Some remarks on the equivalence of Kirkwood’s diffusion equation and the coupled fluctuating polymer and solvent kinetic equations of Oono and Freed

Abstract

Two broad approaches to the mathematical modelling of dilute solutions of hydrodynamically interacting macromolecules using bead-spring or bead-rod chains have emerged over the last 60 years or so: the diffusion equation of Kirkwood [23] and the coupled stochastic equations describing the evolution of polymer conformations and of the solvent, first elaborated by Oono and Freed [38]. In this paper we prove, using elementary arguments, that Kirkwood’s diffusion equation may be derived from the Oono–Freed equations provided one assumes that the solvent velocity satisfies the quasi-steady Stokes equations and makes the correct interpretation of the bead stochastic equations. In the appendix to this paper we show that provided the friction coefficient is set equal to the Stokesian value the equation of motion that we derive for the special case of a single point particle is the same, to leading order, as that of a small sphere moving slowly through a Newtonian fluid at a distance greatly exceeding its radius from the nearest solid boundary. This is illustrated for a particle moving in a semi-infinite expanse of fluid in which case the classical results of Lorentz [31] are recovered.