Quantification of entropy generation in defected crystals

Defects such as vacancies, impurities, dislocations, and grain boundaries modify phonon transport in real crystals. Among these, vacancies represent one of the simplest and most prevalent forms, acting as scattering centers that fundamentally alter phonon dynamics and thermal behavior. Conventional assessments emphasize thermal conductivity as a key metric for phonon transport—estimated via equilibrium Green–Kubo or nonequilibrium approaches based on Fourier's law—but this quantity does not explicitly capture the microscopic thermodynamic evolution stemming from phonon–defect interactions. In our work, we combine nonequilibrium molecular dynamics (NEMD) with stochastic-thermodynamic analysis to study vacancy-defected crystalline silicon over a range of temperatures. From the microscopic heat-current obtained from the simulations, we use the thermodynamic uncertainty relation (TUR) to estimate lower bounds on the entropy production. Lower bounds for the thermal conductivity are also competed and compared to the thermal conductivity extracted from Fourier's-law fits to the same data. We report the temperature- and vacancy-fraction dependence of both measures and identify regimes—particularly at low temperatures and high vacancy concentrations—where the fluctuation-based bound and the Fourier estimate diverge. These results indicate that a purely diffusive picture of phonon–defect scattering does not fully capture the observed stochastic behavior. Our approach seeks to overcome this limitation and elucidate the interplay between noncoherent and coherent scattering mechanisms.