Universal Foundations of Thermodynamics: Entropy and Energy Beyond Equilibrium and Without Extensivity

Thermodynamics is commonly presented as a theory of macroscopic systems in stable equilibrium, built upon assumptions of extensivity and scaling with system size. In this paper, we present a universal formulation of the elementary foundations of thermodynamics, in which entropy and energy are defined and employed beyond equilibrium and without assuming extensivity. The formulation applies to all systems-large and small, with many or few particles-and to all states, whether equilibrium or nonequilibrium, by relying on carefully stated operational definitions and existence principles rather than macroscopic idealizations. Key thermodynamic concepts, including adiabatic availability and available energy, are developed and illustrated using the energy-entropy diagram representation of nonequilibrium states, which provides geometric insight into irreversibility and the limits of work extraction for systems of any size. A substantial part of the paper is devoted to the analysis of entropy transfer in non-work interactions, leading to precise definitions of heat interactions and heat-and-diffusion interactions of central importance in mesoscopic continuum theories of nonequilibrium behavior in simple and complex solids and fluids. As a direct consequence of this analysis, Clausius inequalities and the Clausius statement of the second law are derived in forms explicitly extended to nonequilibrium processes. The resulting framework presents thermodynamics as a universal theory whose concepts apply uniformly to all systems, large and small, and provides a coherent foundation for both teaching and modern applications.