  Particles interacting with short-ranged potentials have attracted increasing
interest, partly for their ability to model mesoscale systems such as colloids
interacting via DNA or depletion. We consider the free energy landscape of such
systems as the range of the potential goes to zero. In this limit, the
landscape is entirely defined by geometrical manifolds, plus a single control
parameter. These manifolds are fundamental objects that do not depend on the
details of the interaction potential, and provide the starting point from which
any quantity characterizing the system -- equilibrium or non-equilibrium -- can
be computed for arbitrary potentials. To consider dynamical quantities we
compute the asymptotic limit of the Fokker-Planck equation, and show that it
becomes restricted to the low-dimensional manifolds connected by "sticky"
boundary conditions. To illustrate our theory, we compute the low-dimensional
manifolds for n<=8 identical particles, providing a complete description of the
lowest-energy parts of the landscape including floppy modes with up to 2
internal degrees of freedom. The results can be directly tested on colloidal
clusters. This limit is a novel approach for understanding energy landscapes,
and our hope is that it can also provide insight into finite-range potentials.
