Energy and entropy: Path from game theory to statistical mechanics

Abstract

Statistical mechanics is based on the interplay between energy and entropy. Here we formalize this interplay via axiomatic bargaining theory (a branch of cooperative game theory), where entropy and negative energy are represented by utilities of two different players. Game-theoretic axioms provide a solution to the thermalization problem, which is complementary to existing physical approaches. We predict thermalization of a nonequilibrium statistical system employing the axiom of affine covariance, related to the freedom of changing initial points and dimensions for entropy and energy, together with the contraction invariance of the entropy-energy diagram. Thermalization to negative temperatures is allowed for active initial states. Demanding a symmetry between players determines the final state to be the Nash solution (well known in game theory), whose derivation is improved as a by-product of our analysis. The approach helps to retrodict nonequilibrium predecessors of a given equilibrium state.