A rough measure of the average size of the polymer is given by the mean square end-to-end vector , which we shall calculate in this section. Related properties are the radius of gyration , and the persistence length . Both of them may be calculated using methods similar to the ones in this section.

The end-to-end vector is given by

(1.27) |

The mean square then reads

= | |||

= | |||

= | (1.28) |

Again assuming the chain is infinitely long, we may put independent on

where (

We now set forth to calculate
.
In order to do so we need to calculate the scalar product
as a function of the angles
.
To this end we associate with every monomer *i* a
Cartesian coordinate system
.
Every vector may then be expanded like

The precise definition of the local coordinate system is given in Appendix A. Here we only mention that

= | |||

= | (1.31) |

A particular example of Eq. (1.30) is

(1.32) |

The matrix is calculated in Appendix B.

The scalar product
now reads

= | |||

= | |||

= | (1.33) |

and in general

(1.34) |

from which we get

We finally calculate the remaining average using the methods of the last section

(1.36) |

where again we have omitted the subscript ''max''. We may write this in a
concise form like

(1.37) |

where

(1.38) |

S |
= | ||

= | |||

= | (1.39) |

Introducing everything into Eq. (1.35) we get

(1.40) |

For infinitely long chains we can analytically sum, obtaining

where

Similar equations, but much more complicated, may be derived for . For these and other equations we refer to P.J. Flory, Statistical Mechanics of Chain Molecules .