The Geometric Foundations of Microcanonical Thermodynamics: Entropy Flow Equation and Thermodynamic Equivalence

Abstract: We develop a geometric foundation of microcanonical thermodynamics in which entropy and its derivatives are determined from the geometry of phase space, rather than being introduced through an a priori ensemble postulate. Once the minimal structure needed to measure constant -- energy manifolds is made explicit, the microcanonical measure emerges as the natural hypersurface measure on each energy shell. Thermodynamics becomes the study of how these shells deform with energy: the entropy is the logarithm of a geometric area, and its derivatives satisfy a deterministic hierarchy of entropy flow equations driven by microcanonical averages of curvature invariants (built from the shape/Weingarten operator and related geometric data). Within this framework, phase transitions correspond to qualitative reorganizations of the geometry of energy manifolds, leaving systematic signatures in the derivatives of the entropy. Two general structural consequences follow. First, we reveal a thermodynamic covariance: the reconstructed thermodynamics is invariant under arbitrary descriptive choices such as reparametrizations and equivalent representations of the same conserved dynamics. Second, a geometric microcanonical equivalence is found: microscopic realizations that share the same geometric content of their energy manifolds (in the sense of entering the curvature sources of the flow) necessarily yield the same microcanonical thermodynamics. We demonstrate the full practical power of the formalism by reconstructing microcanonical response and identifying criticality across paradigmatic systems, from exactly solvable mean-field models to genuinely nontrivial short-range lattice field theories and the 1D long-range XY model with $1/r^α$ interactions.