From Data-Driven Models to Physical Insight: Vibrational Entropy and Its Dependence on Atomic Volume

Vibrational entropy plays a central role in determining phase stability and temperature-dependent behavior in materials, yet its calculation from first-principles phonon methods remains computationally demanding. In this work, we combine data-driven modeling with physically motivated analysis to develop an efficient and interpretable framework for predicting vibrational entropy. Using a data set derived from PhononDB, we systematically evaluate linear regression, random forest and feedforward neural network models trained on Materials Project and composition-based descriptors. All three approaches provide good predictive accuracy, with the random forest model showing a marginal improvement over the other models. SHAP analysis consistently identifies atomic volume as the dominant factor governing vibrational entropy. Guided by this insight, simplified analytical models are constructed, revealing a logarithmic dependence of vibrational entropy on atomic volume consistent with lattice dynamical considerations. A logarithmic-linear model is shown to provide an accurate and physically interpretable description across the full range of materials. To extend the analysis to finite temperatures, a temperature-dependent formulation is introduced that incorporates T3 scaling at low temperatures and logarithmic dependence at higher temperatures, consistent with Debye- and Einstein-type behavior. This unified model captures both structural and thermal contributions to vibrational entropy with good accuracy. Overall, the proposed framework demonstrates that vibrational entropy can be predicted using simple, physically meaningful relationships, offering a computationally efficient alternative to full phonon calculations and enabling entropy-informed materials screening.