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# The partition function

The RIS model can be treated in considerable detail. As a first step we calculate the partition function

 (1.3)

Introducing the energy (1.2) and using a short hand notation for the summations we get

 (1.4)

 (1.5)

We recognize a sequence of matrix products in Eq. (1.4), which makes it possible to write

 (1.6)

T is a matrix with elements . In the case of polyethylene  it is given by

 (1.7)

where and except for . The zero of energy has been chosen such that .

Eq. (1.6) may also be written as

 (1.8)

Moreover, using

 (1.9)

we may write

 Z = YT TN-2 X (1.10)

with and .

It is useful to decompose the matrix T in terms of its eigenvectors:

 (1.11)

 (1.12)

where is the diagonal matrix containing the eigenvalues of T. For notational convenience we have introduced B = A-1.

Eq. (1.12) may be written like

 (1.13)

where the Ai are the columns of A and the BiT the rows of B. Then
 TN = TN = (1.14)

The partition function then reads

 (1.15)

In practice we are usually interested in the free energy  per monomer

 (1.16)

where is the largest eigenvalue of T. In the limit of N going to infinity the second term goes to zero

 (1.17)

In the case of matrix (1.7) one easily calculates the eigenvectors. From one gets

 (1.18)

Next: Some probabilities Up: The Rotational Isomeric State Previous: The model
W.J. Briels