Nonextensive Statistical Mechanics

60 3 Generalizing What We LearntThe correctness of the present generalized thermodynamical scaling has alreadybeen specifically checked in many physical systems, such as a ferrofluid-like model[869], Lennard-Jones-like fluids [870], magnetic systems [174, 175, 177, 871, 872],anomalous diffusion [873], percolation [878, 879]. It has been also argued analytically[807].In addition to this, if a phase transition occurs in the system at a temperature T c ,it is expected to happen for a finite value of T c /Ñ. This implies that (i) in the limitα/d → 1 + 0, T c ∝ 1/(α/d − 1), thus recovering a result known since long (forinstance for the n-vector ferromagnet, including the Ising one); (ii) for 0 ≤ α/d 1). It is indeed so in the present case. Forexample, for N >> 1wehaveU(λ)U()= λ (λN)1−α/d − 1N 1−α/d − 1⎧⎪⎨ λ if α/d ≥ 1;∼⎪⎩λ 2−α/d if 0 0, 0 ≤ α/d < 1). The critical temperature is given byT c = μ JN 1−α/d /[(1 − α/d) k B ], where the pure number μ ≃ 1. This is the thermodynamicallycorrect result. What is instead customary to do in the literature is to (unphysically) replace J byJ/N 1−α/d , thus obtaining T c = μ J/[(1 − α/d) k B ], which remains finite for N →∞.