Liquid Hopfield model: Retrieval and localization in multicomponent liquid mixtures

Biological mixtures, such as the cellular cytoplasm, are composed of a large number of different components. From this heterogeneity, ordered mesoscopic structures emerge, such as liquid phases with controlled composition. The competition of these structures for the same components raises several questions: what types of interactions allow the retrieval of multiple ordered mesoscopic structures, and what are the physical limitations for the retrieval of said structures. In this work, we develop an analytically tractable model for multicomponent liquids capable of retrieving states with target compositions. We name this model the liquid Hopfield model in reference to corresponding work in the theory of associative neural networks. In this model, we show that nonlinear repulsive interactions are a general requirement for retrieval of target structures. We demonstrate that this is because liquid mixtures at low temperatures tend to transition to phases with few components, a phenomenon that we term localization. Taken together, our results reveal a trade-off between retrieval and localization phenomena in liquid mixtures, and pave the way for other connections between the phenomenologies of neural computation and liquid mixtures.