范德华-卡茨极限下液晶向列一级相变
Nematic first order phase transition for liquid crystals in the van der Waals-Kac limit
摘要：
在本文中，我们重新审视和扩展Onsager的液晶理论的一些数学方面，近年来已经调查了不同的社区（统计力学，分析和概率）。我们介绍了一个模型的各向异性分子的三维取向相互作用通过Kac型相互作用。我们证明，在限制中的相互作用的范围被发送到无穷大后的热力学极限，自由能趋于下确界的一个有效的能量功能à la Onsager。然后，我们证明，如果球谐变换的角相互作用有一个负的最小值，这个有效的自由能泛函显示一个一阶相变作为系统的总密度的增加。
英文摘要：
In this paper we revisit and extend some mathematical aspects of Onsager's theory of liquid crystals that have been investigated in recent years by different communities (statistical mechanics, analysis and probability). We introduce a model of anisotropic molecules with three-dimensional orientations interacting via a Kac-type interaction. We prove that, in the limit in which the range of the interaction is sent to infinity after the thermodynamic limit, the free energy tends to the infimum of an effective energy functional à la Onsager. We then prove that, if the spherical harmonic transform of the angular interaction has a negative minimum, this effective free energy functional displays a first order phase transition as the total density of the system increases.
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