Abstract
We present an internally consistent, iterative solution to the systematically truncated BBGKY hierarchy for a specific transport process, self‐diffusion. Our method is an elaboration of one proposed by Stillinger and Suplinskas. We consider the decay of a particular nonequilibrium displacement—a single Fourier component corresponding to self‐diffusion. Then, using the hierarchy, we obtain particularly simple forms for the equations which govern the behavior of the nonequilibrium contributions to the low order reduced distribution functions. We show that closure, by means of the physically plausible ``dynamical superposition'' approximation, is consistent with the general form of the equations. We develop truncation procedures which lead to less complicated equations which are also internally consistent. We then develop the complete, formal, systematic solution to the most highly truncated problem which yields an expression for the self‐diffusion coefficient. The method we use seems applicable to the untruncated problem as well. We find that a direct momentum expansion of the perturbation functions which describe the nonequilibrium correlations is inconsistent with our form of the closed, truncated hierarchy equations. Instead, in a first approximation, we find that
f(1)(Pz,r;t)=exp(−12P2z)[1+σ (t)exp(−12α P2z)sinkz]/[(2π)1/2V]f(1)(Pz,r; t)=exp(−12Pz2)[1+σ (t)exp(−12α Pz2)sinkz]/[(2π)1/2V]
describes the one‐particle reduced distribution function averaged over the x and ymomenta. The parameter α varies with the density. We outline some approaches which seem fruitful for further study.