A Novel Mixing Index Based on Directional Branch Length Similarity Entropy on Delaunay Networks

Quantitative evaluation of particle mixing is a fundamental task in science and industry, but existing indices often fail under spatial heterogeneity and irregular boundaries. This study introduces the Directional Branch Length Similarity entropy-based Mixing Index (DMI), which represents particles as nodes in a Delaunay triangulation network and evaluates mixing through directional branch length similarity entropy. To verify its performance, DMI was compared with five conventional indices across three scenarios: Gaussian-distributed dual-cluster mixing, banded dual-cluster mixing, and diffusion-driven mixing. Results show that DMI achieves higher sensitivity to local variations in small particle systems and maintains both accuracy and computational efficiency in large-scale systems through representative node sampling. Furthermore, DMI demonstrated strong discriminative power in quantifying heterogeneous spatial patterns, suggesting its applicability beyond classical particle mixing to broader complex systems.