Appendix A

In the case of a freely rotating chain  it is rather easy to calculate the mean square end-to-end vector, the radius of gyration and the persistence length.

The freely rotating chain is defined by saying that the angles $\vartheta _{i}$are fixed and that all $\varphi _{i}$ may randomly take a value in $[0,2\pi]$. In order to treat this chain mathematically we introduce the probability density in configuration space $P_N ( \vec{r}_1,\ldots,\vec{r}_N)$ and define

  

$$P_n (\vec{r}_1,\ldots,\vec{r}_n) \int d^3r_{n+1} \ldots \int d^3r_N P_N(\vec{r}_1,\ldots,\vec{r}_N)$$ = (3.34)

$$P_n (\vec{r}_1,\ldots,\vec{r}_n) Q(\vec{r}_n;\vec{r}_1,\ldots,\vec{r} _{n-1}) P_{n-1}(\vec{r}_1,\ldots,\vec{r}_{n-1}) .$$ = (3.35)

So $P_n(\vec{r}_1,\ldots,\vec{r}_n)$ is the probability density to find the first n bond vectors in the configuration $\vec{r}_1,\ldots,\vec{r}_n$, and $Q(\vec{r}_n;\vec{r}_1,\ldots,\vec{r}_{n-1})$ is the conditional probability density to find bond vector n in state $\vec{r}_n$, given that the preceding vectors are in configuration $\vec{r}_1,\ldots,\vec{r}_{n-1}$. Integrating Eq. (3.35) over $\vec{r}_n$ and using Eq. (3.34) we obtain

$$\int d^3r_n Q(\vec{r}_n;\vec{r}_1,\ldots,\vec{r}_{n-1}) =1.$$ (3.36)

In our case

$$Q(\vec{r}_n;\vec{r}_1,\ldots,\vec{r}_{n-1}) = q(\vec{r}_n;\vec{r}_{n-1})$$ (3.37)

i.e. the conditional probability to have vector n in state $\vec{r}_n$depends only on the state of vector n-1. In mathematics it is said that the chain has the Markov property. Using the definitions given so far we may write

$$P_n (\vec{r}_1,\ldots,\vec{r}_n) = q(\vec{r}_n;\vec{r}_{n-1})... ...};\vec{r}_{n-2}) \ldots q(\vec{r}_2;\vec{r}_1) P_1(\vec{r}_1).$$ (3.38)

We may now easily calculate all quantities of interest.

We first notice, that in the present case

$$\int d^{3}r_{n}\vec{r}_{n}q(\vec{r}_{n};\vec{r}_{n-1})=\cos \vartheta \vec{r}_{n-1}.$$ (3.39)

Repeatedly using this equation we calculate

$${\langle \vec{r}_{i}\cdot \vec{r}_{i+n}\rangle }$$

$$\int d^{3}r_{1}\ldots \int d^{3}r_{i+n}\vec{r}_{i}\cdot \vec{r}_{... ...( \vec{r}_{i+n};\vec{r}_{i+n-1})\ldots q(\vec{r}_{2};\vec{r}_{1})P(\vec{r}_{1})$$ =

$$l^{2}(\cos \vartheta )^{n}.$$ = (3.40)

Using this in Eqs. (1.29) and (1.44) we get for large N  :

$$\langle R^2 \rangle Nl^{2} \frac{1+\cos \vartheta }{1-\cos \vartheta }$$ = (3.41)

$$\frac{1}{6}Nl^{2}\frac{1+\cos \vartheta }{1-\cos \vartheta }.$$ R_g² = (3.42)

Another quantity characteristic of the chain is the persistence length . It is defined by

$$a=\sum_{i=1}^{N}\langle \vec{r}_{i}\cdot \hat{r}_{1}\rangle ... ...ft( \sum_{i=1}^{N}\vec{r} _{i}\right) \cdot \hat{r}_{1}\rangle$$ (3.43)

and it gives the average of the projection of the end-to-end vector on the direction of the first bond. For large N we may write Eq. (1.29 ) like $\langle R^{2}\rangle =Nl^{2}+2N(la-l^{2})$ from which we get

$$a=\frac{1}{2}(l+\frac{\langle R^{2}\rangle }{Nl})=l\frac{1}{1-\cos \vartheta }.$$ (3.44)

We see that for a rod the persistence length is infinite.