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# The central limit theorem

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 (3.3)

The central limit theorem then states that (3.4)

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

In order to prove Eq. (3.4) we write = = = (3.5)

For k=0, the Fourier transform of will be equal to one. Because has zero mean and finite second moment, the Fourier transform of will have its maximum around k=0 and go to zero for large values of k. Raising such a function to the N'th power leaves us with a function that differs from zero only very close to the origin, and which may be approximated by     = (3.6)

for small values of k, and by zero for the values of k where is negative. This again may be approximated by for all values of k. Then = = I(Rx) I(Ry) I(Rz) (3.7)

 I(Rx) = = (3.8)

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.      Next: The Gaussian chain Up: The Gaussian chain Previous: Simple models
W.J. Briels