Entropy drives changes in macroscopic and quantum systems, which include magnetic and superconducting materials. The challenge is how to quantify the entropy of materials and the entropy change due to internal processes that dictate their macroscopic transformative functionalities with respect to external stimuli. This is because macroscopic functionalities of materials stem from assemblies of microscopic states (microstates) at all scales *at and below* the scale of the macroscopic state of the experimental observation (macrostate), while the state-of-the-art first-principles calculations based on density functional theory (DFT) predict only the properties of individual microstates. The “zentropy” theory developed by the PI’s team provides a nested formalism that integrates three fundamental scientific domains: (1) DFT-based quantum mechanics for prediction of the entropy of each microstate, (2) Statistical mechanics among ergodic microstates, and (3) macroscopic thermodynamics represented by the integration of experimentally measured heat capacity under given constraints from the surroundings. The zentropy theory has been successfully applied to predict temperature-pressure and temperature-volume phase diagrams of simple magnetic materials such as Ce and Fe_{3}Pt, including first- and second-order phase transitions, critical points, the divergence of thermal expansion, the anomaly in heat capacity, and the negative thermal expansion in Fe_{3}Pt, with all input data predicted from DFT-based calculations and without empirical models and parameters.

The proposed research aims to further develop the zentropy theory through applications to complex magnetic materials and superconductors under the *hypothesis that the emergent properties of complex magnetic materials and superconductors can be predicted by statistical mechanics of ergodic microstates with their partition function Z computed from DFT-predicted ***free energies**. The key objective is to develop approaches to systematically determine the types and number of microstates and the supercell size in DFT-based calculations through convergence of macroscopic functionalities. In addition to using scientific intuitions to guide the design of important microstates, the key innovation of the proposed research is to integrate the domain knowledge and the material-property-descriptor database with four million microstates, which is supported by our deep neural network machine learning models and integrated with our high-throughput DFT Tool Kit (DFTTK). For complex magnetic materials, one of the objectives is to develop approaches to calculate short-range ordering from the statistical distribution of each microstate. For superconductors, the superconducting and non-superconducting microstates will be delineated.

The accomplishment of the proposed research objectives will not only enable us to have a better fundamental understanding of magnetic and superconducting transitions and their predictions through the integration of quantum mechanics and statistical mechanics, but also provide a systemized theoretical and computational framework to understand and predict other transformative functionalities. A new open-source software tool will be developed for the zentropy theory based on our existing open-source high-throughput thermodynamic modeling tools along with new short-courses and workshops, which will be incorporated into our existing global outreach cloud-based teaching portfolio through the nonprofit Materials Genome Foundation.