Two-Dimensional Einstein Manifolds in Geometrothermodynamics

Antonio C Gutiérrez-Piñeres,Cesar S López-Monsalvo,Francisco Nettel

doi:10.1155/2013/967618

Abstract

We present a class of thermodynamic systems with constant thermodynamic curvature which, within the context of geometric approaches of thermodynamics, can be interpreted as constant thermodynamic interaction among their components. In particular, for systems constrained by the vanishing of the Hessian curvature we write down the systems of partial differential equations. In such a case it is possible to find a subset of solutions lying on a circumference in an abstract space constructed from the first derivatives of the isothermal coordinates. We conjecture that solutions on the characteristic circumference are of physical relevance, separating them from those of pure mathematical interest. We present the case of a one-parameter family of fundamental relations that—when lying in the circumference—describe a polytropic fluid.

Highlights

The study of physical systems that admit a geometric description in terms of Riemannian manifolds is an interesting and timely subject
We present a class of thermodynamic systems with constant thermodynamic curvature which, within the context of geometric approaches of thermodynamics, can be interpreted as constant thermodynamic interaction among their components
Analogously to the case of field theories, it has been argued that the curvature of the appropriate manifold should be linked to the notion of thermodynamic interaction [1]

Summary

Introduction

The study of physical systems that admit a geometric description in terms of Riemannian manifolds is an interesting and timely subject. There are the conformally related metric theories of Ruppeiner and Weinhold, where the metric takes the form of a Hessian of the extensive parameters in the entropy and energy representations, respectively [2, 3] Both fail to comply with the spirit of the geometric construction of field theories; that is, those are not invariant under the natural set of transformations in thermodynamics. In the GTD programme one posits that the physical information about a thermodynamic system cannot depend on the potential used to describe it and that such information is encoded in the curvature of the maximal integral manifold of the Pfaffian system defining the first law of thermodynamics (cf (1)) We call such a manifold the space of equilibrium states.

Two-Dimensional Thermodynamic Systems of Constant Curvature

Isothermal Coordinates

R 10 5

Closing Remarks