Appendix C

In this appendix we shall derive the Smoluchowski equation , starting from the Langevin equation  (4.27), (4.28).

Using exactly the same method as in appendix A, but now applied to $G(\vec{r};\vec{r}_{0};t)$ we arrive at

$${ \int d^3r F(\vec{r}) G(\vec{r};\vec{r}_0;t+\Delta t) = \int d^3r' \{ d^3r G(\vec{r};\vec{r}';\Delta t) \} F(\vec{r}') }$$

$$+ \sum_{\alpha} \int d^3r^{\prime}\{ \int d^3r (r_{\alpha}-r^{\pr... ...;\vec{r}_0;t) \frac{\partial}{\partial r^{\prime}_{\alpha}} F(\vec{r}^{\prime})$$

$$+\frac{1}{2} \sum_{\alpha}\sum_{\beta} \int d^3r^{\prime} \{ \int... ...\alpha})(r_{\beta}-r^{\prime}_{\beta}) G(\vec{r}; \vec{r}^{\prime};\Delta t) \}$$

$$\times G(\vec{r}^{\prime};\vec{r}_0;t) \frac{\partial^2}{ \partial r^{\prime}_{\alpha} \partial r^{\prime}_{\beta}} F(\vec{r} ^{\prime}).$$ (4.75)

Using

$$\int d^3r G(\vec{r};\vec{r}^{\prime};\Delta t)$$ = 1 (4.76)

$$\int d^3r (\vec{r}-\vec{r}^{\prime}) G(\vec{r};\vec{r} ^{\prime};\Delta t) -\frac{1}{\xi m} \vec{\nabla}^{\prime}\Phi \Delta t + \vec{\nabla}^{\prime}D \Delta t$$ = (4.77)

$$\int d^3r (\vec{r}-\vec{r}^{\prime})(\vec{r}-\vec{r}^{\prime}) G(\vec{ r};\vec{r}^{\prime};\Delta t) 2 D \mathbf{1} \Delta t$$ = (4.78)

which follow from Eqs. (4.27) and (4.28), and following the same strategy as in appendix A we arrive at

$${ \frac{\partial}{\partial t} G(\vec{r};\vec{r}_0;t) = \vec{\nabla} \cdot \frac{1}{\xi m} G(\vec{r};\vec{r}_0;t) \vec{\nabla} \Phi (\vec{r}) }$$

$$-\vec{\nabla}\cdot G(\vec{r};\vec{r}_0;t)\vec{\nabla} D + \nabla^2 D G( \vec{r};\vec{r}_0;t)$$ (4.79)

It is not difficult to show that the last two terms together yield $\vec{ \nabla}\cdot D \vec{\nabla} G(\vec{r};\vec{r}_0;t)$, which proves the equivalence of Eqs. (4.27), (4.28) and Eqs. (4.32), (4.33).