Bargaining with entropy and energy

TL;DR

This paper formalizes the interplay of energy and entropy in statistical mechanics using cooperative game theory axioms, predicting thermalization and extending maximum entropy principles to uncertain constraints.

Contribution

It introduces a novel axiomatic framework based on Nash bargaining to model non-equilibrium thermodynamics and generalizes maximum entropy inference under uncertain constraints.

Findings

Predicts thermalization to negative temperatures from active initial states.

Demonstrates that symmetry between players leads to a unique positive-temperature equilibrium.

Provides a solution to the open problem of maximum entropy inference with uncertain constraints.

Abstract

Statistical mechanics is based on interplay between energy minimization and entropy maximization. Here we formalize this interplay via axioms of cooperative game theory (Nash bargaining) and apply it out of equilibrium. These axioms capture basic notions related to joint maximization of entropy and minus energy, formally represented by utilities of two different players. We predict thermalization of a non-equilibrium 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. Whenever the initial non-equilibrium state is active, this mechanism allows thermalization to negative temperatures. Demanding a symmetry between players fixes the final state to a specific positive-temperature (equilibrium) state. The approach solves an important open problem in the maximum entropy inference principle, {\it viz.} generalizes it to the case when the constraint is not known precisely.