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

TL;DR

The paper provides a universal, nonextensive formulation of thermodynamics in which energy and entropy are defined for all states, including nonequilibrium and nanoscale systems, without relying on macroscopic extensivity. It develops an operational framework around adiabatic availability and available energy, introduces the energy–entropy ($E$–$S$) diagram to visualize nonequilibrium states, and derives Clausius inequalities and the second law extended to nonequilibrium processes. A key contribution is a rigorous, size-agnostic treatment of entropy transfer in non-work interactions, clarifying heat and heat–diffusion processes and their role in mesoscopic continua. The work unifies teaching and modern applications by presenting a coherent, universal thermodynamics that applies uniformly to all systems, from single particles to large engines, and shows how extensivity, equilibrium, and macroscopic behavior emerge as special cases.

Abstract

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.

Figures (41)

• Figure S1: The term "process" refers to the description of the initial state, the final state, and the effects caused on the environment, related to a given temporal evolution of the state of a system.
• Figure S2: A process is a weight process if the only external effect caused by the system's interactions is a change in the elevation of a weight.
• Figure S3: A process is reversible if there is a way to return both the system and its environment to their respective initial states.
• Figure S4: Energy differences are additive, $E_{11}^C-E_{00}^C=(E_1^A-E_0^A)+(E_1^B-E_0^B)$. Energy values can be made additive by selecting reference values for composite systems so that $E_{00}^C=E_0^A+E_0^B$. As a result, $E_{11}^C=E_1^A+E_1^B$.
• Figure S5: Energy can be exchanged between two systems $A$ and $B$ through interaction. In this example, the composite system $C=AB$ is isolated.