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# Some probabilities

It is instructive to calculate some probabilities occurring in the RIS model. An important probability for computer simulations is the conditional probability  defined below. We shall always assume that the chain is infinitely long, and that end effects may be neglected. In this case is the only eigenvalue we need.

We define the probability that a given bond is in state by

 (1.19)

Explicitly introducing the averaging procedure yields
 (1.20)

Using the same method as in the previous section we find
 = = = (1.21)

Dividing numerator and denominator by , and assuming N, i, and N-i are large we get

 (1.22)

Because here and in the remaining part of the chapter we only need and the corresponding vectors Amax and B maxT, we shall omit the subscript ''max'' and write

 (1.23)

Similarly one calculates

 = = (1.24)

An important quantity is , the conditional probability to find bond i in state , given that bond i-1 is in state

 (1.25)

Introducing Eqs. (1.23) and (1.24) we get

 (1.26)

This quantity may be used to generate chain conformations on a computer.

Next: The mean square end-to-end Up: The Rotational Isomeric State Previous: The partition function
W.J. Briels