  Equilibrium molecular dynamics (EMD) simulations along with the Green-Kubo
formula have been widely used to calculate lattice thermal conductivities.
Previous studies, however, focused primarily on the calculated thermal
conductivities, with the uncertainty of the thermal conductivities remaining
poorly understood. In this paper, we study the quantification of the
uncertainty by using solid argon, silicon, and germanium as model material
systems, and examine the origin of the observed uncertainty. We find that the
uncertainty increases with the upper limit of the correlation time,
$t_{\text{corre, UL}}$, and decreases with the total simulation time,
$t_{\text{total}}$, whereas the velocity initialization seed, simulation domain
size, temperature, and type of material have minimal effects. The relative
uncertainties of the thermal conductivities for solid argon, silicon, and
germanium under different simulation conditions all follow a similar trend,
which can be fit with a "universal" square-root relation, as
$\sigma_{k_x}/k_{x, \text{ave}} = 2(t_{\text{total}}/t_{\text{corre,
UL}})^{-0.5}$. We have also conducted statistical analysis of the EMD-predicted
thermal conductivities and derived a formula that correlates the relative error
bound ($Q$), confidence level ($P$), $t_{\text{corre, UL}}$,
$t_{\text{total}}$, and number of independent simulations ($N$). We recommend
choosing $t_{\text{corre, UL}}$ to be 5-10 times the effective phonon
relaxation time, $\tau_{\text{eff}}$, and choosing $t_{\text{total}}$ and $N$
based on the desired relative error bound and confidence level. This study
provides new insights into understanding the uncertainty of EMD-predicted
thermal conductivities. It also provides a guideline for running EMD
simulations to achieve a desired relative error bound with a desired confidence
level and for reporting EMD-predicted thermal conductivities.
