Game Theory and Topological Phase Transition

Abstract

Phase transition is a war game. It widely exists in different kinds of complex system beyond physics. Where there is revolution, there is phase transition. The renormalization group transformation, which was proved to be a powerful tool to study the critical phenomena, is actually a game process. The phase boundary between the old phase and new phase is the outcome of many rounds of negotiation between the old force and new force. The order of phase transition is determined by the cutoff of renormalization group transformation. This definition unified Ehrenfest's definition of phase transition in thermodynamic physics. If the strategy manifold has nontrivial topology, the topological relation would put a constrain on the surviving strategies, the transition occurred under this constrain may be called a topological one. If the strategy manifold is open and noncompact, phase transition is simply a game process, there is no table for topology. An universal phase coexistence equation is found, it sits at the Nash equilibrium point. Inspired by the fractal space structure demonstrated by renormalization group theory, a conjecture is proposed that the universal scaling law of a general phase transition in a complex system comes from the coexistence equation around Nash equilibrium point. Game theory also provide us new understanding to pairing mechanism and entanglement in many body physics.