Matrix algebra and eigenvalues for the bead/spring model of polymer solutions

Some twenty years ago, Zimm [1] 1 formulated ~ linear, second-order partial differential eq uation for a distribution function IjJ which depends on time and ~N + 1) coordinates Xo. Yo. Zoo .. , , XN, YN, Z N, of the N + 1 beads, for modeling the bulk behavior of very dilute polymer s0lutions under the influe nce of external force, Brownian motion, and hydrodynamic interaction among the beads of the neckJace model. The mechanical model for each polymer molec ule is that of a c hain of N identical, ideally elastic segments joining N + 1 identical beads with comple te flexibility at each bead. Two length parameters are of interest in thi s model: ah, the so-called hydrodynamic radius of the bead, and bo, the root mean square of the segment length. The ratio a of the two length parameters (a = a,/ bo), and the number N of elastic segments completely charac terize the mathematical problem