Imagine agents moving according to simple rules that disfavor close crowding but favor a mutual intermediate range proximity. Systems of this type appear across the sciences, from molecular dynamics to microbiology and ecology. The competition between the underlying short-range repulsion and intermediate-range attraction can lead to a phase transition, where the preferred state changes from a "crystalline" equidistribution to the formation of clusters (or colonies) separated by vacuum regions. I will describe recent work that analyzes this transition (or bifurcation) in some detail, emphasizing a curious aspect that makes this transition "non-hysteretic" or "reversible" in an infinite system-size limit, thus allowing for easy switching from crystal to cluster -- and back. Results include predictions for sizes and shapes of vacuum regions, corrections due to noise, and expansions for finite system size corrections. This is based on joint work with Angela Stevens and on a summer REU project mentored by Olivia Clifton.

Xmas PDE Colloquium - coffee break before (14:45 - 15:15, SRZ Lounge)