# A solvable problem in statistical mechanics: the dipole-type Hamiltonian mean field model

## TL;DR

This paper analytically solves a mean field dipole interaction model in both canonical and microcanonical ensembles, deriving thermodynamic properties and demonstrating the equivalence of the schemes in equilibrium.

## Contribution

It provides an exact analytical solution for a dipole-type Hamiltonian mean field model, including free energy and entropy calculations, showing ensemble equivalence.

## Key findings

- Exact solutions for free energy and entropy.
- Canonical and microcanonical schemes coincide at equilibrium.
- Thermodynamic properties are graphically represented.

## Abstract

The present study regards the zeroth order mean field approximation of a dipole-type interaction model, which is analytically solved in the canonical and microcanonical ensembles. After writing the canonical partition function, the free and internal energies, magnetization and the specific heat are derived and graphically represented. A crucial derivation is the calculation of the free energy, which is variationally evaluated, and it is shown that the exact result coincides with the approximate trend when $N$ tends to infinity. In the microcanonical ensemble, the entropy as other thermodynamic properties are calculated. We notice that both schemes coincide in equilibrium.