  The motility-induced phase separation (MIPS) phenomenon in active matter has
been of great interest for the past decade or so. A central conceptual puzzle
is that this behavior, which is generally characterized as a nonequilibrium
phenomenon, can yet be explained using simple equilibrium models of
thermodynamics. Here, we address this problem using a new theory, statistical
teleodynamics, which is a conceptual synthesis of game theory and statistical
mechanics. In this framework, active agents compete in their pursuit of maximum
effective utility, and this self-organizing dynamics results in an arbitrage
equilibrium in which all agents have the same effective utility. We show that
MIPS is an example of arbitrage equilibrium and that it is mathematically
equivalent to other phase-separation phenomena in entirely different domains,
such as sociology and economics. As examples, we present the behavior of Janus
particles in a potential trap and the effect of chemotaxis on MIPS.
