$). ($T_c,P_c$) are, in the $N\to \infty$ limit ($N\equiv$ number of molecules), {\em finite} for $\alpha > D$ and {\em diverge} for $\alpha \leq D$ (as $(\alpha - D)^{-1}$ for $\alpha/D \to 1 + 0$). However, the variables $T_c^* \equiv T_c/N^*$ and $P_c^* \equiv P_c/N^*$ with $N^* \equiv [N^{1-\alpha/D} -1]/[1-\alpha/D]$ remain {\em finite for all} $\alpha$. Thus, the extensive and nonextensive regions become unified, as recently conjectured. These results should be useful for discussing gravitation and some special fluidsComment: 7 pages and 6 PS-figures, RevTeX, to appear in Physics Letters