Mean Square Displacement Short-Time Behavior Between a Pair of Monomers

2 Mean Square Displacement Short-Time Behavior Between a Pair of Monomers Before considering the mean square displacement of the same beads (monomers) in Section 3.4.9.3, we look at the evolution of for a pair of 

There is a distinct difference between a different pair (m + n) and the same pair (m = n self-diffusion of each bead). For the same pair, ([r (0] = 6Dq(N + l)r = 6 k T/C)t. It means that each bead moves freely with its own friction coefficient as if the other beads were absent or not connected. For a pair of different beads, the short-time mean square displacement increases as 6Dot, the same as the center of mass diffusion. Different beads are uncorrelated. 

When n = m, cosilinir/N) is a rapidly changing function of i. The sum will be much smaller compared with the first term, Thus, 

When n 9 m, the sum is dominated with the first term because cos[i(n + my-jr/N] changes between positive and negative more rapidly compared with cos[i( - myir/N]. From Eq. A3.3 in Appendix A3, we have 

The hydrodynamic interactions allow the distance between a nearby pair of beads to grow more rapidly compared with a distant pair. For the latter, the short-time mean square displacement increases as in the Rouse model. 

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