MD scheme based on the Gutzwiller method, and the Hubbard model on a deformable lattice to be studied in this work. Section \[sec:thermal-mott\] discusses the finite-temperature phase diagram of the Hubbard model and outlines the Gutzwiller MD simulations of the thermal Mott transition driven by temperature quenches. In Section \[sec:nucleation\], the evolution of the cluster-size distribution from our MD simulations is discussed within the classical theory of nucleation-and-growth kinetics [@becker35; @zeldovich43; @frenkel46]. Section \[sec:interface\] presents the numerical evidence of an interface-controlled cluster-growth mechanism at the early stage of the phase transformation. Comparison of the transition kinetics with the classical Kolmogorov, Johnson, Mehl, and Avrami model [@kolmogorov37; @avrami39; @avrami40; @johnson39] is also discussed. In Section \[sec:avalanche\], we discuss the novel intermediate stage of the transformation and the associated avalanche cluster-growth behavior. Section \[sec:structure\] presents the numerical structure factor and two-point correlation function of the mixed-phase states during the Mott transition. And finally, Section \[sec:outlook\] concludes the paper with an outlook of future directions.

Gutzwiller molecular dynamics {#sec:GMD}
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We consider the half-filled Hubbard model on a compressible triangular lattice $$\begin{aligned}
    \label{eq:H}
    \mathcal{H} = \sum_{\langle ij \rangle, \,\alpha } t(\mathbf u_i - \mathbf u_j) \left(c^\dagger_{i,\alpha} c^{\;}_{j, \alpha}  +  c^\dagger_{j,\alpha} c^{\;}_{i, \alpha} \right) 
    