Coherent application of a contact structure to formulate Classical Non-Equilibrium Thermodynamics

This contribution presents an outline of a new mathematical formulation for
Classical Non-Equilibrium Thermodynamics (CNET) based on a contact
structure in differential geometry. First a non-equilibrium state space is introduced as the third key element besides the first and second law of thermodynamics.
This state space provides the mathematical structure to generalize
the Gibbs fundamental relation to non-equilibrium thermodynamics. A
unique formulation for the second law of thermodynamics is postulated and it
showed how the complying concept for non-equilibrium entropy is retrieved.
The foundation of this formulation is a physical quantity, which is in nonequilibrium thermodynamics nowhere equal to zero. This is another perspective compared to the inequality, which is used in most other formulations in the literature. Based on this mathematical framework, it is proven that the
thermodynamic potential is defined by the Gibbs free energy. The set of conjugated coordinates in the mathematical structure for the Gibbs fundamental
relation will be identified for single component, closed systems. Only in the
final section of this contribution will the equilibrium constraint be introduced
and applied to obtain some familiar formulations for classical (equilibrium)
thermodynamics.