| Abstract Scope |
Thermodynamics has evolved from an equilibrium framework to encompass nonequilibrium phenomena and data-driven materials modeling. Building on this progression, we present a unified formulation introducing partial internal energy, entropy, and volume into the first law, enabling explicit expressions for chemical potential and thermodynamically grounded transport relations. Extending these concepts, zentropy establishes an all-scale framework that unifies quantum mechanics and statistical thermodynamics through Helmholtz energy-based partitioning, reconstructing Helmholtz -energy landscapes with stable states and transition pathways. By combining configurational (Gibbs-Shannon) and intrinsic entropies, zentropy captures the recursive nature of complexity and emergent behavior across materials and complex systems. For data-driven applications, a zentropy-enhanced neural network (ZENN) embeds thermodynamic principles into machine learning, achieving improved generalization and robustness. This framework connects first-principles calculations, CALPHAD, and AI, offering predictive insights into phase stability and materials design while ensuring physically consistent, intrinsically stable modeling. (https://doi.org/10.1007/s11669-026-01248-0) |