Nonhydrodynamic States

We very briefly consider the effects of the terms that were neglected in the course of projecting onto the diffusion mode. Once again, the analysis closely parallels the derivation of Stokes law. To explicitly consider the effects of nonhydrodynamic (nondiffusion mode) states on the rate kernel, we use an analysis similar to that of van Beijeren and Dorfman and introduce a projection operator 

Given the integral equation (10.23), we now apply projection operator techniques to obtain formal solutions for 1 and Qp. The equation for 

The quantity S(r) may also be written in terms of a modified collision operator 5 

The rate kernel expression then takes the form 

This reformulation in terms of diffusive propagation and microscopic dynamics in the boundary layer is reminiscent of Noyes s encounter formulation that we briefly described earlier. Now each diffusive encounter is interrupted by sequences of collisions and very short excursions into the fluid. The analysis of nonhydrodynamic effects on the rate kernel can, therefore, be discussed naturally in terms of the encounter formalism. 

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The presence of forces and spatial dependence of D will, of course, modify these results (cf. Section XII). In addition, according to the arguments presented throughout this chapter, we expect these results to lose all validity in the short-time region. Here the nonhydrodynamic states, discussed in Section X.C, will play a crucial role. 

The choice of nonhydrodynamic states may seem somewhat arbitrary until one realizes that it is logical to choose those states which are most directly associated with the description of transport coefficients. It is obviously desirable to have kinetic models that give the proper values for transport coefficients and thus ensure the correct behavior of the fluctuation spectra at long wavelengths. 

There are other ways of analyzing nonhydrodynamic contributions. Projections onto finite sets of velocity states, in combination with kinetic modeling techniques, have proved useful in the analysis of the small molecule velocity autocorrelation function. These techniques can also be used to calculate the rate kernel. ... 

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