Definition of the Subject
Matter in the universe is organized in a hierarchical structure. At the bottom (if there is one) we have elementary particles, atoms andmolecules from which we get macro molecules like proteins and DNA, these are the building blocks of organelles, which together form the cells. From cellswe get organs, which put together form organisms: animals and plants of a great variety of species. One level of structure emerges from the levelbelow. Is it possible to scientifically describe, let alone, predict emergence. Sometimes emergence is described as a phenomena beyond analysis. Theperplexity with which this concept is sometimes met, is well illustrated by the following quote from a recent call for participation ina meeting, held by the British research council EPSRC, to look at ways to explore emergence in complex systems. Emergence is described in thefollowing words: “For the first time since the enlightenment in the western tradition we have started to understand...
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Abbreviations
- Correlations :
-
is the degree, to which events at different positions and at different times depend on or influence each other, is measured by correlation functions. If two events are statistically independent, the correlation between them is zero. The opposite is not necessarily the case, but one will often expect that if correlations decay, the mutual dependence does likewise.
- Correlation function:
-
describes correlations between two quantities and depends on their separation in time and space.
- Complex systems:
-
consist of a large number of interacting components. The interactions give rise to emergent hierarchical structures. The components of the system and properties at systems level typically change with time. A complex system is inherently open and its boundaries often a matter of convention.
- Equilibrium:
-
In statistical mechanics the prototype equilibrium system consists of a “small” system in thermal contact with another system, the latter being big enough to act as a heat bath. A heat bath is defined as a system so big that when it exchanges energy with the small system the temperature of the heat bath remains the same. The statistical properties of equilibrium systems are independent of time.
- Generalized rigidity:
-
is a term introduced by P.W. Anderson [1] to describe the situation, when a many component system acts as a globally connected unit, in the sense that if one apply a force at one point, the effect can be transmitted across the system. Ice has rigidity, if we push at one point, the entire piece of ice will start moving. If the ice, on the other hand melts to water, a force applied locally will only have an effect locally.
- Hamiltonian:
-
expresses the energy of a system as a function of the degrees of freedom, in terms of which the system is defined at the considered level of description. Emergence in physical systems can sometimes be understood in terms of lumping degrees of freedom, in the Hamiltonian, together in sets of effective degrees of freedom, e. g. the center of mass of a solid body.
- Non‐equilibrium systems:
-
is a term used to describe any system that is not in equilibrium. Needless to say this is a characterization of limited value, since there are many very different types of systems included in this category.
- Order parameter:
-
is a quantity that allows one to discriminate between two phases of a physical system. The order parameter changes from zero to non-zero as one passes from one phase to the other. To identify the relevant order parameter is often non‐trivial and, is in itself, a first important step.
- Renormalization groupanalysis:
-
is a systematic mathematical procedure that enables a derivation of the emergent behavior at the macroscopicsystems level. The behavior at long length and time scales is obtainedfrom the underlying microskopic short length scales and fast dynamics.
- Statistical mechanics :
-
seeks to understand how properties at systems level emerge from the level of the system‐components and their interactions. This often involves the application of probability theory, and a number of mathematical techniques. Throughout, we draw a distinction between statistical mechanics and statistical physics. The latter is mainly concerned with the microscopic foundation of thermodynamics and, e. g., phenomena such as phase transitions and superconductivity.
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Acknowledgments
I am deeply grateful to many people with whom I have collaborated and learnt from during many yearsconcerning matters relevant to the topic of the present article. In particular it is a pleasure to mention Hans Fogedby, Peter Minnhagen, and HansWeber.
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Jeldtoft Jensen, H. (2009). Complex Systems and Emergent Phenomena. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_85
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