Sum rules for Sαβ(k,ω) for two diffusing species

Starting from the Smoluchowski equation without hydrodynamic interactions for two species of spherical diffusing particles, sum rules are derived here for the first three moments of Sαβ(k,ω), i.e., for the initial value of the first, second and third time-derivatives of Fαβ(k, t) (the time-dependent correlations between the fluctuations in the local concentration of diffusing particles of species α and β). These sum rules are written in terms of the potential of interaction uαβ(r) between the diffusing particles and the two- and three-particles distribution functions. This derivation is motivated by its potential use in the study of counterion effects on the diffusion of highly charged colloidal particles. Thus, we propose to approximate the memory function involved in the time evolution equation for Fαβ(k, t by a two-parameter model, with its (k-dependent) parameters being determined by the sum rules derived here. This procedure, along with Kirkwood's superposition approximation, reduces the dynamical problem to the knowledge of the radial distribution functions gαβ(r).