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Bachmann, Sven and Genoud, Francois (2017): Mean-Field Limit and Phase Transitions for Nematic Liquid Crystals in the Continuum. In: Journal of Statistical Physics, Vol. 168, No. 4: pp. 746-771

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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 -> infinity, 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.

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