Resolving the Debate Between Boltzmann and Gibbs Entropy: Relative Energy Window Eliminates Thermodynamic Inconsistencies and Allows Negative Absolute Temperatures

Abstract: Small systems consisting of a few particles are increasingly technologically relevant. In such systems, an intense debate in microcanonical statistical mechanics has been about the correctness of Boltzmann's surface entropy versus Gibbs' volume entropy. Both entropies have shortcomings -- while Boltzmann entropy predicts unphysical negative/infinite absolute temperatures for small systems with unbounded energy, Gibbs entropy entirely disallows negative absolute temperatures, in disagreement with experiments. We consider a relative energy window, motivated by the Heisenberg energy-time uncertainty principle and eigenstate thermalization in quantum mechanics. The proposed entropy ensures positive, finite temperatures for systems without a maximum limit on their energy and allows negative absolute temperatures in bounded energy systems, e.g., with population inversion. It also closely matches canonical ensemble predictions for prototypical systems, thus correctly describing the zero-point energy of an isolated quantum harmonic oscillator. Overall, we enable accurate thermodynamic models for isolated systems with few degrees of freedom.