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 a 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.

中文翻译：

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范德华-卡克极限下液晶的向列一阶相变

在本文中，我们重新审视并扩展了 Onsager 液晶理论的一些数学方面，近年来不同社区（统计力学、分析和概率）对此进行了研究。我们介绍了一种通过 Kac 型相互作用相互作用的具有三维取向的各向异性分子模型。我们证明，在热力学极限之后相互作用的范围被发送到无穷大的极限内，自由能趋向于有效能量泛函 a la Onsager 的下限。然后我们证明，如果角相互作用的球谐变换具有负最小值，则该有效自由能泛函随着系统总密度的增加而显示出一阶相变。