[Nesting in] phase transitions

Nesting in systems like rule 184 (see page 273) is closely related to the phenomenon of scaling studied in phase transitions and critical phenomena since the 1960s. As discussed on page 983 ordinary equilibrium statistical mechanics effectively samples configurations of systems like rule 184 after large numbers of steps of evolution. But the point is that when the initial number of black and white cells is exactly equal—corresponding to a phase transition point—a typical configuration of rule 184 will contain domains with a nested distribution of sizes. The properties of such configurations can be studied by considering invariance under rescalings of the kind discussed on page 955, in analogy to renormalization group methods. A typical result is that correlations between colors of different cells fall off like a power of distance—with the specific power depending only on general features of the nested patterns formed, and not on most details of the system.