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.