  We study experimentally what is arguably the simplest yet non-trivial
colloidal system: two-dimensional clusters of 6 spherical particles bound by
depletion interactions. These clusters have multiple, degenerate ground states
whose equilibrium distribution is determined by entropic factors, principally
the symmetry. We observe the equilibrium rearrangements between ground states
as well as all of the low-lying excited states. In contrast to the ground
states, the excited states have soft modes and low symmetry, and their
occupation probabilities depend on the size of the configuration space reached
through internal degrees of freedom, as well as a single "sticky parameter"
encapsulating the depth and curvature of the potential. Using a geometrical
model that accounts for the entropy of the soft modes and the diffusion rates
along them, we accurately reproduce the measured rearrangement rates. The
success of this model, which requires no fitting parameters or measurements of
the potential, shows that the free-energy landscape of colloidal systems and
the dynamics it governs can be understood geometrically.
