Concept of the Long-Range Order in Percolation Problems

Abstract

It is proved that the dilute alloy ferromagnet (Ising model) at the absolute zero and the site percolation problem are mathematically identical by introducing the uniqueness theorem (there is only one infinitely extended connected cluster of one atomic species) and the symmetry theorem (the number of clusters of a certain finite shape and size with a plus spin is the same as that with a mimus spin). It is then shown that the long-range order parameter in the site problem is the density of atoms belonging to the infinitely extending cluster. It is suggested that the site percolation problem when atoms interact and the atomic distribution is not random can be treated using the cluster variation method by placing a + or a − tag on the atomic species whose connectively is being sought, with the restriction that no (+ −) connection occurs and by maximizing the entropy (not minimizing the free energy) of the system. These signs are mathematical devices and have no physical meaning. Some results of higher order approximations of the bond and site percolation problems are discussed.