Dynamics Solved by the Three-Point Formula: Exact Analytical Results for Rings

In this paper, we study in the framework of the Gaussian model, the relaxation dynamics, and diffusion process on structures which show a ring-shape geometry. In order to extend the classical connectivity matrix to include interactions between more distant nearest neighbors, we treat the second derivative with respect to position by using the three-point formula. For this new Laplacian matrix, we determine analytical solutions to the eigenvalue problem. The relaxation dynamics is described by the mechanical relaxation moduli and for diffusion we focus on the behavior of the residual concentration at the initial node. Additionally, we investigate the scaling behaviors of the mean squared radius of gyration and of the smallest eigenvalue. To calculate the residual concentration, we consider that initially the whole material is concentrated only in one node and then it spreads over the ring. We compare our results with the ones obtained from the incremental ratio method. We observe that the results of the two methods for the considered quantities are slightly different. At any intermediate time/frequency domain, the results obtained by using the incremental ratio method underestimate the ones obtained by using the three-point formula. This finding can turn important for many applications in polymer systems or in other systems where diffusive motion occurs.