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The indefinite metric of R. Mrugala and the geometry of the thermodynamical phase space

Serge Preston, James Vargo

Abstract

We study the indefinite metric $G$ in the contact phase space $(P,θ)$ of a homogeneous thermodynamical system introduced by R. Mrugala. We calculate the curvature tensor, Killing vector fields, second fundamental form of Legendre submanifolds of $P$ - constitutive surfaces of different homogeneous thermodynamical systems. We established an isomorphism of the space $(P,θ,G)$ with the Heisenberg Lie group $H_{n}$ endowed with the right invariant contact structure and the right invariant indefinite metric. The lift $\tG$ of the metric $G$ to the symplectization $\tP$ of contact space $(P,θ)$, its curvature properties, and its Killing vector fields are studied. Finally we introduce the "hyperbolic projectivization" of the space $(\tP,{\tilde θ}, \tG)$ that can be considered as the natural {\bf compactification} of the thermodynamical phase space $(P,θ, G).$

The indefinite metric of R. Mrugala and the geometry of the thermodynamical phase space

Abstract

We study the indefinite metric in the contact phase space of a homogeneous thermodynamical system introduced by R. Mrugala. We calculate the curvature tensor, Killing vector fields, second fundamental form of Legendre submanifolds of - constitutive surfaces of different homogeneous thermodynamical systems. We established an isomorphism of the space with the Heisenberg Lie group endowed with the right invariant contact structure and the right invariant indefinite metric. The lift of the metric to the symplectization of contact space , its curvature properties, and its Killing vector fields are studied. Finally we introduce the "hyperbolic projectivization" of the space that can be considered as the natural {\bf compactification} of the thermodynamical phase space

Paper Structure

This paper contains 25 sections, 17 theorems, 320 equations.

Table of Contents

  1. Introduction.
  2. The contact structure of homogeneous thermodynamics.
  3. Gibbs space. Legendre surfaces of equilibrium.
  4. Thermodynamical metrics of Weinhold and Ruppeiner.
  5. Indefinite thermodynamical metric $G$ of R. Mrugala.
  6. Non-holonomic frame $(\xi, P_{i},X_{i})$.
  7. Compatibility of contact structure and metric $G$.
  8. Levi-Civita Connection.
  9. Ricci and scalar curvatures.
  10. Sectional curvatures.
  11. Killing vector fields, isotropy Lie algebra.
  12. Second fundamental form of Legendre surfaces.
  13. Constitutive hypersurface.
  14. The Heisenberg Group as the thermodynamical phase space.
  15. Symplectization of manifold $(P,\theta,G)$.
  16. ...and 10 more sections

Key Result

Theorem 1

The diffeomorphism $\chi$ defined by determines an isomorphism of the "thermodynamical metric contact manifold" $(P,\theta ,G)$ with the Heisenberg group $H_{n}$ endowed with the right invariant contact from $\theta_{H}$ and the right invariant metric $G_{H}$ of signature $(n+1,n).$

Theorems & Definitions (32)

  • Theorem 1
  • Example 1
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Proposition 3
  • Example 2
  • Remark 2
  • Theorem 2
  • Remark 3
  • ...and 22 more