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

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.

arXiv commentSubstantial revision due to a mistake in the proof of the 'Gross-Pitaevskii' limit. The paper now focuses on the mean-field regime. Title and abstract modified. Remark 2 comments on the restriction to homogeneous systems made throughout the phase transition analysis. This version will appear in J. Stat. Phys