  Diffusion in colloidal suspensions can be very slow due to the cage effect,
which confines each particle within a short radius on one hand, and involves
large-scale cooperative motions on the other. In search of insight into this
cooperativity, here the authors develop a formalism to calculate the
displacement correlation in colloidal systems, mainly in the two-dimensional
case. To clarify the idea for it, studies are reviewed on cooperativity among
the particles in the one-dimensional case, i.e. the single-file diffusion
(SFD). As an improvement over the celebrated formula by Alexander and Pincus on
the mean-square displacement (MSD) in SFD, it is shown that the displacement
correlation in SFD can be calculated from Lagrangian correlation of the
particle interval in the one-dimensional case, and also that the formula can be
extended to higher dimensions. The improved formula becomes exact for large
systems. By combining the formula with a nonlinear theory for correlation, a
correction to the asymptotic law for the MSD in SFD is obtained. In the
two-dimensional case, the linear theory gives description of vortical
cooperative motion.
