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.
Highlights
Entropy and energy are fundamental for statistical mechanics, because they define the concept of equilibrium [1,2,3,4,5,6,7]
The second method reflects the postulate of maximal work [3,4]: The maximal work extracted from a given nonequilibrium state during a cyclic process is achieved for a fixed entropy and leaves the system in an equilibrium state
This includes solving the thermalization problem: when and how a statistical system that starts its evolution from a sufficiently macroscopic nonequilibrium state relaxes to thermal equilibrium [4,5]
Summary
Entropy and energy are fundamental for statistical mechanics, because they define the concept of equilibrium [1,2,3,4,5,6,7]. Utilities of players Joint actions of players Feasible set of utility values Defection point Pareto set Statistical mechanics entropy and negative energy probabilities of states for the physical system entropy-energy diagram initial state maximum entropy curve for positive inverse temperatures β > 0 started with zero-sum games, where the interests of players are strictly opposite to each other. Such agents have no reason to cooperate, i.e., meaningful actions are noncooperative [8,9,10,11].
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