The first dependable thermodynamic laws -- in the early 19th century -- were those of Fourier and Navier-Stokes for heat conduction and viscous friction. They were purely phenomenological and did not require the understanding of the nature of heat, let alone the concepts of energy and entropy.
Those concepts emerged in the works of Mayer, Joule, Helmholtz, Clausius and Boltzmann in the latter half of the 19th century and -- apart from making energy conversion more efficient -- they had a deep impact on natural philosophy: It was recognized that nature is governed by a universal conflict between determinacy, which forces energy into a minimum, and stochasticity, by which entropy tends to a maximum. The outcome of the competition is determined by temperature so that miminal energy requires cold systems and maximal entropy occurs in hot systems. Thus phenomena like phase transitions, planetary atmospheres, osmosis, phase diagrams and chemical reactions were all understood in terms of the universal competition of energy and entropy. Mixtures, alloys and solutions were systematically incorporated into thermodynamics by Gibbs. And so the early 20th century saw the development of a second vast field of industrial application, -- apart from energy conversion --, the rectification of naturally occuring mixtures like natural gas and mineral oil.
By the work of Eckart the phenomenological laws of Fourier, Navier-Stokes and Fick -- the latter for diffusion -- received a systematic derivation in terms of the Gibbs equation for the entropy. Thus equilibrium thermodynamics and thermodynamics of irreversible processes had come to a kind of conclusion by the mid 20th century. At the same time the limits of those fields were recognized: They required fairly dense matter which means that they could not describe rapidly changing processes and processes with steep gradients.
Some wishful thinking occured concerning stationary processes: The hypothesis of Onsager about the mean regression of fluctuations, and Prigogine´s principle on minimal entropy production.
Boltzmann´s kinetic theory of gases provides macroscopic balance laws in its hierarchy of moments of the distribution function. These have furnished the blueprint for extended thermodynamics which is capable of describing rapid changes and steep gradients, if enough moments are taken into account. The theory assumes that all field equations are balance laws with local and instantaneous constitutive relations for fluxes and productions. Thus follows a system of quasi-linear field equations and the entropy principle ensures that the system is symmetric hyperbolic, if written in the proper variables. That property guarantees
- existence and uniqueness of solutions,
- continuous dependence of solutions on the initial data and
- finite speeds of characteristic waves.
That is to say: The entropy principle ensures well-posedness of initial value problems.
Extended thermodynamics permits the calculation of light scattering spectra down to very rarefied gases. Moreover, it offers a deep insight into the structure of shocks and, above all, it reveals the very narrow limitations of linear irreversible thermodynamics based on the laws of Fourier, Fick and Navier-Stokes. Thus for instance extended thermodynamics shows that a gas in the gap between two cylinders cannot rotate rigidly, if heat conduction occurs between the inner and outer surface. Moreover, the temperature fields in the gap does not obey the Fourier law, even if there is no rotation.