Dwivedi, Ripunjay (2026) A Fiber Bundle Foundation for Classical Thermodynamics. [Preprint]

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Abstract

The thermodynamic decomposition of energy transfer into work and heat is conventionally postulated as part of the first law. We show that it is instead a theorem of Newtonian mechanics and differential geometry, once the spatial confinement of an N-particle system is formalised as a fiber bundle over a manifold of macroscopic control parameters. A smooth family of diffeomorphisms φ_α : Ω_0 → Ω(α) trivialises the bundle and induces a flat connection whose horizontal-vertical decomposition splits the total differential of the internal energy into thermodynamic work (boundary deformation at frozen internal coordinates) and heat (internal rearrangement at fixed boundary): dU = δW + δQ. This microscopic first law is the chain rule on a product space; it requires no equations of motion and no statistical input. To pass from microscopic to macroscopic, we show that linearity, positivity, and normalisation force the projection from the total phase space to the base manifold to be integration against a probability measure. Four physical constraints—support on the energy shell, flow-invariance, absolute continuity, and normalisation—together with the ergodic hypothesis, select the microcanonical measure uniquely. The canonical and grand canonical ensembles are then derived by marginalisation over bath degrees of freedom, with no additional postulates. Fiber averaging yields the macroscopic first law dU = δW̅ + δQ̅, exploiting the fact that U is constant on the energy shell while the generalized forces fluctuate. Re-expressing the macroscopic heat form as δQ̅ = dΩ/Σ—where Ω is the cumulative phase volume and Σ the density of states—reveals 1/T = k_B Σ/Ω as the integrating factor and S = k_B ln Ω as the entropy, yielding the Gibbs fundamental relation dU = T dS + δW̅ without postulation. The Clausius inequality and the second law follow upon supplementing this framework with the Stosszahlansatz. Three assumptions beyond Newton's laws are required, each stated explicitly: a regularity hypothesis on the bundle structure, ergodicity, and molecular chaos. The framework delineates which features of classical thermodynamics are theorems of geometry and Hamiltonian mechanics, which depend on the choice of trivialisation, and which require statistical hypotheses.

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Item Type:	Preprint
Creators:	Creators Email ORCID Dwivedi, Ripunjay ripunjay.cmi@gmail.com 0009-0007-7269-9382
Keywords:	Fiber Bundles; Classical Thermodynamics; Work-Heat Decomposition; Ehresmann Connection; Microcanonical Ensemble
Subjects:	Specific Sciences > Physics > Classical Physics Specific Sciences > Physics > Statistical Mechanics/Thermodynamics General Issues > Structure of Theories
Depositing User:	Mr. RIPUNJAY DWIVEDI
Date Deposited:	10 Apr 2026 13:02
Last Modified:	10 Apr 2026 13:02
Item ID:	29000
Subjects:	Specific Sciences > Physics > Classical Physics Specific Sciences > Physics > Statistical Mechanics/Thermodynamics General Issues > Structure of Theories
Date:	10 April 2026
URI:	https://philsci-archive.pitt.edu/id/eprint/29000

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