Editing Entropy (section)

=== Entropy and adiabatic accessibility ===
A definition of entropy based entirely on the relation of [[adiabatic accessibility]] between equilibrium states was given by [[Elliott H. Lieb|E.&nbsp;H.&nbsp;Lieb]] and [[Jakob Yngvason|J.&nbsp;Yngvason]] in 1999.<ref>{{cite journal |last1=Lieb |first1=Elliott H. |last2=Yngvason |first2=Jakob |title=The physics and mathematics of the second law of thermodynamics |journal=Physics Reports |date=March 1999 |volume=310 |issue=1 |pages=1–96 |doi=10.1016/S0370-1573(98)00082-9 |arxiv=cond-mat/9708200 |bibcode=1999PhR...310....1L |s2cid=119620408}}</ref> This approach has several predecessors, including the pioneering work of [[Constantin Carathéodory]] from 1909<ref>{{cite journal |last1=Carathéodory |first1=C. |title=Untersuchungen über die Grundlagen der Thermodynamik |journal=Mathematische Annalen |date=September 1909 |volume=67 |issue=3 |pages=355–386 |doi=10.1007/BF01450409 |s2cid=118230148 |url=https://zenodo.org/record/1428268 |language=de}}</ref> and the monograph by R.&nbsp;Giles.<ref>{{cite book |author=R. Giles |title=Mathematical Foundations of Thermodynamics: International Series of Monographs on Pure and Applied Mathematics |url=https://books.google.com/books?id=oK03BQAAQBAJ |date=2016 |publisher=Elsevier Science |isbn=978-1-4831-8491-3}}</ref> In the setting of Lieb and Yngvason, one starts by picking, for a unit amount of the substance under consideration, two reference states <math display="inline">X_0</math> and <math display="inline">X_1</math> such that the latter is adiabatically accessible from the former but not conversely. Defining the entropies of the reference states to be 0 and 1 respectively, the entropy of a state <math display="inline">X</math> is defined as the largest number <math display="inline">\lambda</math> such that <math display="inline">X</math> is adiabatically accessible from a composite state consisting of an amount <math display="inline">\lambda</math> in the state <math display="inline">X_1</math> and a complementary amount, <math display="inline">(1 - \lambda)</math>, in the state <math display="inline">X_0</math>. A simple but important result within this setting is that entropy is uniquely determined, apart from a choice of unit and an additive constant for each chemical element, by the following properties: it is monotonic with respect to the relation of adiabatic accessibility, additive on composite systems, and extensive under scaling.