Onsager模型的解景观识别非轴对称临界点
Solution landscape of the Onsager model identifies non-axisymmetric critical points
摘要：
我们研究了球上Onsager自由能模型的临界点，这些模型具有不同的势核，包括偶极势、Maier-Saupe势、偶极/Maier-Saupe势和Onsager势.采用均匀采样方法对Onsager模型进行离散化，并利用鞍型动力学结合向下/向上搜索算法构造Onsager模型的解景观.我们首先构造的解决方案的景观与偶极和Maier-Saupe潜力，其中所有的临界点是轴对称的。对于耦合偶极/Maier-Saupe势，解的景观显示了一个新的非轴对称的临界点，命名为网球，它存在于广泛的参数范围。我们进一步证明了各种非轴对称的临界点的Onsager模型与Onsager势，包括正方形，六边形，八面体，立方，四分之一，二十面体}，和十二面体状态。给出了各向同性态的一次和二次分岔图，揭示了临界点的出现。解景观提供了一种有效的方法来显示模型系统的全局结构以及临界点的分叉，它不仅可以支持以前的理论结果，而且可以提出新的基于数值结果的结构。
英文摘要：
We investigate critical points of the Onsager free-energy model on a sphere with different potential kernels, including the dipolar potential, the Maier-Saupe potential, the coupled dipolar/Maier-Saupe potential, and the Onsager potential. A uniform sampling method is implemented for the discretization of the Onsager model, and solution landscapes of the Onsager model are constructed using saddle dynamics coupled with downward/upward search algorithms. We first construct the solution landscapes with the dipolar and Maier-Saupe potentials, for which all critical points are axisymmetric. For the coupled dipolar/Maier-Saupe potential, the solution landscape shows a novel non-axisymmetric critical point, named tennis, which exists for a wide range of parameters. We further demonstrate various non-axisymmetric critical points in the Onsager model with the Onsager potential, including square, hexagon, octahedral, cubic, quarter, icosahedral}, and dodecahedral states. The bifurcation diagram is presented to show the primary and secondary bifurcations of the isotropic state and reveal the emergence of the critical points. The solution landscape provides an efficient approach to show the global structure of the model system as well as the bifurcations of critical points, which can not only support the previous theoretical conjectures but also propose new conjectures based on the numerical findings.
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