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Phase Transitions as Emergent Geometric Phenomena: A Deterministic Entropy Evolution Law
by Loris Di Cairano
Submission summary
| Authors (as registered SciPost users): | Loris Di Cairano |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202512_00024v1 (pdf) |
| Date submitted: | Dec. 10, 2025, 9:12 a.m. |
| Submitted by: | Loris Di Cairano |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
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 $\phi^4$ 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.
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