Anomalous diffusion of distinguished particles in bead-spring networks

Abstract

We consider the anomalous sub-diffusion of a class of Gaussian processes that can be expressed in terms of sums of Ornstein-Uhlenbeck processes. As a generic class of processes, we introduce a single parameter such that for any $\nu \in (0,1)$ the process can be tuned to produce a mean-squared displacement with $\E{x^2(t)} \sim t^\nu$ for large $t$. The motivation for the specific structure of these sums of OU processes comes from the Rouse chain model from polymer kinetic theory. We generalize the model by studying the general dynamics of individual particles in networks of thermally fluctuating beads connected by Hookean springs. Such a set-up is similar to the study of Kac-Zwanzig heat bath models. Whereas the existing heat bath literature places its assumptions on the spectrum of the Laplacian matrix associated to the spring connection graph, we study explicit graph structures. In this setting we prove a notion of universality for the Rouse chain's well-known $\E{x^2(t)} \sim t^{1/2}$ scaling behavior. Subsequently we demonstrate the existence of other anomalous behavior by changing the dimension of the connection graph or by allowing repulsive forces among the beads.