Form of the exact partition function for the generalized Ising Model

TL;DR

This paper derives an exact, closed-form expression for the partition function of the generalized Ising model, linking it to cluster energy levels and their temperature-dependent degeneracies, crucial for understanding phase transitions.

Contribution

It introduces a new exact expression for the partition function per spin in the generalized Ising model, emphasizing the role of temperature-dependent degeneracies.

Findings

Partition function expressed in terms of cluster energy levels

Degeneracies are functions of temperature

Common form of partition functions for Ising-like systems

Abstract

The problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.