Axiomatic Definition of Entropy for Nonequilibrium States

In introductory courses and textbooks on elementary thermodynamics, entropy is often presented as a property defined only for equilibrium states, and its axiomatic definition is almost invariably given in terms of a heat to temperature ratio, the traditional Clausius definition. Teaching thermodynamics to undergraduate and graduate students from all over the globe, we have sensed a need for more clarity, unambiguity, generality and logical consistency in the exposition of thermodynamics, including the general definition of entropy, than provided by traditional approaches. Continuing the effort pioneered by Keenan and Hatsopoulos in 1965, we proposed in 1991 a novel axiomatic approach which eliminates the ambiguities, logical circularities and inconsistencies of the traditional approach still adopted in many new books. One of the new and important aspects of our exposition is the simple, non-mathematical axiomatic definition of entropy which naturally extends the traditional Clausius definition to all states, including non-equilibrium states (for which temperature is not defined). And it does so without any recourse to statistical mechanical reasoning. We have successfully presented the foundations of thermodynamics in undergraduate and graduate courses for the past thirty years. Our approach, including the definition of entropy for non-equilibrium states, is developed with full proofs in the treatise E. P. Gyftopoulos and G. P. Beretta, Thermodynamics. Foundations and Applications, Dover Edition, 2005 (First edition, Macmillan, 1991) [1]. The slight variation we present here illustrates and emphasizes the essential elements and the minimal logical sequence to get as quickly as possible to our general axiomatic definition of entropy valid for nonequilibrium states no matter how “far” from thermodynamic equilibrium.