Landau-de Gennes弹性能量的有效复杂度降低：一个定量框架和数值验证
Effective Complexity Reduction of the Landau-de Gennes Elastic Energy: A Quantitative Framework and Numerical Validation
AI总结：
该研究重新审视向列型液晶Landau-de Gennes模型弹性能量公式，基于广义最优缩放程序确定参数范围，通过数值测试验证解析缩放预测，为简化弹性模型有效性提供定量标准，建立最优缩放与数值观测的联系。
AI中文摘要：
我们重新审视向列型液晶的Landau-de Gennes模型的弹性能量公式，重点关注多常数弹性能量的定量降低。基于Rusconi等人在2025年引入的广义最优缩放程序（GOS），我们确定了明确的参数范围，其中三常数模型$(L_1,L_2,L_3)$可简化为$(L_1,L_2,0)$，以及两常数模型$(L_1,L_2,0)$如何简化为常用的单常数配置$(L_1,0,0)$。使用openQmin模拟框架对解析缩放预测进行了数值测试，证实了在GOS给出的$L_3$或$L_2$的临界阈值以下，与简化模型的偏差与缩放理论预测的量级相同。这些结果为简化弹性模型的有效性提供了定量标准，并在Landau-de Gennes框架内建立了最优缩放论点与数值观测之间的直接联系。
英文摘要：
We revisit the elastic energy formulation of the Landau-de Gennes model for nematic liquid crystals, focusing on quantitative reductions of the multi-constant elastic energy. Building on the generalized optimal scaling procedure (GOS) introduced by Rusconi et al. in 2025, we identify explicit parameter regimes in which the three-constant model $(L_1,L_2,L_3)$ can be reduced to $(L_1,L_2,0)$ and how the two-constant model $(L_1,L_2,0)$ can be reduced to the commonly used one-constant configuration $(L_1,0,0)$. The analytical scaling predictions are tested numerically using the openQmin simulation framework, confirming that below a critical threshold for $L_3$ or $L_2$, given by GOS, the deviation from the reduced model remains of the same order of magnitude as predicted by the scaling theory. These results provide a quantitative criterion for the validity of reduced elastic models and establish a direct connection between optimal scaling arguments and numerical observations within the Landau-de Gennes framework.
