Thermodynamics as Geometry

Abstract

Title: The Geometric Foundation of Thermodynamics: From Phase Transitions to Black Holes

Authors: Heidi Peterson (ORCID: 0009-0005-5630-6804)

Date: February 10, 2026

This research presents a geometric framework that unifies thermodynamics, critical phenomena, and information theory through a single principle: dimensional collapse. We demonstrate that the breakdown of mean-field theory at critical points is not merely an algebraic failure but a geometric phase transition where the effective dimensionality collapses from three to two degrees of freedom.

The framework is built upon four geometric postulates:

1. A logarithmic potential field \Phi = \alpha \ln(\xi_0/\xi)

2. A conformal metric f = 1 + \kappa\Phi

3. Temperature as the gradient norm T = \|\nabla\Phi\|/k_B

4. Entropy as S = k_B \ln(\Phi)V

From these foundations, we derive:

· Exact 2D Ising critical exponents (\nu=1, \alpha=0, \eta=1/4, \beta=1/8, \gamma=7/4) without fitting parameters

· Hawking temperature as the geometric gradient at black hole horizons, naturally yielding the area law with quantum corrections

· Finite-size scaling predictions that differ from standard theory: T_c(L) = T_c(\infty)[1 + B/\ln(L)] versus the conventional 1/L scaling

· Unification of Ruppeiner and Fisher information metrics as emergent projections of the underlying geometric structure

The research program is validated through six computational tests demonstrating mathematical consistency, critical exponent recovery, black hole thermodynamics derivation, falsifiable finite-size predictions, and information geometry unification. A key falsifiable prediction—logarithmic versus power-law finite-size scaling—provides a clear experimental test of the geometric hypothesis.

This work transforms thermodynamics from a phenomenological collection of relations into a geometric theory where heat, phase transitions, and information entropy emerge from curvature and dimensionality in an abstract thermodynamic manifold. The framework bridges microscopic statistical systems, gravitational thermodynamics, and information theory under a single geometric language.

Keywords: Geometric Thermodynamics, Critical Phenomena, Dimensional Collapse, Hawking Radiation, Information Geometry, Phase Transitions, 2D Ising Model, Finite-Size Scaling

DOI: 10.5281/zenodo.18541473

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This abstract:

1. States the core innovation (dimensional collapse as geometric principle)

2. Summarizes the mathematical framework concisely

3. Lists the key derived results

4. Highlights the falsifiable prediction

5. Positions the work as transformative for the field

6. Includes all necessary metadata for Zenodo

Not for commercial use contact heidiswork6@gmail.com for licencing.