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
are fixed and that all
may randomly take a value in .
In order to treat this chain mathematically we introduce the probability
density in configuration space
and define

So is the probability density to find the first

(3.36) |

In our case

(3.37) |

i.e. the conditional probability to have vector

(3.38) |

We may now easily calculate all quantities of interest.

We first notice, that in the present case

(3.39) |

Repeatedly using this equation we calculate

= | |||

= | (3.40) |

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

= | (3.41) | ||

R_{g}^{2} |
= | (3.42) |

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

(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

(3.44) |

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