Crossover from extensive to nonextensive behavior driven by long-range interactions

We determine, through numerical energy optimization, the fundamental-state energy E associated with an extended model for ferrofluids (colloidal suspension of magnetic particles assumed to form rings of N dipoles). The extension consists in generalizing, into 1/${\mathit{r}}^{\mathrm{\alpha }}$ (0≤α), a standard 1/${\mathit{r}}^6$ contribution to the attrative potential. We show that E∝N, N lnN, or ${\mathit{N}}^{2\mathrm{-}\mathrm{\alpha }}$, respectively, for α>1, α=1, or 0≤αd=2 XY ferromagnet is exhibited.