A closed form solution for the internal dynamics of polymer chains. I. Bonds with independent rotational potentials

An analytical solution to the master equation governing the conformational dynamics of linear polymer chains is formulated. Symmetric chains with N bonds subject to independent rotational potentials are considered. The eigenvalues of the transition rate matrix, which characterize the frequencies of the various relaxation modes, and the corresponding eigenvectors and eigenrows are obtained in closed form. A simple recurrence equation permits one to express the eigenvalues of the N-bond motion in terms of the nonzero eigenvalues associated with the isomeric transitions of single bonds. This leads to a clear understanding of the increase in conformational mobility with N.