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 

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

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

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

$${F(\vec{z})=F(\vec{z}^{\prime })+\sum_{\alpha }(z_{\alpha }-z_{\a... ...}^{\prime })\frac{\partial }{\partial z\prime _{\alpha }}F(\vec{z }^{\prime })}$$

$$+\frac{1}{2}\sum_{\alpha }\sum_{\beta }(z_{\alpha }-z_{\alpha }^{... ...artial z_{\alpha }^{\prime }\partial z_{\beta }^{\prime }}F(\vec{z}^{\prime }).$$ (4.42)

Introducing this into Eq. (4.41) we get

 

$${\int d^{6}zF(\vec{z})\mathcal{G}(\vec{z};\vec{z}_{0};t+\Delta t) }$$

$$\int d^{6}z^{\prime }\left\{ \int d^{6}z\mathcal{G}(\vec{z};\vec{... ...ta t)\right\} \mathcal{G}(\vec{z}^{\prime };\vec{z} _{0};t)F(\vec{z}^{\prime })$$ =

$$+\sum_{\alpha }\int d^{6}z^{\prime }\left\{ \int d^{6}z(z_{\alpha... ...{z}_{0};t)\frac{\partial }{ \partial z_{\alpha }^{\prime }}F(\vec{z}^{\prime })$$

$$+\frac{1}{2}\sum_{\alpha }\sum_{\beta }\int d^{6}z^{\prime }\left... ...}-z_{\beta }^{\prime }) \mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t)\right\}$$

$$\times \mathcal{G}(\vec{z}^{\prime };\vec{z}_{0};t)\frac{\partial... ...rtial z_{\alpha }^{\prime }\partial z_{\beta }^{\prime }}F(\vec{z} ^{\prime }).$$ (4.43)

Now we make use of

 

$$\int d^{6}z\mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t)$$ = 1 (4.44)

$$\int d^{6}z(\vec{r}-\vec{r}^{\prime })\mathcal{G}(\vec{z};\vec{z} ^{\prime };\Delta t) \vec{v}^{\prime }\Delta t$$ = (4.45)

$$\int d^{6}z(\vec{v}-\vec{v}^{\prime })\mathcal{G}(\vec{z};\vec{z} ^{\prime };\Delta t) -\xi \vec{v}^{\prime }\Delta t$$ = (4.46)

$$\int d^{6}z(\vec{r}-\vec{r}^{\prime })(\vec{r}-\vec{r}^{\prime }) \mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t)$$ = 0 (4.47)

$$\int d^{6}z(\vec{r}-\vec{r}^{\prime })(\vec{v}-\vec{v}^{\prime }) \mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t)$$ = 0 (4.48)

$$\int d^{6}z(\vec{v}-\vec{v}^{\prime })(\vec{v}-\vec{v}^{\prime }) \mathcal{G}(\vec{z};\vec{z}^{\prime };\Delta t) 2\frac{kT}{m}\xi \mathbf{1}\Delta t$$ = (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

$${\int d^{6}zF(\vec{z})\frac{\partial }{\partial t}\mathcal{G}( \vec{z};\vec{z}_{0};t)}$$

$$\int d^{6}z^{\prime }\vec{v}^{\prime }\mathcal{G}(\vec{z}^{\prime... ...{\vec{r}^{\prime }}-\xi \vec{\nabla}_{ \vec{v}^{\prime }}\}F(\vec{z}^{\prime })$$ =

$$+\int d^{6}z^{\prime }\frac{kT}{m}\xi \mathcal{G}(\vec{z}^{\prime }; \vec{z}_{0};t)\nabla _{\vec{v}^{\prime }}^{2}F(\vec{z}^{\prime }).$$ (4.50)

Next we change the integration variable $\vec{z}^{\prime }$ into $\vec{z}$and perform some partial integrations, obtaining

$${\int d^{6}zF(\vec{z})\frac{\partial }{\partial t}\mathcal{G}( \vec{z};\vec{z}_{0};t)}$$

$$\int d^{6}zF(\vec{z})\{\vec{\nabla}_{\vec{v}}\xi -\vec{\nabla}_{\vec{r }}\}\cdot (\vec{v}\mathcal{G}(\vec{z};\vec{z}_{0};t))$$ =

$$+\int d^{6}zF(\vec{z})\nabla _{\vec{v}}^{2}\frac{kT}{m}\xi \mathcal{G}( \vec{r};\vec{r}_{0};t).$$ (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.