Appendix C

We shall first show that for small n we may write

$$G(\vec{R},\vec{R}^{\prime};n) = \exp \{-n\beta\Phi (\vec{R}) ... ...}} \exp \{ -\frac{3}{2nb^2} (\vec{R}- \vec{R}^{\prime})^2 \} .$$ (3.50)

Taking the limit for $n \rightarrow 0$ we then easily check Eq. (3.16). In order to prove Eq. (3.50) we first notice that n being small, all R_i in the exponent in Eq. (3.13) are approximately equal to $\vec{R}$, from which it follows that

$${ G(\vec{R},\vec{R}';n) = \exp\{-n\beta\Phi (\vec{R})\} \left\{ \frac{3}{2\pi b^2} \right\}^{\frac{3}{2}n} }$$

$$\int d^3R_1 \cdots \int d^3R_{n-1} \exp \left\{ - \sum_{i=1}^{n} \frac{3}{2b^2}(\vec{R}_i -\vec{R}_{i-1})^2 \right\} .$$ (3.51)

It is now simply a matter of successively performing the integrals. The simplest way to do this is by induction.

$${ \left\{\frac{3}{2\pi b^2}\right\}^{\frac{3}{2}n} \int d^3R_1 \c... ...3R_{n-1} \exp\{-\sum_{i=1}^n \frac{3}{2b^2} (\vec{R}_i -\vec{R} _{i-1} )^2 \} }$$

$$= \left\{\frac{3}{2\pi b^2}\right\}^{\frac{3}{2}} \left\{ \frac{3}{2\pi (n-1)b^2} \right\}^{\frac{3}{2}} \int d^3R_{n-1}$$

$$\exp \left\{ -\frac{3}{2(n-1)b^2} (\vec{R}_{n-1}-\vec{R}^{\prime})^2 - \frac{3}{2b^2} (\vec{R}_{n-1}-\vec{R})^2 \right\}$$

$$= \left\{ \frac{3}{2\pi nb^2} \right\}^{\frac{3}{2}} \exp \left\{ - \frac{3}{2nb^2} (\vec{R}-\vec{R}^{\prime})^2 \right\} .$$ (3.52)

The first equals sign is the induction step, and the second one is the final proof.

Now look at the Chapman-Kolmogorov equation  (3.14), and take n to be small. Then $G( \vec{R},\vec{R}^{\prime \prime };n)$ differs from zero only if $\vec{R} ^{\prime \prime }$ is in the neighbourhood of $\vec{R}$. We therefore expand the second factor $G(\vec{R}^{\prime \prime },\vec{R}^{\prime };N)$ in Eq. (3.14) with respect to $\vec{R} ^{\prime \prime }$.

 

$$G(\vec{R}^{\prime \prime },\vec{R}^{\prime };N)=G(\vec{R},\vec{R} ^{\prime };N)$$

$$+\sum_{\alpha }(R_{\alpha }^{\prime \prime }-R_{\alpha })\frac{\partial }{\partial R_{\alpha }}G(\vec{R},\vec{R}^{\prime };N)$$

$$+\frac{1}{2}\sum_{\alpha }\sum_{\beta }(R_{\alpha }^{\prime \prim... ...l ^{2}}{ \partial R_{\alpha }\partial R_{\beta }}G(\vec{R},\vec{R}^{\prime };N)$$ (3.53)

where $R_{\alpha }$ denotes the $\alpha$'th Cartesian component of $\vec{R}$. Introducing Eqs. (3.50) and (3.53) into ( 3.14) we get

$$G(\vec{R},\vec{R}^{\prime };N+n)=\exp \{-n\beta \Phi (\vec{R}... ...{6}nb^{2}\nabla _{\vec{R}}^{2}\}G(\vec{R},\vec{R}^{\prime };N)$$ (3.54)

where we have used

$$\left\{ \frac{3}{2\pi nb^{2}}\right\} ^{\frac{3}{2}}\int d^{3}R^{... ...eta })\exp \left\{ -\frac{3}{2nb^{2}}(\vec{R}-\vec{R}^{\prime \prime })\right\}$$

$$=\frac{1}{3}nb^{2}\delta _{\alpha \beta }$$ (3.55)

For small n, we expand the exponent

$$G(\vec{R},\vec{R}^{\prime };N+n)=\{1-n\beta \Phi (\vec{R})\}\... ...} nb^{2}\nabla _{\vec{R}}^{2}\}G(\vec{R},\vec{R}^{\prime };N).$$ (3.56)

Collecting terms of order n, and using

$$\frac{\partial }{\partial N}G(\vec{R},\vec{R}^{\prime };N)=\l... ...ec{R},\vec{R}^{\prime };N+n)-G( \vec{R},\vec{R}^{\prime };N)\}$$ (3.57)

we finally arrive at Eq. (3.15).