A Geometric Representation of the Second Law in Optimal Entropic Dimensionality

抄録

Equilibrium thermodynamics is often organized by minimizing potentials built from competing contributions (e.g. Helmholtz and Gibbs free energies). We introduce a compact protocol-level representation based on Optimal Entropic Dimensionality (OED), where an effective capacity Neff and an effective free payload Hfree (optionally separated from structured components) are treated as declared proxies of a coarse-graining. OED keeps the geometry class explicit through a single class parameter p>0. As a canonical reference point, the class p=2 reproduces the familiar Gaussian/Euclidean backbone via the OED–Gaussian free-energy isomorphism. The central move is a dimensionless filling coordinate ψ ≡ p Hfree / Neff that packages the balance between capacity and free payload into one number. In ψ-space, within the declared protocol, the induced one-dimensional ψ-potential Cp(ψ; Hfree) is strictly convex on ψ>0 with a unique minimizer at ψ=1, so equilibrium becomes a normalized geometric statement (“optimal filling”) once the bookkeeping protocol (estimators for Neff and Hfree) is fixed. Second-law-like directionality is treated as additional structure rather than a consequence of geometry alone. Without postulating a global state-space metric, we use the closed-form excess ΔCp(ψ) = (1/p)(ψ − 1 − ln ψ) as a scale-free departure from ψ=1. Under an illustrative coarse-grained closure in ψ-space, ΔCp decreases monotonically, separating equilibrium geometry from arrow-of-time statements while leaving any domain-canonical metric/length identification as a calibration-dependent extension.