Abstract

Newton’s third law establishes that pair interactions are reciprocal: for every action, there is an equal and opposite reaction. In some nonequilibrium multicomponent systems, however, interactions among different elements seem to evade this law. This occurs for instance when mesoscale forces are mediated by a nonequilibrium environment and is common in the living world, from molecular interactions between proteins to antagonistic interactions between predators and their prey and social forces. This nonreciprocity breaks time-reversal symmetry, resulting in the emergence of traveling and oscillating patterns.   After reviewing familiar theoretical approaches to phase separation of two immiscible fluids, such as oil and water, I will introduce a minimal model for the emergence of both spatial and temporal order observed in a broad class of nonequilibrium settings, from chemical reactions in living cells to active cytoskeletal gels. These systems can be described in terms of two conserved concentration fields that ``chase’’ each through antagonistic cross-diffusivities. The resulting nonreciprocal dynamics yields a rich variety of traveling and oscillating patterns, with many features that can be analyzed analytically. This model provides a generic description for the transition from stationary to self-organized dynamical patterns and identifies a new class of pattern formation.