  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. These
structures compete with each other for the same components. This raises several
questions, such as 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 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. By solving
this model, we show that non-linear repulsive interactions are necessary 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 demonstrate
a trade-off between retrieval and localization phenomena in liquid mixtures.
