{"id":"hexagon/2610.00557","pdf":"https://hexagonmath.org/pdf/2610.00557v1","url":"https://hexagonmath.org/abs/2610.00557v1","tags":["msc:82B03","Gibbs distribution","coarse graining","probability bounds","statistical mechanics","thermodynamic integrability"],"score":0,"title":"Gibbs Approximation from Coarse-Graining of Finite Systems","domain":"math-ph","source":"hexagon","authors":["Zhenyu Sun","Chen Chen"],"reviews":0,"subject":"math-ph","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}