Sheared athermal soft-particle suspensions near jamming: dependence of effective diffusion on packing density and system size

We perform numerical simulations to study diffusion in a model bi-disperse frictionless athermal soft-particle suspension of disks in two dimensions (2D) using the so-called ``mean field'' version of Durian's bubble model. We measure the effective transverse diffusion coefficient $D_{\rm eff}$ in shear flows at various volume fraction $\phi$ and shearing rate $\dot{\gamma}$. For $\phi>\phi_c$, where $\phi_c$ is identified with the random close packing limit, in the quasi-static limit, $D_{\rm eff}$ shows a pronounced linear system size dependence with very weak dependence on $\phi$. For $\phi<\phi_c$, $D_{\rm eff}$, in the quasi-static limit, increases with increasing $\phi$ and shows very little system size dependence. We discuss how the behavior of $D_{\rm eff}$ is related to non-trivial correlations in the spatial structure of the displacement fields \emph{at long times} in the Fickian regime.