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Exactly equivalent thermal conductivity in finite systems from equilibrium and nonequilibrium molecular dynamics simulations

In a previous paper [Physical Review B \textbf{103}, 035417 (2021)], we showed that the equilibrium molecular dynamics (EMD) method can be used to compute the apparent thermal conductivity of finite systems. It has been shown that the apparent thermal conductivity from EMD for a system with domain length $2L$ is equal to that from nonequilibrium molecular dynamics (NEMD) for a system with domain length $L$. Taking monolayer silicence with an accurate machine learning potential as an example, here we show that the thermal conductivity values from EMD and NEMD agree for the same domain length if the NEMD is applied with periodic boundary conditions in the transport direction. Our results thus establish an exact equivalence between EMD and NEMD for thermal conductivity calculations.