Crossover from extensive to nonextensive behavior driven by long-range d=1bond percolation

We present a Monte Carlo study of a linear chain (d = 1) with long-range bo nds whose occupancy probabilities are given by pij = p/r(ij)(alpha) (0 less than or equal to p less than or equal to 1; alpha greater than or equal to 0) where r(ij)=1,2,... is the distance between sites. The alpha --> infini ty (alpha = 0) corresponds to the first-neighbor ("mean field") particular case. We exhibit that the order parameter P-infinity equals unity For All p > 0 for 0 less than or equal to alpha less than or equal to 1, presents a familiar behavior (i.e., 0 for p less than or equal to p(c)(alpha) and fini te otherwise) for 1 < alpha < 2, and vanishes For All p < 1 for alpha > 2. Our results confirm recent conjecture, namely that the nonextensive region (0 less than or equal to alpha less than or equal to 1) can be meaningfully unfolded as well as unified with the extensive region (ce > 1), by exhibit ing P-infinity as a function of p* where (1 - p*)=(1 - p)(N*) (N* equivalen t to (N1-alpha/d -1)/(1-a/d), N being the number of sites of the chain). A corollary of this conjecture, now numerically verified, is that p(c) propor tional to (alpha - 1) in the alpha --> 1 + 0 limit. (C) 1999 Elsevier Scien ce B.V. All rights reserved.