Question

Consider Onsager's exact solution of two-dimensional Ising model for square lattices with nearest neighbour interaction energy ‘J ‘being equal in the horizontal and vertical directions. At the critical temperature, the partition function estimated by numerical integration (e.g using eqn 115 of Onsager’s original paper) is approximately 0.92969. In a few Journal articles, it is mentioned that the integrand appearing in the double integral of Onsager’s exact solution leads to a singularity of the free energy at the critical temperature. Can this be clarified?

Answer 1

The singularity in the free energy of Onsager's exact solution of the two-dimensional Ising model at the critical temperature is due to a property of the integrand. This singularity indicates a phase transition in the system.

Step-by-step explanation:

The singularity in the free energy mentioned in the journal articles is a result of a property of the integrand in Onsager's exact solution of the two-dimensional Ising model. This singularity occurs at the critical temperature, where the partition function is estimated to be approximately 0.92969. The singularity indicates a divergence in the free energy, which implies a phase transition.

To clarify further, let's consider the context of Onsager's model. The Ising model is a mathematical model that describes the behavior of magnetic spins on a lattice. Onsager developed an exact solution for this model in two dimensions.

The partition function is a fundamental concept in statistical mechanics, which is used to calculate thermodynamic properties of a system. In Onsager's exact solution, the partition function involves a double integral with an integrand that exhibits a singularity at the critical temperature. This singularity arises from the fact that the system undergoes a phase transition at the critical temperature, which is a point of divergence in thermodynamic quantities like the free energy.