For a given height, we continuously varied the spheres' initial separation between 2 m and 10 m at 0.025 Hz for a total of 20 minutes. These trajectories were decomposed into cooperative motions and relative motions either perpendicular or parallel to the initial separation vector, and binned according to the initial separation, . 1 shows typical data for one mode of motion at one height and starting separation. Diffusion coefficients extracted from least squares fits to Eq. Dashed curves result from linear superposition of drag coefficients.
Solid curves result from the theory described here, with no adjustable parameters.
Horizontal dashed lines indicate the asymptotic self-diffusion coefficient from Eq. In the absence of other interactions, the observed trends reflect hydrodynamic coupling between the spheres and the bounding surface. These flows couple distant particles' motions, so that each particle's dynamics depends on the particular configuration of the entire collection. The probability to find particles at equilibrium in a particular configuration depends on their interaction through Boltzmann's distribution, . The corresponding force drives a probability flux , where is the particles' mobility tensor.