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

TL;DR

This work develops a geometric foundation for microcanonical thermodynamics by introducing a phase-space metric that induces a natural measure on energy shells, making entropy S(E) the log-area of the energy leaf Σ_E. It demonstrates thermodynamic covariance under metric reparametrizations and defines a geometric microcanonical equivalence, whereby different metrics yielding the same energy-shell measure produce identical thermodynamics. The entropy satisfies deterministic, geometry-driven Entropy Flow Equations (EFEs) whose sources are curvature invariants of the energy leaves; phase transitions correspond to qualitative geometric reorganizations of Σ_E, detectable via these curvature terms. The framework is validated on paradigmatic models, including the 1D mean-field XY model, the 2D φ^4 lattice, and the 1D XY long-range chain, showing accurate reconstruction of entropy derivatives and robust identification of finite-size precursors to criticality, even in genuinely long-range regimes. Overall, geometry generates thermodynamics and, through the induced measure, naturally encompasses probabilistic notions, providing a unified, finite-N, ensemble-independent description of phase behavior with broad applicability to complex, constrained, and long-range systems.

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.

Figures (13)

• Figure 1: Pictorial representation of the orthogonality relation $\nabla_\eta H\perp\bm{X}_H$. The Hamiltonian flow represented by the red curve $\bm{x}_S(t)$ follows the direction of the Hamiltonian vector field $\bm{X}_H$ which lies in turn on the energy level set $\Sigma_E$ at each time.
• Figure 2: Energetic Clock. The parameter $\epsilon$ directly measures energy: $dE = d\varepsilon$. We have the freedom to fix the energy step, say, $dE=1$, so that the geometric step varies point by point: $d\ell = 1/\|\nabla H\|$. This is an energy flow since we fix the target hypersurface (through the energy step $dE=1$) and vary $d\ell$ point by point in order to map all points of $\Sigma_E$ onto $\Sigma_{E+dE}$.
• Figure 3: Phase-space flows on the energy foliation $\Lambda=\bigcup_E \Sigma_E$. (a.1) Energy flow$\Phi^{\rm diff}_{\bm\xi_\eta}$ generated by $\bm\xi_\eta=\nabla_{\!\eta} H/\|\nabla_{\!\eta} H\|_\eta^2$ that maps a energy hypersurfaces onto another. Once the initial condition $\bm x_0$ is fixed, it remains on the selected energy level set $\Sigma_E$, i.e. $H(\bm x_0)=E$ for all $t$. Then, on this level set, a Hamiltonian (symplectic) flow$\Phi^{\rm sym}_t$ is established: the trajectory $\bm x_S(t)=\Phi^{\rm sym}_t(\bm x_0)$ is an integral curve of the Hamiltonian vector field $\bm X_H$ and, $H(\bm x_S(t))=H(\bm x_0)$ The introduction of the phase-space metric $\eta$ turns energy conservation into the orthogonality condition $\eta(\bm X_H,\bm\xi_\eta)=0$. (a.2) Pictorial deformation of the energy hypersurfaces as $E$ varies (e.g., round $\to$ distorted $\to$ necked), illustrating the qualitative geometric change that is captured by the energy-derivatives of geometric observables. (a.3) Geometric (diffeomorphic) flow$\Phi^{\rm diff}_{\bm\xi_\eta}$, transversal to $\Phi^{\rm sym}_t$ and generated by $\bm\xi_\eta$, which maps one leaf into a neighboring one, $\Sigma_E\mapsto\Sigma_{E'}$, providing the controlled displacement along the energy direction in $(\Lambda,\eta)$.
• Figure 4: Pictorial meaning of the Weingarten (shape) operator $W_{\bm\xi}$ and its principal curvatures for a two-dimensional hypersurface $\Sigma_E$ embedded in an ambient manifold. (a) At a point $\bm x\in\Sigma_E$, with unit normal $\bm\xi$ and tangent plane $T_{\bm x}\Sigma_E$, the principal directions $\bm e_1,\bm e_2$ diagonalize $W_{\bm\xi}$ and define the principal curvatures $\kappa_{1,2}$ via the normal sections; the corresponding osculating radii are $1/\kappa_1$ and $1/\kappa_2$. (b) Two representative local geometries: an elliptic (convex) point (on the left) with $\kappa_1,\kappa_2>0$ and a hyperbolic (saddle) point (on the right) with opposite signs. The invariants $\text{Tr}[W_{\bm\xi}]=\kappa_1+\kappa_2$ (mean curvature) and $K_G=\kappa_1\kappa_2$ (Gaussian curvature) summarize the local curvature signature.
• Figure 5: Pictorial visual representation of the geometric change triggering the second-order phase transition in the 1D XY mean-field model. Schematic decomposition of the Hamiltonian in canonical normal coordinates into: (i) a flat mode $(p_1,q_1)$ with $\lambda_1=0$ (top boxes), (ii) a set of weakly mixed bulk modes $(p_k,q_k)$ (middle boxes), and (iii) two collective modes $(p_\pm,q_\pm)$ (bottom boxes). In the ordered phase (left) the bulk sector is elliptic with positive curvature $\lambda_k=J>0$, while the collective sector is absent and the reduced sections are compact. At criticality (center, $\epsilon_c\simeq 3J/4$) positive and negative curvature coexist: the bulk eigenvalues soften to $\lambda_k=\mathcal{O}(JM)$, whereas the collective eigenvalues become negative, $\lambda_\pm=-J/2+\mathcal{O}(JM^2)$, producing a "neck" (one--sheet hyperboloid) in the reduced sections. In the disordered phase (right) the bulk becomes flat ($N-2$ flat modes) and the nontrivial curvature is concentrated in the collective hyperbolic sector with $\lambda_\pm=-J/2$. The horizontal lines at $\pm\sqrt{2E}$ indicate the kinematic bounds from the energy constraint in the corresponding reduced coordinates.