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# 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 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

 = (3.34) = (3.35)

So is the probability density to find the first n bond vectors in the configuration , and is the conditional probability density to find bond vector n in state , given that the preceding vectors are in configuration . Integrating Eq. (3.35) over and using Eq. (3.34) we obtain

 (3.36)

In our case

 (3.37)

i.e. the conditional probability to have vector n in state 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

 (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 N  :
 = (3.41) Rg2 = (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 N we may write Eq. (1.29 ) like from which we get

 (3.44)

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

Next: Appendix B Up: The Gaussian chain Previous: One Gaussian chain in
W.J. Briels