[Paper Review] Representation invariant Geometrothermodynamics: applications to ordinary thermodynamic systems

Hernando Quevedo, Francisco Nettel+2 more|arXiv (Cornell University)|Mar 6, 2013

Black Holes and Theoretical Physics11 references3 citations

TL;DR

This paper introduces a representation-invariant metric within Geometrothermodynamics (GTD) that maintains invariance under both Legendre transformations and changes of thermodynamic representation, ensuring consistent geometric description across different thermodynamic potentials. The metric, derived from homogeneous fundamental equations, correctly captures thermodynamic interactions and phase transitions in systems like the ideal gas, van der Waals fluid, one-dimensional Ising model, and generalized Chaplygin gas, with curvature singularities signaling phase transitions and constant curvature indicating non-interacting or polytropic fluids.

ABSTRACT

In this work we employ a recently devised metric within the Geometrothermodynamics program to study ordinary thermodynamic systems. The new feature of this metric is that, in addition to Legendre symmetry, it exhibits invariance under a change of representation. This metric was derived in a previous work by the authors while addressing the problem of the conformal structure of the thermodynamic metrics for different representations. Here, we present an thorough analysis for the ideal gas, the van der Waals fluid, the one dimensional Ising model and some other systems of cosmological interest.

Motivation & Objective

• To develop a thermodynamic geometric formalism invariant under both Legendre transformations and changes of thermodynamic representation.
• To address the limitation of existing GTD metrics that fail to preserve geometric structure when switching between thermodynamic potentials.
• To ensure consistent description of thermodynamic interactions and phase transitions regardless of the chosen representation (e.g., entropy, energy, free energy).
• To clarify the role of homogeneity in fundamental equations for maintaining representation invariance in geometric thermodynamics.
• To classify GTD metrics based on their invariance properties and applicability to different physical systems.

Proposed method

• The paper employs a recently derived metric, the 'natural metric', which is invariant under total Legendre transformations and changes of representation.
• It applies the metric to systems where the fundamental equation is a homogeneous function of the extensive variables, ensuring thermodynamic equivalence across representations.
• The curvature scalar of the metric is computed in various representations (e.g., entropy, Helmholtz free energy) to detect thermodynamic interactions and phase transitions.
• The formalism relies on the conformal structure of GTD metrics, with the natural metric preserving geometric invariance under representation changes.
• For systems like the one-dimensional Ising model, the metric's invariance under total Legendre transformations allows curvature analysis even when the internal energy is not explicitly known.
• The analysis uses the Hessian of thermodynamic potentials and the resulting Riemann curvature tensor to extract physical information, with singularities indicating phase transitions.

Experimental results

Research questions

• RQ1 Can a geometric metric in thermodynamics be invariant under both Legendre transformations and changes of thermodynamic representation?
• RQ2 What conditions must a fundamental equation satisfy to allow consistent geometric description across different thermodynamic potentials?
• RQ3 How does the curvature scalar of the invariant metric reflect thermodynamic interactions and phase transitions in standard systems like the ideal gas and van der Waals fluid?
• RQ4 Why does the curvature fail to detect phase transitions when computed from non-canonical representations (e.g., Helmholtz free energy) for the van der Waals gas?
• RQ5 What types of equations of state yield spaces of constant curvature in the invariant geometric formalism?

Key findings

• The curvature scalar of the invariant metric correctly identifies phase transitions in the ideal gas and van der Waals fluid when computed in the canonical (entropy) representation, but fails when computed from the Helmholtz free energy representation, confirming the necessity of starting from a homogeneous fundamental relation.
• For the one-dimensional Ising model, the metric's invariance under total Legendre transformations allows correct detection of thermodynamic interaction even though the fundamental relation depends on intensive variables only.
• The generalized Chaplygin gas model yields a space of constant curvature when the metric is applied, with the curvature scalar given by $ R^{ atural}_{\alpha=\beta} = -\frac{1}{2}\frac{(1+\alpha)^2}{\alpha} $, indicating a non-interacting or polytropic fluid depending on $ \alpha $.
• The metric fails to describe the thermodynamic geometry correctly when applied to a non-homogeneous fundamental relation, as demonstrated by the incorrect phase transition structure in the van der Waals gas when using the Helmholtz free energy.
• The natural metric provides a consistent geometric description across all thermodynamic potentials only when the fundamental equation is homogeneous of a definite order, establishing homogeneity as a necessary condition for representation invariance.
• The analysis classifies GTD metrics into those invariant under partial Legendre transformations (for multi-potential analysis) and those invariant under total Legendre transformations (for representation-invariant analysis), with the latter being essential for consistent geometric thermodynamics.