Statistical physics game: convergence to invariant measures

Demonstrate, under precise dynamical conditions (e.g., ergodicity, chaos, topological mixing) and in the limit where the observation time interval goes to infinity, that in the proposed statistical-physics game the equilibrium sequence-selection strategy of the initial-state chooser converges to the invariant measure, and that the observer’s equilibrium betting strategy converges to ensemble averages with respect to that invariant measure.

Background

The authors sketch a game-theoretic foundation for statistical physics: one player selects initial conditions of a dynamical system (subject to constraints), while the other chooses when (or how often) to observe and places bets on coarse-grained states. They conjecture that, for suitably chaotic or ergodic systems and long observation times, equilibrium strategies match standard statistical-physics objects.

Formally establishing convergence to the invariant measure for the selector and the emergence of ensemble averages for the bettor would connect the game-theoretic framework rigorously to equilibrium statistical mechanics.