Relaxation of a Single Knotted Ring Polymer

Abstract

The relaxation of a single knotted ring polymer is studied by Brownian dynamics simulations. The relaxation rate λ q for the wave number q is estimated by the least square fit of the equilibrium time-displaced correlation function \(\hat{C}_{q}(t) = N^{-1} \sum_{i} \sum_{j}(1/3) \langle \mathbf{R}_{i} (t) \cdot \mathbf{R}_{j} (0) \rangle \exp [\mathrm{i} 2\pi q( j-i)/N]\) to a double exponential decay at long times. Here, N is the number of segments of a ring polymer and R i denotes the position of the i th segment relative to the center of mass of the polymer. The relaxation rate distribution of a single ring polymer with the trivial knot appears to behave as λ q ≃ A (1/ N ) x for q =1 and λ q ≃ A '( q / N ) x ' for q >1, where x ≃2.10, x ' ≃2.17, and A 1, which does not appear for a linear polymer chain. In the case of the trefoil knot...