## The Thermodynamic Coordinate Transformations and the Thermodynamic Covariance Principle

From the thermodynamic viewpoint, the idea of equivalent systems was originally proposed by Th. I and De Donder. Prigogine. Prigogine. However, the De Donder-Prigogine concept of thermodynamic invariance based solely on the invariance of the output of entropy is not sufficient to guarantee that two sets of thermodynamic forces and conjugate thermodynamic fluxes are identical in character. Moreover, it is understood that a broad class of flux-force transformations exist such that they can lead to such paradoxes, even if they leave the meaning of the output of entropy unaltered. The main objective of this series of works is to define non-linear closure equations valid for thermodynamic systems outside the Onsager area that do not involve contradictions (i.e. flux force relationships). A thermodynamic theory for irreversible procedures [referred to as the Thermodynamic Field Theory (TFT)] has been developed for this reason. The principle of equivalence between thermodynamic systems rests on the TFT. More specifically, the identical character of two alternative definitions of a thermodynamic system is ensured if, and only if, the so-called Thermodynamic Coordinate Transformations (TCT) connect the two sets of thermodynamic forces with each other. In this paper , we define the TCT-associated Lie group. The TCT guarantees the validity of the so-called Thermodynamic Covariance Principle (TCP): “The nonlinear closure equations, i.e. flux-force relationships, must be covariant under TCT everywhere and particularly outside the Onsager area.” In other words, under transformations between the admissible thermodynamic powers, i.e. under TCT, the fundamental laws of thermodynamics should be manifestly covariant. For systems far from equilibrium, the TCP guarantees the validity of simple theorems. We can see that the TCP validity criterion will enforce strict restrictions allowing, for example, the expression of a collision operator to be calculated for magnetically confined plasmas.

**Author(s) Details**

**Giorgio Sonnino
**Department of Theoretical Physics and Mathematics, Université Libre de Bruxelles (ULB), Campus Plaine CP 231, Boulevard de Triomphe, 1050 Brussels, Belgium.

**Alberto Sonnino**

Faculty of Engineering Sciences, University College London, Gower Street, London WC1E 6BT, UK.

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covariance principles Lie algebras and Lie groups Thermodynamics of irreversible processes transport processes in Tokamak-plasmas.