In science, emergence is a theory that meaningful order can "emerge" all on its own, spontaneously, in complex systems made of many interacting parts. [1] Emergence, according to another definition, is the process of what happens when an interconnected system or relatively simple elements self-organizes to form more intelligent, more adaptive higher-level behavior. [2]

Thermodynamics
The subject of emergence is often found interwoven with thermodynamics, being that the latter studies order and disorder in terms of energy and entropy. In 2003, American mathematician Steven Strogatz gave his summary of the understanding of connection of the two, emergence and thermodynamics, in modern science: [3]

“Scientists have long been baffled by the existence of spontaneous order in the universe. The laws of thermodynamics seem to dictate the opposite, that nature should inexorably degenerate toward a state of disorder, greater entropy. Yet all around us we see magnificent structures—galaxies, cells, ecosystems, human beings—that have somehow managed to assemble themselves. This enigma bedevils all of science today.”

In yet another instance or point of view, Strogatz tells us: [3]

“The chemist Ilya Prigogine and his colleagues feel that the key to unlocking the mysteries of self-organization lies in a deeper understanding of thermodynamics … they see the emergence of order as a victorious uphill battle against entropy, as a complex system feeds itself on energy flowing in from the environment.”

In 2008, Swiss mathematician Claes Johnson argued that computational thermodynamics, based on his own Euler equation type formulation of the first two laws, could explain emergence. [4]

Reference
1. Buchanan, Mark. (2002). Nexus: Small Worlds, and the Groundbreaking Science of Networks, (pg. 198, 207). W.W. Norton & Co.
2. Johnson, Steven. (2001). Emergence: the Connected Lives of Ants, Brains, Cities, and Software, (front-matter, pg. 52). Scribner.
3. Strogatz, Steven. (2003). Sync - the Emerging Science of Spontaneous Order, (pgs. 1, 125, 287). New York: Theia.
4. (a) Hoffman, Johan and Johnson, Claes. (2008). Computational Thermodynamics (PDF), Applied mathematics: Body & Soul, Vol. 5., 217-pages, Aug. 14.
(b) Second Law of Thermodynamics (Section: Emergence) – Knol by Claes Johnson.