Phase Transitions as Emergent Geometric Phenomena: A Deterministic Entropy Evolution Law
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918227594969088 |
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| author | Di Cairano, Loris |
| author_facet | Di Cairano, Loris |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_02242 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase Transitions as Emergent Geometric Phenomena: A Deterministic Entropy Evolution Law Di Cairano, Loris Statistical Mechanics 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. |
| title | Phase Transitions as Emergent Geometric Phenomena: A Deterministic Entropy Evolution Law |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2512.02242 |