the lattice system becomes essentially a collection of free harmonic oscillator, and both energy densities $E_l$ and $E_k$ approach $k_B T_f$ consistent with the equipartition theorem.

Nucleation and growth {#sec:nucleation}
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To further investigate the nature of the phase transformation, Fig. \[fig:snapshots\] shows the spatial profile of the local double-occupancy $D(\mathbf r_i) = \langle n_{i, \uparrow} n_{i, \downarrow} \rangle$ at four different times during the relaxation. These snapshots clearly show that the transition to the new phase is initiated by the nucleation of Mott insulating puddles, i.e. regions with vanishing double-occupancy, in the metallic matrix. Importantly, the double-occupation in the metallic regions remains roughly the same. This observation also means that the gradual decrease of the average $\langle D \rangle$ in Fig. \[fig:trace\](b) results mostly from the increasing volume fraction of the Mott phase in a heterogeneous electronic state, instead of the decrease of a a homogeneously distributed double-occupancy.

This scenario is further confirmed by the histogram or probability distribution function $h(D)$ of the double-occupancy, shown in FIg. \[fig:hist\_db\](a), at various Mott volume fractions. The distribution function consists mainly of a delta-function at $D = 0$ corresponding to the Mott clusters, and a broad hump centered at $D \approx 0.055$, 