the quantum Liouville equation (QLE) over the bath DOF and taking a semiclassical limit of the resulting equations. To this end, let us consider the fully quantum counterpart to the Hamiltonian in Equation (\[eq:H\_W\]): $$\hat H = \frac{\hat P^2}{2M} + \frac{\hat p^2}{2m} + V(\hat q,\hat Q)\;.
\label{eq:H}$$

The quantum statistical state of the system is described by the density matrix (or statistical operator) $\hat \rho(t)$. The time dependence of the density matrix is dictated by the QLE: $$\begin{aligned}
\frac{d}{dt}\hat\rho(t) &=& -\frac{i}{\hbar}\left[\hat H,\hat\rho(t)\right]
=-\frac{i}{\hbar}\left[\begin{array}{cc}\hat\rho & \hat H\end{array}\right]
\mbox{\boldmath$\mathcal B$}
\left[\begin{array}{c}\hat\rho \\ \hat H\end{array}\right]
\;,
\label{eq:qle}\end{aligned}$$ where $[..., ...]$ denotes the commutator, and $\mbox{\boldmath${\cal B}$}$ is the symplectic matrix [@mccauley; @goldstein]: $$\mbox{\boldmath$\cal B$}
= \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right]
\;.
\label{eq:B_matrix}$$

The average of an operator $\hat\chi$ defined on the same Hilbert space of the system is calculated by $$\langle\hat\chi\rangle(t) = {\rm Tr}\left(\hat\rho(t)\hat\chi\right) \;,
\label{eq:Trq}$$ where ${\rm Tr}$ denotes the trace operation. Now, in order to derive a classical-like description of the bath, one introduces the partial Wigner transform of the density matrix $\hat\rho$ over the $\hat X$’s: $$\begin{aligned}
\hat W(X,t) &=& \frac{1}{2\pi\hbar} \int dZ e^{iP\cdot Z/\hbar}
\langle Q-\frac{Z}{2}\arrowvert\hat\rho(t)
\arrowvert Q+\frac{Z}{2}\rangle\;.
\label{eq:partWtrho}\end{aligned}$$

The symbol $\hat W$ 