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The physics of active biological matter, such as bacterial colonies and birdflocks, exhibiting interesting self-organizing dynamical behavior has gainedconsiderable importance in recent years. Recent theoretical advances usetechniques from hydrodynamics, kinetic theory, and non-equilibrium statisticalphysics. However, for biological agents, these don't seem to recognizeexplicitly their critical feature, namely, the role of survival-driven purposeand the attendant pursuit of maximum utility. Here, we propose a novelgame-theoretic framework and show a surprising result that the bird-likeagents, garuds, self-organize dynamically into flocks to approach a stablearbitrage equilibrium of equal effective utilities. While it has beenwell-known for three centuries that there are constants of motion for passivematter, it comes as a surprise to discover that the dynamics of active matterpopulations could also have an invariant. This is essentially the invisiblehand mechanism of Adam Smith's in a biological context. What we demonstrate isfor ideal systems, similar to the ideal gas or Ising model in thermodynamics.The next steps would involve examining and learning how real swarms behavecompared to their ideal versions. Our theory is not limited to just birdsflocking but can be adapted for the self-organizing dynamics of other activematter systems.

Arbitrage Equilibrium, Invariance, and the Emergence of Spontaneous Order in the Dynamics of Birds Flocking