which is often used in the study of phase transformation kinetics. The time dependence of these two quantities during the thermal Mott transition is shown in Fig. \[fig:trace\](b); here $\langle D \rangle = \sum_i D_i / N$ is the spatial average of the local double occupation. Numerically, we label a given site as the Mott insulator if its double-occupancy $D_i < 10^{-8}$. It is worth noting that the transformed volume $\phi_M$ is negligible in the initial period when $t < t_1$. In fact, the average $\langle D \rangle$ even slightly increases due to the lowered electron energy.

After this initial period, the system is trapped in a meta-stable metallic state with a temperature $T \approx T_f$. The transformation to the Mott insulator mainly takes place in the temporal window $[t_1, t_2]$, in which the Mott volume fraction $\phi_M$ gradually increases from zero to its equilibrium value at $T_f$ while the average $\langle D \rangle$ slowly decays to zero, see Fig. \[fig:trace\](b). Once the system reaches the new equilibrium at $T_f$ for both the lattice and electrons, the electron binding energy is renormalized to almost zero. This also indicates a vanishing electronic force $\mathbf f_i^e \approx 0$; see Eq. (\[eq:f\_e\]). Consequently, 