Suppose one sprinkles points uniformly at random on the infinite Euclidean plane,
with some density $\rho$ per unit area.
View the points as disks of radius zero.
Now the radii $r$ of all disks grows simultaneously at the same (slow) rate,
with the disks pushing one another when they come into contact,
with never two disk interiors overlapping.
Each disk-disk contact pushes both disks apart along the line connecting
More complex clusters move the disks by summing the disk-disk force-separation
vectors, although I want the disks to move only when pushed, i.e.,
they never gain momentum to drift without pushing.
Three time-snapshots for a region containing $8$ points
might look like this (just a hand simulation):
Here the disks do not yet touch at $t=1$ (blue), but at $t=2$ (tan), $\lbrace 1,2 \rbrace$ and $\lbrace 7,8 \rbrace$ are in contact, and by $t=3$ (pink), $\lbrace 1,2,3,4 \rbrace$ and $\lbrace 5,6,7,8 \rbrace$ are touching.
Although the model is not fully specified, perhaps analogous processes have been studied in the literature? Essentially I want to know:
Q. What happens? Is there some limit configuration as $r$ grows?
Intuitively, it feels like all empty space would be squeezed out. Perhaps the behavior for points initially confined to a finite region $R$, as that region grows larger, would hint at the situation for the infinite plane? I'd appreciate pointers or observations—Thanks!
Update. Here is one of the many images from the Lubachevsky, Graham, Stillinger work to which Henry Cohn points, "Spontaneous Patterns in Disk Packings":