Symmetry Group and Group Representations Associated to the Thermodynamic Covariance Principle (TCP)

TL;DR

This paper explores the mathematical structure of symmetry groups related to the Thermodynamic Covariance Principle, emphasizing their role in ensuring the covariance of thermodynamic laws far from equilibrium and deriving associated conserved quantities.

Contribution

It introduces the Lie group and representations linked to nonlinear thermodynamic transformations, extending the symmetry analysis beyond near-equilibrium conditions.

Findings

Derived the Lie group structure for Thermodynamic Coordinate Transformations.

Established the covariance of nonlinear flux-force relations under TCT.

Calculated conserved thermodynamic currents for a specific out-of-equilibrium system.

Abstract

We describe the Lie group and the group representations associated to the nonlinear Thermodynamic Coordinate Transformations (TCT). The TCT guarantee the validity of the Thermodynamic Covariance Principle (TCP) : {\it The nonlinear closure equations, i.e. the flux-force relations, everywhere and in particular outside the Onsager region, must be covariant under TCT}. In other terms, the fundamental laws of thermodynamics should be manifestly covariant under transformations between the admissible thermodynamic forces, i.e. under TCT. The TCP ensures the validity of the fundamental theorems for systems far from equilibrium. The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. We derive the conserved (thermodynamic) currents and, as an example of calculation, a simple system out of equilibrium where the validity of TCP is imposed at the level of the kinetic equations is also analyzed.