🤖 AI Summary
This study addresses the limited applicability of the “typical state” concept in statistical mechanics to non-equilibrium, small-scale, and strongly correlated systems. To overcome this, we introduce a generalized entropy functional defined via a typicality measure—free from conventional ergodicity assumptions and thermodynamic (large-system) limits—thereby establishing a unified thermodynamic framework for non-equilibrium states, finite-size systems, and localized quantum many-body systems. Our methodology integrates stochastic process analysis, large-deviation theory, information geometry, and non-equilibrium fluctuation theorems. The resulting formalism yields state-selection criteria applicable to quantum many-body localization, active matter, and nanoscale heat engines. Numerical validation demonstrates that predictions based on our generalized entropy exhibit over 40% lower error compared to existing approaches, substantially extending both the domain of applicability and the universality of statistical mechanics.