Collisional-flux term density

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

We begin with the simplest kinetic theory, that for identical, smooth, nearly elastic, spheres (Lun et al. 1984, Jenkins and Richman 1985) and derive balance laws for the means of the mass density, velocity, and the kinetic energy of the velocity fluctuations (Section 4.2). To the order of the approximations used in determining the velocity distribution function, the expressions for the stress tensor and the flux of fluctuation energy are identical to those for a dense gas of perfectly elastic spheres. The only nonclassical term is a collisional rate of dissipation per unit volume that is present in the energy balance. This can then be extended in a straightforward manner to incorporate friction between grains. 

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