[Paper Review] Invariant geometry of the ideal gas

Hernando Quevedo, Alberto Sánchez+1 more|arXiv (Cornell University)|Nov 3, 2008

Cosmology and Gravitation Theories16 references13 citations

TL;DR

This paper introduces a Legendre-invariant metric in the equilibrium space of an ideal gas, deriving a geometric structure where thermodynamic geodesics—representing quasi-static processes—split the space into disconnected regions separated by adiabatic geodesics. This structure mirrors the causal cone in special relativity, suggesting a deep geometric link between thermodynamics and relativity.

ABSTRACT

We analyze in the context of geometrothermodynamics a Legendre invariant metric structure in the equilibrium space of an ideal gas. We introduce the concept of thermodynamic geodesic as a succession of points, each corresponding to a state of equilibrium, so that the resulting curve represents a quasi-static process. A rigorous geometric structure is derived in which the thermodynamic geodesics at a given point split the equilibrium space into two disconnected regions separated by adiabatic geodesics. This resembles the causal structure of special relativity, which we use to introduce the concept of adiabatic cone for thermodynamic systems. This result might be interpreted as an alternative indication of the inter-relationship between relativistic physics and classical thermodynamics.

Motivation & Objective

• To develop a Legendre-invariant geometric framework for the equilibrium space of an ideal gas.
• To define thermodynamic geodesics as curves representing quasi-static processes.
• To investigate how these geodesics partition the equilibrium space into disconnected regions.
• To explore the analogy between thermodynamic causality and the causal structure of special relativity.
• To propose the concept of an adiabatic cone as a geometric representation of thermodynamic irreversibility.

Proposed method

• The paper constructs a Legendre-invariant metric in the equilibrium space of the ideal gas using differential geometry.
• It defines thermodynamic geodesics as curves that represent quasi-static processes in the equilibrium space.
• The analysis reveals that at each point, thermodynamic geodesics divide the space into two disconnected regions.
• The paper identifies adiabatic geodesics as the boundary separating these regions, analogous to light cones in relativity.
• It introduces the concept of an adiabatic cone, structurally mirroring the light cone in special relativity.
• The geometric structure is derived using invariant differential geometry, emphasizing Legendre invariance to ensure thermodynamic consistency.

Experimental results

Research questions

• RQ1 How can a Legendre-invariant metric be constructed in the equilibrium space of an ideal gas?
• RQ2 What is the geometric meaning of thermodynamic geodesics representing quasi-static processes?
• RQ3 How do thermodynamic geodesics partition the equilibrium space into disconnected regions?
• RQ4 What is the significance of adiabatic geodesics in defining a causal structure in thermodynamics?
• RQ5 To what extent does the geometric structure of thermodynamic geodesics resemble the causal cone in special relativity?

Key findings

• The geometric structure derived is invariant under Legendre transformations, ensuring thermodynamic consistency.
• Thermodynamic geodesics, representing quasi-static processes, form curves that split the equilibrium space into two disconnected regions.
• Adiabatic geodesics act as boundaries separating these regions, analogous to light cones in special relativity.
• The resulting structure forms an adiabatic cone, providing a geometric interpretation of thermodynamic irreversibility.
• The analogy with special relativity suggests a deeper geometric relationship between classical thermodynamics and relativistic physics.