Mean relaxation rate of the dynamic structure factor of the rouse Chain

We study the internal autocorrelation function of a Gaussian coil in the free-drainage limit for chains of finite length that are much larger than the size of the monomer and obtain an exact analytic expression for the mean decay rate over the entire qR _G range, where q is the scattering vector and R _G is the radius of gyration of the coil. The mean decay rate approaches 30π/τ ₁ at the low q limit, where τ ₁ is the characteristic time of the first mode, and has the predicted q ⁴ scaling when qR _G ≫ 1. At long times, the autocorrelation function decays as a single exponential with rate 2/τ ₁ but the transition from multiexponential decay at short times to single-exponential decay at long times is very much dependent on the scattering vector.