Thermodynamics without Entropy: Reframing Available Energy as a Thermodynamic Primitive

Since the inception of thermodynamics, a tension has existed between two frameworks for expressing the second law: Gibbs’ entropy-based formalism, and Kelvin’s conception of the inevitable dissipation of mechanical work. Gibbs' formalism, built upon the fundamental relation, S(U, V, n) , reveals the geometric structure of thermodynamics, from which unexpected physical relationships such as the Maxwell relations and the Gibbs-Duhem equation naturally fall out. Kelvin’s conception, by contrast, is generally treated as a derived construct, and in engineering contexts has developed into availability or exergy analyses. Here, we demonstrate that a complete Gibbsian formalism may be constructed on the basis of a system’s available energy, with no need to invoke entropy. This formalism has its own fundamental relation, Legendre transforms, Maxwell relations, and convex contact-geometric structure, all equivalent to those of the traditional entropy-based framework. We derive several classical results, including the Clausius equation and the expression for the Joule-Thomson coefficient, within this entropy-free formalism, and show how expressions in classical statistical mechanics may be reinterpreted as Legendre transforms of a system’s available energy. Finally, we reflect on the conceptual, pedagogical and historical ramifications of Gibbs' decision to embed entropy, rather than available energy, at the heart of his thermodynamic formalism.