Scaling Symmetry in Symplectic Thermodynamics

TL;DR

This paper explores the role of scaling symmetry in thermodynamics using symplectic and contact geometry, linking geometric formalism to physical systems like gases and black holes.

Contribution

It introduces a geometric framework unifying thermodynamic descriptions via contactization and symplectization, highlighting the importance of scale symmetry in physical models.

Findings

Diffeomorphism established between ideal and van der Waals gases.

Scale symmetry breaking is necessary for non-isothermal black hole dynamics.

Extended scale variables recover standard thermodynamics in a geometric setting.

Abstract

This paper investigates scaling symmetry in thermodynamics by unifying constrained Hamiltonian dynamics with symplectic and contact geometries. Through the mathematical processes of contactization and symplectization, we demonstrate that fixing an extended global scale variable effectively recovers the standard thermodynamic description in terms of scale-invariant quantities. The geometric formalism is illustrated by establishing the diffeomorphism between the Lagrangian submanifolds of ideal and van der Waals gases. Finally, applying this framework to a Schwarzschild black hole reveals that breaking the scale symmetry between internal energy and entropy is a fundamental physical requirement to accommodate non-isothermal dynamics.