We shall discuss here the central limit theorem as applied to the calculation of the distribution function of the end-to-end vector .

Consider a chain consisting of *N* independent bond vectors .
By this we mean that the probability density in configuration
space
may be written as

(3.1) |

Assume further that the bond vector probability density depends only on the length of the bond vector, and has zero mean. For the second moment we write

(3.2) |

The distribution of the end-to-end vector may be calculated according to

The central limit theorem then states that

i.e. that the end-to-end vector has Gaussian distribution.

In order to prove Eq. (3.4) we write

For

= | (3.6) |

for small values of

Combining Eqs. (3.7) and (3.8) we get Eq. ( 3.4).

Now apply the above result to a general RIS chain . To this end we write

(3.9) |

i.e. we combine bond vectors into one new vector as in Fig. ( 3.1) where . For large enough the distribution of these new vectors meets the conditions on used to derive the central limit theorem. Notice that this holds true because there are no long range excluded volume interactions in the RIS chain. Writing

(3.10) |

we find that the distribution of the end-to-end vector is again given by Eq. (3.4). We conclude that every model which does not incorporate the long range excluded volume effect will have a Gaussianly distributed end-to-end vector. It is an easy exercise to check that in this case Eq. (1.45) holds true.