quasi-particle behavior in the Mott transition is well captured, at least qualitatively, by the Brinkman-Rice theory, although it cannot describe the incoherent electronic excitations and the Hubbard bands.

At finite temperatures, the Mott transition becomes first-order with the double-occupancy showing a discontinuity $\Delta D$ at the transition point. This is demonstrated in Fig. \[fig:phase-diagram\](b) which shows the temperature dependence of the spatially averaged double-occupancy $\langle D \rangle$ serving as an order parameter of the Mott transition. In addition to a clear discontinuity in $\langle D \rangle$, the first-order nature of the MIT also manifests itself in the hysteresis loop from the heating and cooling curves.

Here we consider the evolution of the Hubbard model subject to a temperature quench. For simplicity, we assume the temperature of the system is controlled by a reservoir which only interacts with the lattice through the stochastic forces $\bm\xi_i(t)$ in the Langevin equation (\[eq:langevin\]). The system is initially equilibrated at a temperature $T_i$ in the metallic phase. At $t = 0$, the reservoir temperature is suddenly increased to a higher temperature $T_f$ across the phase boundary. Interestingly, due to the negative slope of the phase boundary $dU_c/dT < 0$ [@note-dUdT], such a thermal quench drives 