Reduction to an isolated pair equation

Necessarily for any number of particles more than two, eqn. (211) cannot be solved exactly, even if v° = 0 and U = 0. When there are more than two particles, the motion of one particle, say j, causes both k and / to move. Now because k and / are perturbed by j, then the perturbation to the motion of k is felt by /. The motion of j affects / directly and also indirectly through k. These indirect effects are not usually very important, especially in chemical kinetics, because the particles most likely to react are those which are closest together. Under such circumstances, the direct effect is stronger than the transmitted and reflected components. These effects have been considered by Adelman [481], Freed and Muthukumar [482] and Allison et al. [483]. Adelman draws an interesting parallel between the screening of hydrodynamic repulsion and the electrolyte screening of a coulomb interaction [481]. 

Reverting to the diffusion of two particles in solution, Deutch and Felderhof have solved for the density in the steady-state. Restricting N to 2 with v° = 0, eqn. (211) becomes 

The spherically symmetric component of this diffusion equation is of prime concern because the boundary and initial conditions are themselves spherically symmetric, for instance the Smoluchowski conditions (eqns. (3)—(5)]. Selecting the spherical equation for a coulomb potential U = rc/r [see eqn. 39)] 

This equation [70] should be compared with eqn. (44) as, for instance, in Chap. 3, Sect. 1.1. By such comparison, the diffusion coefficient is 

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