We developed a Smoluchowskihyphen;level theory of timehyphen;dependent ion concentration fluctuations at equilibrium, including higherhyphen;order correlations, but with hydrodynamic interactions suppressed. A partial sum over the higherhyphen;order correlations, a chain sum, eliminates divergences. From the resulting renormalized selfhyphen;van Hove function we recover Onsagerrsquo;s limiting law for the selfhyphen;diffusion coefficients of the ions. The limiting law comes from the chain sums that are neglected in a lsquo;lsquo;linearizationrsquo;rsquo; approximation used in some earlier work. The corresponding development for the distinct van Hove functions is merely indicated. We apply these results to a solution of a single spherical polyion with many small ions, to obtain the selfhyphen;diffusion coefficient for the polyion in nearhyphen;limiting law conditions, and to simple models for aqueous NaCl and CuSO4solutions, to calculate the selfhyphen;diffusion coefficients of the ions and the Maxwell (Debyendash;Falkenhagen) relaxation, all in the concentration range up to 1 M.
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