pyzentropy: A Python package implementing recursive entropy for first-principles thermodynamics

While the recursive property of entropy is well known in information theory, it is rarely utilized in thermodynamics, despite entropy originating in this field. Moreover, computational tools to implement this concept within first-principles thermodynamics remain lacking. In this work, we introduce an open-source Python package, pyzentropy, to implement this approach. We demonstrate its effectiveness using $Fe_3Pt$ as a case study, considering a 12-atom supercell with multiple magnetic configurations. By applying the recursive formulation of entropy to compute the total entropy of the system, we reproduce the Invar behavior, along with the anomalous temperature dependence of the linear coefficient of thermal expansion (LCTE), heat capacity $C_P$, and bulk modulus $B$. We also construct the $T$-$V$ and $P$-$T$ phase diagrams in good agreement with experimental observations. Finally, we highlight the importance of determining key high-probability configurations to accurately capture material properties.