Thermodynamic stability and critical points in multicomponent mixtures with structured interactions

It is increasingly recognized that the phase behavior of mixtures with many components plays an important role in biology. But while the thermodynamics of mixtures with random interactions is well understood, functional mixtures often contain a large number of components that interact through a smaller number of features, leading to a structured interaction matrix. Here we consider such solutions with non-random interactions, characterized in terms of a pairwise interaction matrix of variable rank. We derive mean-field conditions for thermodynamic stability and critical behavior that only depend on the distribution of the components in the lower-dimensional space of features, thus strongly reducing the complexity of the problem. This representation in feature space also suggests a principled way for coarse-graining multicomponent mixtures as binary mixtures while preserving the system’s location with respect to the spinodal and critical manifold. More generally, the framework we develop offers an instructive perspective on multicomponent mixtures and might help to elucidate principles of intracellular phase behavior.