math-ph — Mathematical Physics
Gibbs Approximation from Coarse-Graining of Finite Systems
Thermodynamic integrability in finite systems admits concave probability-response potentials beyond the canonical ensemble. We investigate how replacing an unresolved block of energy levels by its effective free energy constrains this class and determines the Gibbs distribution, both exactly and quantitatively. For a response family defined across all finite spectra, exact single-block replacement implies conditional independence within each block and selects the Gibbs family with its canonical free-energy normalization. On a bounded energy window, conditional log-odds discrepancies determine a common reference inverse-temperature scale and yield a dimension-independent total-variation bound. A discrete calibration protocol incorporates grid rounding, interpolation under stated regularity bounds, and finite-sample concentration guarantees. We bound the conditional log-odds discrepancies in terms of coarse-grained block-probability residuals through finite block suppression, a bounded outer-control displacement, and a local internal curvature bound. The estimate separates auxiliary system size, measurement precision, and probability conditioning. Smooth local perturbations show that finitely many observations do not uniquely determine the global response. Applications to finite-capacity reservoirs and fixed-coupling mean-force models distinguish work-form closure from temperature-energy scaling. For two subsystems sharing small, bounded, symmetric log-temperature fluctuations, the leading mutual information is half the squared log-temperature variance times the product of their reference canonical heat capacities.