Unraveling the collinearity in short-range order parameters for lattice configurations arising from topological constraints

In multicomponent lattice problems, for example, in alloys and at crystalline surfaces and interfaces, atomic arrangements exhibit spatial correlations that dictate the kinetic and thermodynamic phase behavior. These correlations emerge from interparticle interactions and are frequently reported in terms of the short-range order (SRO) parameter. Expressed usually in terms of pair distributions and other cluster probabilities, the SRO parameter gives the likelihood of finding atoms/molecules of a particular type in the vicinity of other atoms. This study focuses on fundamental constraints involving the SRO parameters that are imposed by the underlying lattice topology. Using a data-driven approach, we uncover the interrelationships between different SRO parameters (e.g., pairs, triplets, and quadruplets) on a lattice. The main finding is that while some SRO parameters are independent, the remaining are collinear, i.e., the latter are dictated by the independent ones through linear relationships. A kinetic and thermodynamic modeling framework based on these constraints is introduced.