The Thermodynamic Coordinate Transformations and the Thermodynamic Covariance Principle

The concept of equivalent systems from the thermodynamic point of view was originally introduced by
Th. De Donder and I. Prigogine. However, the De Donder-Prigogine definition of thermodynamic
invariance based only on the invariance of the entropy production is not sufficient to guarantee the
equivalence character between two sets of thermodynamic forces and conjugate thermodynamic
fluxes. In addition, it is known that there exists a large class of flux-force transformations such that,
even though they leave unaltered the expression of the entropy production, they may lead to certain
paradoxes.
Main objective of this series of works is to determine the non-linear closure equations (i.e. the fluxforce
relations), valid for thermodynamic systems outside the Onsager region, which do not contain
inconsistencies. To this aim, a thermodynamic theory for irreversible processes [referred to as the
Thermodynamical Field Theory (TFT)] has been developed. The TFT rests upon the concept of
equivalence between thermodynamic systems. More precisely, the equivalent character of two
alternative descriptions of a thermodynamic system is ensured if, and only if, the two sets of
thermodynamic forces are linked with each other by the so-called Thermodynamic Coordinate
Transformations (TCT). In this work, we describe the Lie group associated to the TCT. The TCT
guarantee the validity of the so-called Thermodynamic Covariance Principle (TCP): “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. We shall see that the requirement of the validity of the TCP will impose strict restrictions
allowing determining, for example, the expression of the collisional operator for magnetically confined
plasmas.