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Next: Appendix B Up: Stochastic processes Previous: The Smoluchowski equation

Appendix A

We shall derive the Fokker-Planck equation  by looking at $\int d^{6}zF(
\vec{z})\mathcal{G}(\vec{z};\vec{z}_{0};t)$ for any function $F(\vec{z})$. Because we are always interested in averages like this, equations that may be derived using this object are all we need. (In mathematical terms $
\mathcal{G}(\vec{z},\vec{z}_{0};t)$ is a distribution or generalized function, not an ordinary function).

Our proof very much resembles the one in Appendix 3.C. Our starting point is again the Chapman-Kolmogorov equation 

\vec{z};\vec{z}_{0};t+\Delta t)=\int d^{6}z^{\pr... };\Delta t)\mathcal{G}(\vec{z}^{\prime };\vec{z}_{0};t).
\end{displaymath} (4.40)

Multiplying by $F(\vec{z})$ and integrating yields

 \begin{displaymath}\int d^{6}zF(\vec{z})\mathcal{G}(\vec{z};\vec{z}_{0};t+\Delta... };\Delta t)\mathcal{G}(\vec{z}^{\prime };\vec{z}_{0};t).
\end{displaymath} (4.41)

Now we shall perform the integral with respect to z on the right hand side. Because $\mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t)$ differs from zero only when $\vec{z}$ is in the neighbourhood of $\vec{z}^{\prime
} $, we expand $F(\vec{z})$ around $\vec{z}^{\prime
} $
$\displaystyle {F(\vec{z})=F(\vec{z}^{\prime })+\sum_{\alpha }(z_{\alpha
...}^{\prime })\frac{\partial }{\partial z\prime _{\alpha }}F(\vec{z
}^{\prime })}$
    $\displaystyle +\frac{1}{2}\sum_{\alpha }\sum_{\beta }(z_{\alpha }-z_{\alpha }^{...
...artial z_{\alpha
}^{\prime }\partial z_{\beta }^{\prime }}F(\vec{z}^{\prime }).$ (4.42)

Introducing this into Eq. (4.41) we get
$\displaystyle {\int d^{6}zF(\vec{z})\mathcal{G}(\vec{z};\vec{z}_{0};t+\Delta t)
  = $\displaystyle \int d^{6}z^{\prime }\left\{ \int d^{6}z\mathcal{G}(\vec{z};\vec{...
...ta t)\right\} \mathcal{G}(\vec{z}^{\prime };\vec{z}
_{0};t)F(\vec{z}^{\prime })$  
    $\displaystyle +\sum_{\alpha }\int d^{6}z^{\prime }\left\{ \int d^{6}z(z_{\alpha...
...{z}_{0};t)\frac{\partial }{
\partial z_{\alpha }^{\prime }}F(\vec{z}^{\prime })$  
    $\displaystyle +\frac{1}{2}\sum_{\alpha }\sum_{\beta }\int d^{6}z^{\prime }\left...
...}-z_{\beta }^{\prime })
\mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t)\right\}$  
    $\displaystyle \times \mathcal{G}(\vec{z}^{\prime };\vec{z}_{0};t)\frac{\partial...
...rtial z_{\alpha }^{\prime }\partial z_{\beta }^{\prime }}F(\vec{z}
^{\prime }).$ (4.43)

Now we make use of
$\displaystyle \int d^{6}z\mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t)$ = 1 (4.44)
$\displaystyle \int d^{6}z(\vec{r}-\vec{r}^{\prime })\mathcal{G}(\vec{z};\vec{z}
^{\prime };\Delta t)$ = $\displaystyle \vec{v}^{\prime }\Delta t$ (4.45)
$\displaystyle \int d^{6}z(\vec{v}-\vec{v}^{\prime })\mathcal{G}(\vec{z};\vec{z}
^{\prime };\Delta t)$ = $\displaystyle -\xi \vec{v}^{\prime }\Delta t$ (4.46)
$\displaystyle \int d^{6}z(\vec{r}-\vec{r}^{\prime })(\vec{r}-\vec{r}^{\prime })
\mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t)$ = 0 (4.47)
$\displaystyle \int d^{6}z(\vec{r}-\vec{r}^{\prime })(\vec{v}-\vec{v}^{\prime })
\mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t)$ = 0 (4.48)
$\displaystyle \int d^{6}z(\vec{v}-\vec{v}^{\prime })(\vec{v}-\vec{v}^{\prime })
\mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t)$ = $\displaystyle 2\frac{kT}{m}\xi
\mathbf{1}\Delta t$ (4.49)

which hold true to the first order in $\Delta t$. The first three of these equations are obvious. The last three easily follow from the Langevin equation  in section 4.1 together with the fluctuation-dissipation theorem  Eq. (4.21). 1 in Eq. (4.49) denotes the 3-dimensional unit matrix. Introducing everything into Eq. (4.43), dividing by $\Delta t$and taking the limit $\Delta t\rightarrow 0$, we get
$\displaystyle {\int d^{6}zF(\vec{z})\frac{\partial }{\partial t}\mathcal{G}(
  = $\displaystyle \int d^{6}z^{\prime }\vec{v}^{\prime }\mathcal{G}(\vec{z}^{\prime...
...{\vec{r}^{\prime }}-\xi \vec{\nabla}_{
\vec{v}^{\prime }}\}F(\vec{z}^{\prime })$  
    $\displaystyle +\int d^{6}z^{\prime }\frac{kT}{m}\xi \mathcal{G}(\vec{z}^{\prime };
\vec{z}_{0};t)\nabla _{\vec{v}^{\prime }}^{2}F(\vec{z}^{\prime }).$ (4.50)

Next we change the integration variable $\vec{z}^{\prime
} $ into $\vec{z}$and perform some partial integrations, obtaining
$\displaystyle {\int d^{6}zF(\vec{z})\frac{\partial }{\partial t}\mathcal{G}(
  = $\displaystyle \int d^{6}zF(\vec{z})\{\vec{\nabla}_{\vec{v}}\xi -\vec{\nabla}_{\vec{r
}}\}\cdot (\vec{v}\mathcal{G}(\vec{z};\vec{z}_{0};t))$  
    $\displaystyle +\int d^{6}zF(\vec{z})\nabla _{\vec{v}}^{2}\frac{kT}{m}\xi \mathcal{G}(
\vec{r};\vec{r}_{0};t).$ (4.51)

Because this has to hold true for all possible $F(\vec{z})$ we conclude that the Fokker-Planck equation (4.14) must hold true.

next up previous contents index
Next: Appendix B Up: Stochastic processes Previous: The Smoluchowski equation
W.J. Briels