The equivalence between the Entropy Maximization Principle and the Energy Minimization Principle

Abstract

In this paper, we present Gibbs’ original discussion on the equivalence between equilibrium conditions, i.e., a maximum entropy condition for a given energy and the minimum energy condition for a given entropy – which are presented as theorems by the author. Next, we show how such a discussion appears in contemporary textbooks that present Thermodynamics from its laws (law zero plus the three laws). We compare this presentation with the discussion made by Callen, in his postulational approach to Thermodynamics. We show that part of Gibbs’ discussion has been lost even in textbooks dealing with the laws. Furthermore, we argue that Callen’s approach does not emphasize the fact that although the principles are the same, thermalizing a system composed of distinct processes (one with constant energy and the other with constant temperature) leads to different temperature values. In addition to solving the problem analytically, we present a graphical analysis of the problem. At the end, we discuss implications for thermodynamics teaching.