Order parameters of dense phases

Dense phases of emergent systems, such as constrained complex networks,
exhibit distinct characteristics, the most studied being broken symmetry.
However for practical purposes “rigidity”, the resistance to change, is
also of wide interest. There are difficulties analyzing rigidity since
when perturbed a system can easily move out of its phase. A new approach to
overcome this contradiction has been initiated by David Aristoff and
Charles Radin: see the discussion in Quanta Magazine. Another characteristic
of dense phases are their nonspherical `Wulff’ shapes, polyhedral for ordinary crystals.
This is examined in this expository paper by Charles Radin, and in a related direction in this paper by Charles Radin, Kui Ren, and Lorenzo Sadun.

In a different direction, the process of nucleation is a dynamical signature of the creation of a dense phase, which can appear even in systems far removed from ordinary atomic materials, as shown in this paper by Frank Rietz, Charles Radin, Harry Swinney and Matthias Schroeter. All the above characteristics make essential use of finite systems, a nonstandard approach to understanding emergent phases.