Quantifying Degeneracy in Two-Point Statistics for Small Two-Phase Composite Structures

Volume fraction, or one-point statistics, is commonly used to homogenize composites. However, it contains no geometric information regarding the spatial distribution of the phases. The spatial distribution can be characterized using higher-order statistics. Two-point statistics (f₂) quantify average relative phase positions, and the geometric features encoded in f₂ influence material properties. However, just as a single volume fraction can describe multiple unique microstructures, some f₂ map to multiple distinct microstructures. The existence of multiple microstructures possessing the same f₂ is termed ‘degeneracy’ and is problematic for microstructure-sensitive design because unique microstructures may map to the same f₂ yet exhibit different properties. This study quantifies how pervasive degeneracy is in f₂ through exhaustive enumeration of all 2³⁶ ≈ 6.9¹⁰ possible 6 × 6 binary microstructures, and tests other metrics as ways to uniquely characterize microstructures with degenerate f₂. We determined that using nondirectional f₂ (i.e., orientation-averaged f₂) substantially increases degeneracy, nearly doubling the probability that a randomly selected microstructure will share the same f₂ as some other symmetry-inequivalent microstructure. Notably, the fraction of nontrivially degenerate microstructures does not increase monotonically with system size—a counterintuitive finding that challenges prior theoretical predictions. Finally, for the small microstructures examined, we determined that three-point statistics will fully resolve the degeneracy at a computational cost that scales as n⁴ (where n is side length), while two-point cluster functions resolve the majority of degeneracies with substantially lower computational overhead.