  In colloidal systems, Brownian motion emerges from the massive separation of
time and length scales associated to characteristic dynamics of the solute and
solvent constituents. This separation of scales produces several temporal
regimes in the colloidal dynamics when combined with the effects of the
interaction between the particles, confinement conditions, and state variables,
such as density and temperature. Some examples are the short- and long-time
regimes in two- and three-dimensional open systems and the diffusive and
sub-diffusive regimes observed in the single-file dynamics along a straight
line. This work studies the way in which a confining geometry induces new time
scales. We report on the dynamics of interacting colloidal particles moving
along a circle by combining a heuristic theoretical analysis of the involved
scales, Brownian Dynamics computer simulations, and video-microscopy
experiments with paramagnetic colloids confined to lithographic circular
channels subjected to an external magnetic field. The systems display four
temporal regimes in this order: one-dimensional free diffusion, single-file
sub-diffusion, free-cluster rotational diffusion, and the expected saturation
due to the confinement. We also report analytical expressions for the
mean-square angular displacement and crossover times obtained from scaling
arguments, which accurately reproduce both experiments and simulations. Our
generic approach can be used to predict the long-time dynamics of many other
confined physical systems.
