Expressivity of first-order phase transitions

First-order phase transitions act like on/off switches: small changes move a system across a phase boundary that functions as a classifier's decision surface. This can enable compact sensing, computation, and response in physical and cellular systems. We ask how expressive such boundaries can be and what dynamics control that expressivity. For example, is a physical system with N components at temperature T as powerful as a single layer neural network or a deep one? Using a student–teacher protocol, we generate labeled data from increasingly complex teacher models (polynomials and deeper neural networks) and test whether two-phase physical systems can reproduce them. Using a Potts model, we find that equilibrium phase boundaries are restricted to behave essentially like linear classifiers. However, allowing kinetics and competition for shared components greatly increases expressivity. With rare-event sampling to map nucleation pathways, we link decision time in non-equilibrium regimes to boundary complexity. A clear tradeoff emerges: near coexistence (low supersaturation) decisions are slower but more expressive, while higher supersaturation yields faster, simpler boundaries. These results provide design rules for physical computing with collective phenomena: tune supersaturation and exploit competition to achieve compact, robust information processing.