Observable Statistical Mechanics - Progress on long-range interacting systems and in the presence of non-commuting charges

Observable Statistical Mechanics (OSM) reframes equilibration as a property of measurements rather than full quantum states. The central tenet is that stationary observable statistics can be predicted by maximizing Shannon entropy of the observable, subject to constraints imposed by conserved quantities. This measurement-centric approach yields accurate predictions for stationary distributions without requiring diagonalization of the Hamiltonian, and applies beyond weak-coupling regimes and outside the thermodynamic limit. After briefly reviewing the OSM framework and its numerical validation in integrable and chaotic spin models, I will discuss three recent advances. First, I will present progress on long-range interacting systems, which have historically challenged statistical mechanics and thermalization. Second, for systems with non-commuting charges, I will show how OSM generalizes using weak values and Kirkwood-Dirac quasiprobabilities to characterize equilibrium distributions. In SU(2)-symmetric spin chains, this approach matches the non-Abelian thermal state predictions for one-body observables while outperforming them for two-body observables, without assuming weak coupling or locality. Third, I will outline a classical version of OSM that extends the observable-wise maximum-entropy construction to classical phase space. Overall, OSM provides a unified approach to predicting stationary behavior across quantum and classical systems, accommodating both commuting and non-commuting charges while focusing on the information that measurements actually provide.