Are the geometries of the first and second laws of thermodynamics compatible?

Gerardo Hernández; Ernesto A. Lacomba

doi:10.3934/dcds.2013.33.1113

Article Contents

2013, Volume 33, Issue 3: 1113-1116. Doi: 10.3934/dcds.2013.33.1113

This issue Previous Article Transition map and shadowing lemma for normally hyperbolic invariant manifolds Next Article Discrete dynamics in implicit form

Are the geometries of the first and second laws of thermodynamics compatible?

Gerardo Hernández^1, and
Ernesto A. Lacomba^2,

1.
Sección de Metodología y Teoría de la Ciencia, Cinvestav, Av. IPN 2508, C.P. 07360, México, D.F.
2.
Departamento de Matemáticas, Universidad Autónoma Metropolitana–Iztapalapa, Av. San Rafael Atlixco 186, C.P. 09340 México, D.F.

Received: March 31, 2011

Revised: October 31, 2011

Published: February 2013

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Abstract

First and second laws of thermodynamics are naturally associated, respectively, to contact and Hessian geometries. In this paper we seek for a unique geometric setting that might account for both thermodynamic laws. Using Riemannian metrics that are compatible with the contact structure, we prove that the Hessian manifold of thermodynamic states cannot isometrically be embedded as Legendre submanifold of a contact manifold. Well known fibrations suggest the nature of the obstruction for such embedding.

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Mathematics Subject Classification: Primary: 53D10, 53Z05; Secondary: 80A05.

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