Effective Complexity Reduction of the Landau-de Gennes Elastic Energy: A Quantitative Framework and Numerical Validation

Abstract

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.