  Using Stokesian dynamics simulations, we examine the flow of a monodisperse,
neutrally buoyant, homogeneous suspension of non-Brownian solid spheres in
simple shear, starting from a large number of independent hard-sphere
distributions and ensemble averaging the results. We construct a novel method
for computing the gradient diffusivity via simulations on a {\em homogeneous}
suspension and, although our results are only approximate due to the small
number of particles used in the simulations, we present here the first values
of this important parameter, both along and normal to the plane of shear, which
have ever been obtained directly either experimentally or numerically. We show
furthermore that, although the system of equations describing the particle
motions is deterministic, the particle displacements in the two directions
normal to the bulk flow have Gaussian distributions with zero mean and, a
variance which eventually grows linearly in time thereby establishing that the
system of particles is diffusive. In addition we show that although the
particle evolution equations are, in principle, reversible, the suspension has
in fact a finite correlation time $T_c$ of the order of the inverse shear rate.
For particle concentrations up to 45%, we compute the corresponding tracer
diffusivities both from the slope of the mean square particle displacement as
well as by integrating the corresponding velocity autocorrelations and find
good agreement between the two sets of results.
