Accurate heat flux formula and thermal conductivity calculation in molecular dynamics simulations with machine learning potentials

An accurate formula for the atomic heat flux that can be used in molecular dynamics (MD) simulations driven by machine learning potentials is derived and discussed. The equivalence of the Torii and Fan atomic heat flux formulas for any two- or many-body potential is first demonstrated. For copper and silicon modeled with three machine learning potentials Spectral Neighbor Analysis Potential (SNAP), atomic cluster expansion potential, and moment tensor potential, the default heat flux formula implemented in the large-scale atomic/molecular massively parallel simulator is shown to over- or underestimate the lattice thermal conductivity compared with the accurate formula, with no systematic error based on the material or potential. The accuracy of the heat flux formula and its implementation are further demonstrated by comparing the temperature dependence of the lattice thermal conductivity of copper modeled with a SNAP potential with that obtained with anharmonic lattice dynamics calculations by solving the phonon Boltzmann transport equation incorporating up to four-phonon scattering. This study will facilitate accurate thermal conductivity calculations using MD simulations with machine learning potentials.