Fig. present the incorrect dynamics when the random phases θ ξ (in Eq. 69b) are intentionally ignored by setting them to be zero. In Fig. , the longtime diagonal populations appear correct, but the diagonal populations oscillate with a much larger magnitude than Lindblad dynamics. This is caused by the fact that the coherences are larger than they should be because there is no random phase to reduce the size of the coherences. Consequentially, the larger magnitude of coherence causes a larger magnitude of population oscillations. In Fig. , not only are the oscillation magnitudes of the diagonal populations too large, but the diagonal populations converge to incorrect longtime populations. In fact, the L-MFE method with no random phase shows the excited state |1⟩ with a larger longtime population, while Lindblad dynamics show that the ground state |0⟩ should have the larger longtime population. This is again caused by the incorrect coherences without random phase which cause a longtime shift in the diagonal populations. Fig. presents the population dynamics of Model 4 (Eq. 76d) with different numbers of trajectories to examine the convergence of the L-MFE method. The numbers of trajectories used in Fig. are (a) 10, (b) 100, (c) 1,000, and (d) 10,000, respectively. For the result using 10 trajectories, the magnitudes and phases of the oscillations of the L-MFE dynamics do not match the exact ones. However, the longtime populations of the states are approximately correct, in contrast to the no-randomphase case in Fig. where the relative magnitude of the longtime populations are flipped versus the exact populations. For the result using 100 trajectories, the magnitudes of the oscillations are almost correct while there are some deviations at later times. For 1,000 trajectories, the relative error of the L-MFE dynamics versus the exact dynamics is only a few percent, and there is little visual difference between the L-MFE dynamics and the exact Lindblad dynamics. For 10,000 trajectories, the L-MFE dynamics and exact Lindblad dynamics are nearly indistinguishable. These results give a better sense of how many trajectories are required to achieve a desired level of accuracy using the L-MFE method.