formulation will be particularly useful for MIT in multi-orbital systems, where a large number of local electron states and potential order-parameters have to be considered [@bunemann98; @lanata15; @lanata17].

Phenomenologically, the dynamics of the order-parameter field depends on the conservation laws as well as the relevance of other hydrodynamic variables [@hohenberg77]. The simplest dynamics for a non-conserved order parameter $\Phi(\mathbf r, t)$, as in the case of Mott MIT, is the time-dependent Ginzburg-Landau (TDGL) equation [@binder73; @valls90; @iwamatsu08] $$\begin{aligned}
    \label{eq:TDGL}
    \frac{\partial \Phi}{\partial t} = -\Gamma \frac{\partial \mathcal{E}}{\partial \Phi^\dagger} + \eta(\mathbf r, t).\end{aligned}$$ Here $\mathcal{E}[\Phi]$ is an effective free-energy functional, $\Gamma$ denotes the inverse damping coefficient, and $\eta(\mathbf r, t)$ is a Gaussian white noise. In the case of the kinetic Ising model with Glauber dynamics, the TDGL corresponds to the coarse-grained approximation of the mean-field master equation [@puri09], and is often used in simulations of liquid-gas or magnetic transitions with non-conserved kinetics [@binder87; @onuki02]. The stochastic $\eta$ term, which plays an important role in the nucleation process, comes from the thermal fluctuations of the microscopic variables that are “integrated” out in the above formulation. Although the TDGL has also been used in modeling the hysteresis dynamics of the MIT [@bar18], its 