Phase transition in the spherical model using a direct solution of the partition function

The spherical model was introduced and solved by Berlin and Kac in 1952. However, the solution in their paper makes use of the method of steepest descent and so gives the solution in terms of a saddle point whose location will in general be difficult or impossible to ascertain. We present a new analysis of the general spherical model, which approaches the contour integral directly. Our solution thus has broader applicability. We find evidence for a phase transition and predict the form of the temperature-energy relation on both sides of the transition. In the last part of the paper, we show a numerical experiment which samples from the canonical ensemble of an instance of the spherical model based on the Hasegawa Mima equations. We compare the results to the analytical predictions and find a high degree of agreement.