Universalities in a constrained motion of a particle with memory friction: A bead of a Rouse chain on a periodic wire

In our previous work , we considered the mean-squared displacement (MSD) of particles whose free motion is subdiffusive and is described by a generalized Langevin equation (GLE) with power-law memory kernel under geometric constraints, e.g., when the particles are moving in channels of different shape. The MSD at long times was found to be independent of the channel's shape. Motivated by this finding, we discuss a simple physical model corresponding to such a situation. Here the particle is a tagged end monomer of a Rouse polymer chain, and the bath is represented by the rest of the chain and its surroundings. The constraints correspond to the motion of a monomer on a static, sinusoidally meandering wire. For the case of long chains we find the same independence of the long-time behavior on the wire's modulation as in the case of GLE: The MSD at longer times does not depend on the amplitude of the wire's modulation and is the same as for the straight wire. For shorter chains whose Rouse time lays in the observation domain, for moderate modulation, if the MSD already approached the one for a straight wire at times smaller or comparable to the Rouse time, the independence on the modulation is carried over to the final, diffusion regime at long times. In the case when constraints apply not only to the tagged monomer, but also to the rest of the chain (i.e., to the bath), the sensitivity to the wire's shape never disappears. A simple analytical approach based on homogenization gives a quite satisfactory description of our findings.