Multifractal complexity of liquid-crystal textures and its role in phase classification

One of the foremost methods to quantify the complexity of nonlinear systems developed in the last few years is multifractal formalism. In this study, we apply multifractal analysis for the first time to characterize the complex structures of liquid-crystal textures across different phases. Although the linear correlations of the textures increase as the spatial ordering of the molecules strengthens in the liquid-crystalline phase, the nonlinear characteristics associated with multifractality and structural complexity change markedly across different mesophases. In particular, a sharp decrease in multifractal measures is observed between the SmI and SmC phases, reflecting the pronounced reorganization of the texture accompanying the independently established SmI–SmC phase transition. Specifically, these characteristics undergo a transition from randomness to order and exhibit a sharp drop in multifractal measures leading to predominantly monofractal behavior during the -transition from the smectic I phase to the smectic C phase. These results are consistent with the entropy characteristics independently estimated for each transition. We point out that this behavior is related to the ordering of domains in the liquid crystal, where high molecular mobility at elevated temperatures likely leads to extensive fluctuations in domain shape, orientation, and inter-domain correlations, decreasing with the smaller temperatures and resulting in rich multifractal organization at the edge between randomness and highly ordered structure. We point out the importance of the machine learning model’s nonlinear properties for distinguishing between different phases. We demonstrate that XGBoost performs well on data with inherently nonlinear characteristics, whereas kNN requires additional nonlinear features to achieve acceptable accuracy.