In addition to the excellent answers already added, it is important to note that in addition to Martin Hairer's description, this problem is distinct from circle packing in another way: the system tries to minimize the potential function at every point in time, subject to the physical laws governing the movement of egg yolks in a bowl. This is not generally equivalent to minimizing the potential function, and might lead to a local but not global minimum.
One example of this is a pencil balancing on its tip. This is an equilibrium state, but clearly the potential energy is suboptimal. This example is admittedly rather far from the egg yolk problem though, so an edit from someone with a more similar example would probably be appropriate.