The conditional evolution equation includes the wave function $\Psi$ on the full configuration space in the potentials terms $A$ and $B$, or more precisely in the ratios $\psi'/\psi$ and $\psi''/\psi$. We define

\begin{equation} \phi_i(t,x) = \left. \frac{\nabla^i_y\Psi(t,x,y)}{\Psi(t,x,y)} \right|_{y=Y(t)} \end{equation}

and thus have $\phi_1 = \psi'/\psi$, $\phi_2 = \psi''/\psi$ and so on. Note that this potential quantities with $i$ even are scalar valued while those with $i$ odd are 3d vector valued. To get a general evolution equation for the potentials $\phi_i$ we take the time derivative again obeying the chain rule for the time dependent Bohmian trajectory $Y(t)$ that substitutes the coordinate $y$.

\begin{equation} \partial_t \phi_i = \left. \partial_t \frac{\nabla^i_y \Psi}{\Psi} \right|_{y=Y(t)} + \dot{Y}(t) \cdot\left. \nabla_y \frac{\nabla^i_y \Psi}{\Psi} \right|_{y=Y(t)} \end{equation}

A calculation using the Schrödinger equation with external potential $V$ and which includes the general product rule at one point yields the following lengthy expression. \begin{equation} \begin{aligned} \mathrm{i}\hbar\partial_t \phi_i = -\frac{\hbar^2}{2m} \left( \Delta_x\phi_i + 2\nabla_x\phi_i \cdot \frac{\nabla_x\psi}{\psi} + \phi_{i+2}-\phi_i \phi_2\right) + \sum_{k=1}^i {i \choose k} \left.\left( \nabla_y^k V \right)\right|_{y=Y(t)} \phi_{i-k} + \mathrm{i}\hbar \dot{Y}(t) (\phi_{i+1}-\phi_i\phi_1) \end{aligned} \end{equation}

Note that we define $\phi_0=1$. The sum index really starts at 1 (instead of 0) because the 0th term has been cancelled. This results fits to the one given in ^[1] (35-36) for $i=1$ and $i=2$.