The use of the information-theoretic entropy in thermodynamics

When considering controversial thermodynamic scenarios such as Maxwell's demon, it is often necessary to consider probabilistic mixtures of macrostates. This raises the question of how, if at all, to assign entropy to them. The information-theoretic entropy is often used in such cases; however, no general proof of the soundness of doing so has been given, and indeed some arguments against doing so have been presented. We offer a general proof of the applicability of the information-theoretic entropy to probabilistic mixtures of macrostates that is based upon a probabilistic generalisation of the Kelvin statement of the second law. We defend the latter and make clear the other assumptions on which our main result depends. We also briefly discuss the interpretation of our result.