  Molecular dynamics simulations play an important role in studying heat
transport in complex materials. The lattice thermal conductivity can be
computed either using the Green-Kubo formula in equilibrium MD (EMD)
simulations or using Fourier's law in nonequilibrium MD (NEMD) simulations.
These two methods have not been systematically compared for materials with
different dimensions and inconsistencies between them have been occasionally
reported in the literature. Here we give an in-depth comparison of them in
terms of heat transport in three allotropes of Si: three dimensional bulk
silicon, two-dimensional silicene, and quasi-one-dimensional silicon nanowire.
By multiplying the correlation time in the Green-Kubo formula with an
appropriate effective group velocity, we can express the running thermal
conductivity in the EMD method as a function of an effective length and
directly compare it with the length-dependent thermal conductivity in the NEMD
method. We find that the two methods quantitatively agree with each other for
all the systems studied, firmly establishing their equivalence in computing
thermal conductivity.
