The interaction between a peregrine falcon and a dove is visibly non-reciprocal. What happens to the well-established framework of phase transitions in non-reciprocal systems far from equilibrium?

In this talk, I will answer this question by looking at three archetypal classes of self-organization out of equilibrium: synchronization, flocking, and pattern formation. Simple demonstrations with robots will be presented along with naturally occurring phenomena from various domains of science that share a common feature: reciprocity has no reason to exist. In all these cases, the emergence of unique time-dependent many-body phases can be captured by combining insights from non-Hermitian quantum mechanics and bifurcation theory. This approach lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.