The atomic kinetic energy density is $E_k = m \sum_i |\dot{\mathbf u}|^2 / 2N$. The phase transformation mostly takes place in the temporal regime $t_1 < t < t_2$, marked by the shaded area. Immediately after the quench, the lattice temperature gradually increases toward the final temperature $T_f$. The enhanced thermal fluctuation of displacements in this initial period ($t < t_1$) result in an increase in elastic energy $E_l$. The enlarged fluctuations of $\langle |\mathbf u_i|^2 \rangle$ also lead to an enhanced variance of the nearest-neighbor hopping $t_{ij}$, which in turn gives rise to a slightly reduced electron energy. However, this relatively short transient phase does not play a significant role in terms of phase transformation.

![(Color online) \[fig:snapshots\] Snapshots of the on-site double occupancy $D(\mathbf r_i) = \langle n_{i, \uparrow} n_{i, \downarrow} \rangle$ at four different Mott volume fractions, corresponding to $t = 30$, 50, 58, and 65, during the thermal driven MIT. See Fig. \[fig:trace\](b) for the corresponding time dependence of the Mott volume fraction and the average double-occupancy. ](snapshots.pdf){width="0.99\columnwidth"}

As discussed in Sec. \[sec:intro\], the double occupancy is one of the order-parameters for characterizing the Mott transitions. Another important quantity is the fraction of Mott phase $\phi_M$, 