the system from the correlated metal to a Mott insulator. Within the BO approximation, the electrons are assumed to quickly reach equilibrium at the instantaneous lattice temperature, i.e. $k_B T_e(t) = m \langle \dot{\mathbf u}(t)^2 \rangle /3$. The electron temperature thus increases continuously from $T_i$ to $T_f$ as the lattice is heated up by the reservoir.

![(Color online) \[fig:trace\] (a) The various energy densities versus time in a MIT driven by a temperature quench. Here $E_e$, $E_l$, and $E_k$ denote the electronic, lattice (elastic), and atomic kinetic energies, respectively. The initial and final temperatures are $T_i = 0.09$ and $T_f = 0.13$, respectively. The (red) horizontal dashed line corresponds to $k_B T_f$. (b) The averaged double occupation $\langle D \rangle$ and the volume fraction of the Mott phase $\phi_M$ versus time. The shaded area bounded between $t_1$ and $t_2$ indicates the region where most of the phase transformation takes place. ](MIT-trace.pdf){width="0.99\columnwidth"}

The time dependence of the various energy densities is shown in Fig. \[fig:trace\](a) for a quench from $T_i = 0.09$ to $T_f = 0.13$. The electronic energy $E_{\rm elec}$ includes both the hopping and Hubbard term. The lattice energy $E_l$ comes from the elastic $K$ term in the Hamiltonian. 