Long time tail in the diffusion of a spherical polymer

We study the velocity autocorrelation function for the diffusion of a spherical macromolecule in solution. The diffusion is described by a generalized Langevin equation with memory character derived previously from fluctuation theory applied to the Debye-Bueche-Brinkman equation which describes the polymer-fluid interaction. The long time behaviour of the velocity autocorrelation function is obtained by establishing a low frequency expansion for the drag coefficient ζ(ω). To derive this expansion we prove a number of analyticity properties of ζ(ω). The longest lived contribution to the velocity autocorrelation function goes as t-32 as first discovered by Alder and Wainwright. We also obtain the first correction term of order t-52 which depends explicitly on the polymer structure. By use of a generalization of the Lorentz reciprocity theorem we show that the coefficient of this t-52 term is given in terms of the polymer mass and the two structure dependent coefficients that enter the static Faxén theorem for the total force on the polymer.