Pair diffusion coefficient D(r) for two freely diffusing Brownian particles of diameter 1 with periodic boundary conditions. Results for different grid sizes Δr = 0.004 (plus), 0.032 (cross), 0.024 (star), 0.016 (square) are compared to the exact value 2D 0 = 0.1 (horizontal line). The vertical solid line marks the contact distance r = 1. To assess artifacts from periodic boundary conditions, the vertical dashed lines mark distances r = L/2, , and , where centered spheres touch the faces, edges, and corners of the cubic simulation box, respectively.
Green's functions G(r, t|r ′, 0) from simulations (symbols) and diffusion model (lines). G(r, t|r ′, 0) is shown as a function of the pair distance r at packing fraction ϕ = 0.325. We use an observation time of t = 20 to obtain diffusion model parameters, combining results for lag times Δt = 1, 2, …, 20. The arrow in the top panel reflects increasing r ′ = 1, 2, …, 7. Same color scheme is used for middle and bottom panels.
Dependence of D(r) on diffusion model parameters. Pair diffusion coefficient D(r) versus distance r for a hard sphere fluid at packing fraction ϕ = 0.35 obtained for different (top) grid sizes Δr (with fixed observation time t = 20) and (bottom) observation times t (with fixed grid size Δr = 0.1). The lag time is Δt = 1 consistently.
Pair diffusion for a hard-sphere fluid. (Top) Calculated pair diffusion coefficient D(r) versus distance r with increasing packing fraction ϕ (symbols). Lines are the predictions of hydrodynamic theory (see text). (Bottom) Normalized pair diffusion coefficient D(r)/2D 0, where D 0 is the self-diffusivity for a given ϕ. Symbols are our calculations, the thick line is the exact hydrodynamic theory,25,26 and the dashed line is the Oseen approximation.
Relation between fluid structure and dynamics. (Top) Pair diffusion coefficient D(r) (symbols connected by lines) and scaled pair correlation function g a (r) = g(r)/a (lines) versus distance r where a is an arbitrary scaling factor used to match D(r) and g a (r) at large r. (Bottom) D(r) as a function of the local fractional available volume P 0(r) (symbols; increasing packing fractions from right to left). The line is 2D 0 versus P 0 averaged over the entire system.
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