I have the following Fokker-Planck equation:
$$\frac{\partial\psi}{\partial t} + \nabla_{r_1} \cdot \left[ u(r_1,t)\psi + \frac{1}{\zeta} \textbf{F} (r_2 - r_1) \psi \right] + \nabla_{r_2} \cdot \left[ u(r_2,t)\psi + \frac{1}{\zeta} \textbf{F} (r_1 - r_2) \psi \right] = \frac{k_BT}{\zeta} \nabla^2_{r_1} \psi + \frac{k_BT}{\zeta} \nabla^2_{r_2} \psi$$
changing variables as:
$$x(t) = \frac{r_1(t)+r_2(t)}{2}\,\,\,\mbox{and}\,\,\,q(t) = r_2(t)+r_1(t);$$
I should get:
$$\frac{\partial\psi}{\partial t} + \nabla_{q} \cdot \left( \left[ u(x+\frac{q}{2},t)-u(x-\frac{q}{2},t)\right]\psi - \frac{2}{\zeta} \textbf{F} (q) \psi \right) + \nabla_{x}\cdot \left( \frac{u(x-\frac{q}{2},t)+u(x+\frac{q}{2},t)}{2} \psi \right) = \frac{k_BT}{2\zeta} \nabla^2_{x} \psi + \frac{2k_BT}{\zeta} \nabla^2_{q} \psi$$
how do operators change? Could anyone show step-by-step?