Phase Transitions as Emergent Geometric Phenomena: A Deterministic Entropy Evolution Law

TL;DR

The paper demonstrates that thermodynamics and phase transitions can be derived from the intrinsic geometry of phase space without postulating ensembles; the microcanonical measure emerges as the area of energy shells, and entropy becomes a geometric quantity evolving deterministically via the Entropy Flow Equation $\partial_E^2 S_g(E)+[\partial_E S_g(E)]^2=\Upsilon^{(2)}_g(E)$ with curvature-based sources. By introducing a metric on phase space and an energy clock, it establishes thermodynamic covariance across an equivalence class of geometries and derives the unit-norm gauge in which the energy flow is generated by $\nabla_g H$. The framework is validated on a long-range 1D XY mean-field model and a 2D $\phi^4$ theory, where geometric curvature changes of energy manifolds quantitatively reproduce entropy and its derivatives, identifying the critical energies via a purely geometric mechanism. The results suggest a universal geometric foundation for PTs, applicable to finite systems, long-range interactions, and ensemble-inequivalent regimes, with potential implications for complex systems and spin models where traditional ensemble-based methods falter.

Abstract

We show that thermodynamics can be formulated naturally from the intrinsic geometry of phase space alone-without postulating an ensemble, which instead emerges from the geometric structure itself. Within this formulation, phase transitions are encoded in the geometry of constant-energy manifold: entropy and its derivatives follow from a deterministic equation whose source is built from curvature invariants. As energy increases, geometric transformations in energy-manifold structure drive thermodynamic responses and characterize criticality. We validate this framework through explicit analysis of paradigmatic systems-the 1D XY mean-field model and 2D $φ^4$ theory-showing that geometric transformations in energy-manifold structure characterize criticality quantitatively. The framework applies universally to long-range interacting systems and in ensemble-inequivalence regimes.

Figures (3)

• Figure 1: Comparison of entropy derivatives obtained as the solution of the entropy flow equation and through thermodynamic methods. Empty red circles are associated with the solutions (derivatives of entropy) of the Entropy Flow Equation (EFE). Black disks are the numerical estimations of entropy derivatives obtained from microcanonical simulations with Pearson-Halicioglu-Tiller method pearson1985laplace. The geometric function $\Upsilon^{(2)}_g$ represented by black crosses is computed through numerical simulation according with Eq. (S-27) in SM. In panels a.1-a.3, we report results for the 2D $\phi^4$-model with nearest interactions and coupling parameters $\lambda=3/5$, $\mu=\sqrt{2}$ and $J=1$ and size $N=256^2$. The vertical red dashed line indicates the critical energy detected by MIPA and in agreement with the literature. Panels b.1-b.3 are associated with the comparison of entropy derivatives in the 1D XY mean-field model with coupling parameter $J=1$ and size $N=40000$. Notice that the infinite-size critical energy $\epsilon^\infty_c=3J/4$ is closed to the finite-size critical energy $\epsilon_c^{\text{MIPA}}=0.74$ estimated by MIPA. See SM for more details.
• Figure S1: Comparison of entropy derivatives at different system sizes obtained as the solution of the entropy flow equation and through thermodynamic methods. The quantitative and qualitative agreement is excellent. As the system size increases, the peak in $\Upsilon^{(2)}_g$ and $\partial_\epsilon^2 S$ becomes sharper and sharper indicating that in the thermodynamic limit the peak converges to zero thus giving rise to the divergence of the specific heat.
• Figure S2: Comparison of entropy derivatives at different system sizes obtained as the solution of the entropy flow equation and through thermodynamic methods. The quantitative and qualitative agreement is excellent also in the mean-field case. Here, the transition is sharper and the discontinuity is visible already at finite size. The geometric function $\Upsilon^{(2)}_g$ and the second derivative $\partial_\epsilon^2 S$ show a jump around $\epsilon_{c}\approx0.74$ closed to the infinite-size energy value $\epsilon_c^\infty=3J/4=0.75$ for our choice $J=1$. Notice that the larger the size, the closer the critical point detected by MIPA.