Liquid Hopfield model: retrieval and localization in multicomponent liquid mixtures

TL;DR

This paper introduces the liquid Hopfield model, an analytically tractable framework for understanding how multicomponent liquid mixtures can retrieve and localize specific mesoscopic structures, revealing a trade-off governed by non-linear interactions.

Contribution

It presents a novel liquid Hopfield model that explains the physical mechanisms enabling retrieval and localization in complex liquid mixtures, highlighting the role of non-linear repulsive interactions.

Findings

Non-linear repulsive interactions are essential for structure retrieval.

Liquid mixtures tend to localize into phases with fewer components at low temperatures.

A trade-off exists between retrieval capability and localization phenomena.

Abstract

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.