Mean-field limit and phase transitions for nematic liquid crystals in the continuum

TL;DR

This paper rigorously derives the mean-field limit for nematic liquid crystals, analyzes phase transitions in homogeneous systems, and demonstrates the emergence of nematic order at low temperatures, especially for the Maier-Saupe potential.

Contribution

It provides a rigorous derivation of the mean-field limit and analyzes phase transitions in nematic liquid crystals, including the zero-temperature limit for the Maier-Saupe potential.

Findings

High-temperature isotropic phase

Phase transition via bifurcation at lower temperatures

Convergence to perfect nematic order at zero temperature

Abstract

We discuss thermotropic nematic liquid crystals in the mean-field regime. In the first part of this article, we rigorously carry out the mean-field limit of a system of $N$ rod-like particles as $N\to\infty$, which yields an effective `one-body' free energy functional. In the second part, we focus on spatially homogeneous systems, for which we study the associated Euler-Lagrange equation, with a focus on phase transitions for general axisymmetric potentials. We prove that the system is isotropic at high temperature, while anisotropic distributions appear through a transcritical bifurcation as the temperature is lowered. Finally, 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.