Certain General Order-Disorder Models in the Limit of Long-Range Interactions

We investigate in the limit as the range of the interparticle interactions becomes indefinitely great, but is still small compared to the size of the system, the behavior of a large class of order-disorder models. This class includes, for example, the Ising model, the spherical model, and the Gaussian model. We show that when certain general conditions hold and the interparticle interaction is chosen to be the same for all models, but otherwise arbitrary, the energy per particle above the critical temperature has the same limiting value through terms of order 1R, where R is a measure of the number of spins in the range of the interaction. We further show why the behavior above the critical point in this limit does not necessarily provide information about the behavior below the critical point. Some examples are worked out which illustrate the above results.