Resolving the\nDebate between Boltzmann and Gibbs Entropy:\nRelative Energy Window Eliminates Thermodynamic Inconsistencies and\nAllows Negative Absolute Temperatures

Small systems consisting of a few particles are increasingly\ntechnologically\nrelevant. In such systems, an intense debate in microcanonical statistical\nmechanics has been about the correctness of Boltzmann’s surface\nentropy versus Gibbs’ volume entropy. Both entropies have shortcomingswhile\nBoltzmann entropy predicts unphysical negative/infinite absolute temperatures\nfor small systems with an unbounded energy spectrum, Gibbs entropy\nentirely disallows negative absolute temperatures, in disagreement\nwith experiments. We consider a relative energy window, motivated\nby the Heisenberg energy-time uncertainty principle and eigenstate\nthermalization in quantum mechanics. The proposed entropy ensures\npositive, finite temperatures for systems without a maximum limit\non their energy and allows negative absolute temperatures in bounded\nenergy spectrum systems, e.g., with population inversion. It also\nclosely matches canonical ensemble predictions for prototypical systems,\nthus correctly describing the zero-point energy of an isolated quantum\nharmonic oscillator. Overall, we enable accurate thermodynamic models\nfor isolated systems with few degrees of freedom.