Relaxation of a single knotted ring polymer

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 Ĉ_q(t) = N^-1 ∑_i∑_j(1/3)(R_i(t) · R _j(0)) exp[i2πq(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 ith 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 < A′. These exponents are similar to that found for a linear polymer chain. The topological effect appears as the separation of the power law dependences for q = 1 and q > 1, which does not appear for a linear polymer chain. In the case of the trefoil knot, the relaxation rate distribution appears to behave as λ_q ≃ A(1/N)^x for q = 1 and λ_q ≃ A′(q/N)^x′ for q = 2 and 3, where x ≃ 2.61, x′ ≃ 2:02, and A > A′. The wave number q of the slowest relaxation rate λ_q for each N is given by q = 2 for small values of N, while it is given by q = 1 for large values of N. This crossover corresponds to the change of the structure of the ring polymer caused by the localization of the knotted part to a part of the ring polymer.