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

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### Abstract

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 of the fundamental statistical mechanical model of a system of $$N$$ rod-like particles as $$N\to\infty$$, 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 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.

### Most cited references22

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### The theory of liquid crystals

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### Bifurcation from simple eigenvalues

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### Author and article information

###### Journal
2015-08-20
2016-10-28
1508.05025