Rouse matrix

Equation (3-65), which will be derived in this appendix, represents the eigenvalues used in diagonalizing the matrix [A] of equation (3-53) 

First transform [A] into a form that will be easier to treat. To do this, note that 

In the above equation we took advantage of the fact that (p ][(p = [7], similar to that shown in equation (3-61). Now define normal coordinates 

Equation (1) requires that the corresponding elements in the matrices on both sides must be equal after the multiplication has been carried out. Thus, for each value of A , we obtain a set of linear equations as follows (the reader may wish to verify them for a 3 x 3 matrix)  

To solve the set of equations given above, we note that they all have the same form  

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R(. = r(. r. j is the end-to-end vector of Kuhn segment i and A.. is the Rouse matrix that couples the beads together through the spring force interactions. The spring force coefficient for a single spring in the sequence will be simply NK times the... 

Rouse matrix, (7.55) total absorption tensor of a macromolecule, (5.7). 

Here the normal procedure is to do the transformation from the bond vectors b. to the normal coordinates qt which will diagonalize the Rouse matrix (Agt)- As shown in Appendix 3.A, the Rouse matrix has the eigenvalues ... 

Equation (7.29) for the Rouse chain is equivalent to Eq. (6.14) for the elastic dumbbell. However, Eq. (7.29) indicates that the motions of JV — 1 bond vectors are coupled with each other through the Rouse matrix ( st). 

This equation is equivalent to Eq. (6.19) for the elastic dumbbell and is the Smoluchowski equation for the Rouse chain under the flow field K. In this equation, the Rouse matrix (Ajt) couples the motions of the bond vectors bg. Transforming to normal coordinates is the normal procedure used to decouple the motions. As shown in Chapter 3, the transformation... 

Following a procedure similar to that used in the Rouse matrix case, the normalized eigenvectors C (A = 0,1,2,..., — 1) corresponding to... 

Two other matnces are often used the Rouse matrix with elements Aj, and the Kramers matrix with elements Cj. These matrices are defined as follows ... 

The connectivity matrix (25jj — Sy+i - is called the Rouse matrix. 

In the last expression Cjk is the jk element of the Kramers matrix, which is the inverse of the Rouse matrix. 

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