many DOF, described by many canonically conjugate phase space coordinates, $X=(Q, P)$. We will assume that the Hamiltonian of this hybrid system has the form $$\begin{aligned}
\hat H_{\rm W}(X)
&=&\frac{P^2}{2M} + \frac{\hat p^2}{2m} + V_{\rm W}(\hat q,Q)
\nonumber\\
&=&\frac{P^2}{2M}+\hat h_{\rm W}(Q) \;,
\label{eq:H_W}\end{aligned}$$ where $m$ and $M$ are the masses of the subsystem and bath DOF, respectively, and $V_{\rm W}$ is the potential energy describing the interactions among the subsystem DOF, among the bath DOF, and between these two sets of DOF. The last equality on the right-hand side of Equation (\[eq:H\_W\]) defines the adiabatic Hamiltonian $\hat h_{\rm W}(Q)$ of the system. [It has been known for many years that the statistical mechanics of such hybrid quantum-classical systems may be formulated in terms of an operator-valued quasi-probability function $\hat W(X,t)$]{} [@silin; @rukhazade-silin; @balescu-1968; @zhang-balescu; @balescu-zhang; @osborn]. Specifically, the statistical average of hybrid quantum-classical operators, representing a dynamical property of the system, may be calculated according to $$\langle\hat\chi\rangle(t)={\rm Tr'}\int dX
\hat W(X,t)\hat\chi_{\rm W}(X) \;,
\label{eq:TrW}$$ where ${\rm Tr}'$ denotes the partial trace involving a complete set of states of the quantum subsystem.

The operator-valued quasi-probability function in phase space evolves according to $$\frac{\partial}{\partial t}\hat W(X,t)
=-\frac{i}{\hbar}
\left[\begin{array}{cc} \hat H_{\rm W} & 