One archetypal example of emergent phenomena in nature is found in the self-assembly of complex spatio-temporal structures, either in the relaxation of many-body systems toward their thermal equilibrium or in non-equilibrium steady states characterized by continuous energy throughput.
Examples of the former include crystallization and the formation of higher symmetry liquid crystalline mesophases (e.g., nematic, columnar, and semetic phases of thermotropic liquid crystals), the spontaneous formation of lipids (and multi-block copolymers) into lamellar and bicontinuous phases, in addition to cylindrical and spherical micelles, and the condensation of charged biopolymers into disordered and ordered (e.g. chiral hexatic) bundles.
Recognizing the commonality of these diverse emergent phenomena, in the 1970's researchers began to refer to this set of self-organization as “self assembly” in order to describe the phenomena of molecular and colloidal organization that result from the equilibrium assembly of many copies of one or a small number of constituents via weak (non-covalent) interactions.
Fundamental questions have been explored involving the prediction of the complex equilibrium states of these systems and their low-energy excitations and linear response properties. In addition, much thought has gone into considerations of assembly kinetics and, in particular, the role of kinetic traps or glassy dynamics inhibiting the formation of thermodynamic ground state of the system.
Today, researchers have shown that one may design complex pattern formers using nanofabrication techniques so that one may encode interesting or otherwise desirable complex equilibrium states into the interactions and shapes of the nanoparticles. Examples include DNA origami and pattern formation in colloids of complex shapes or with precisely controlled binding sites.
In non-equilibrium systems, such examples of complex spatio-temporal patterns abound, from patterns formed in convection to contractile waves in active contractile gels and liquid-crystal elastomers. Of course, the quintessential examples of non-equilibrium structure formation are found in biology: viral self-assembly, protein folding (and misfolding), and the dynamic organization of cytoskeletal networks provide frontier problems in statistical and biological physics today.