Effective theories for liquid crystals and the Maier-Saupe phase transition

We discuss effective theories for thermotropic nematic liquid crystals. In the first part of this article, we rigorously carry out two physically different scaling limits as the number of particles \(N\to\infty\), which we call the mean-field and the Gross-Pitaevskii limit. Each of them yields an effective `one-body' free energy functional. In the second part, we study the associated Euler-Lagrange equation, with a focus of phase transitions. We identify two critical points as the temperature is lowered, corresponding first to the appearance of an anisotropic distribution, and secondly to the loss of stability of the isotropic solution. As the temperature goes to zero we also prove, in the concrete case of the Maier-Saupe potential, that the system converges to perfect nematic order.