What is emergence?

The answer to this question is found in the context of philosophy of science where the concept of "emergence" is continuously refined and tested against different situations in physics, chemistry and mathematics. The usage of the term first appeared in the context of philosophy of mind but was later taken by philosophy of science to describe phenomena that arose due to the collective and often complex behaviour of microscopic constituents. Despite many modern notions of "emergence" being in place and no overwhelming consensus existing amongst scientists and philosophers alike, DIEP has chosen to use the following pragmatic, inclusive and moderately subjective definition of emergent phenomena:
Emergent phenomena is behaviour that is novel and robust relative to some comparison class. (introduced by J. Butterfield)
The term "emergent phenomena" in the definition above can also be replaced by "emergent theory" leading to a definition of "emergent theory". To quickly grasp some aspects of this definition, one may think of novel collective behaviour when compared to the microscopic constituents of a system (the comparison class) or of novel behaviour that appears when taking a specific limit of a whole family of systems. For instance, certain polymers can self-assemble forming bigger structures that exhibit new morphologies and topologies. This type of behaviour is new when compared to the behaviour of individual elements of the system and it is also robust in the sense that it doesn't depend on all the attributes of the individual polymers. Similarly, if one thinks of the emergence of thermodynamics from statistical systems,  macro notions such as temperature are not definable for individual molecules in a gas and at the same time are robust, as they are largely independent from the individual molecules' size and velocity.
The notion of "novelty" characterises "emergence" since "to emerge" means something new and non-trivial that arises from a given theory either by some form of coarse-graining/fine-graining or by taking a specific limit. In this context, for example, one may speak of new laws or new behaviour of large-scale objects when compared to the behaviour of its individual constituents.  The notion of "robustness" means that the phenomena has a certain degree of independence from the comparison class, in most situations implying that the behaviour is not easily obtained from the theory describing the microscopic constituents or that the behaviour is obtained by a non-trivial limit of a class of systems.
Though already made explicit, it is important to stress that emergence does not always involve coarse- or fine-graining , it can also arise as a limit of a class of systems such that at that limit there is novel and robust behaviour. A classic example is the emergence of Newtonian physics from quantum mechanics when the number of particles involved is taken to be very large or when Planck's constant is taken to be zero. This possibility enlarges the scope of approaches to the understanding of emergent phenomena.

What is the science of emergence?

The universe is composed of microscopic building blocks and the world we see around us is the result of a combination of millions of billions of billions of those blocks. When we walk through the streets of our cities, we do not see these microscopic elements of the universe but instead cars and buses smoothly driving pass by us. If we would carry our microscopes and particle accelerators with us, we would be able to see some part of that microscopic world but we usually carry nothing more than a pair of Ray-Ban glasses. The world we see with our own eyes is governed by laws that originate from a microscopic world but the laws that govern that microscopic world are completely different. The world we experience is said to have emerged from a world that only microscopes can reach. All the smooth experiences of wind blowing, music, sound or touch are the result of these emergent laws.
There are many examples of emergence around us and in physics, mathematics and chemistry. These include the emergence of Newton's laws from quantum mechanics, the emergence of the heat equation for the collective motion of molecules, the emergence of thermodynamic phase transitions from microscopic statistical systems, the emergence of Van der Waals forces by coarse-graining the interactions of neighbouring molecules, the emergence of new phases of matter such as unconventional superconductivity, the emergence of new spontaneous behaviour such as self-assembly in polymers, criticality and self-organisation in active matter, and the emergence of gravity from holographic field theories such as in the context of the AdS/CFT correspondence and string theory. Emergent phenomena in different areas is often remarkably similar leading to theories with similar structures at very different scales. A well know example is the emergence of phase transitions and critical behaviour in the Ising model, which is applicable to magnetisation in a ferromagnet, as well as to colloidal particles consisting of billions of atoms. 
Tb - lower level description
Tt - higher level description
Between the microscopic world where the fundamental building blocks of matter reside and the macroscopic world that we see around us, there is a very large number of other worlds (or scales) that also emerge from that same microscopic world. These are, for example, the electronic scale, the atomic scale, the molecular scale or the mesoscopic scale. All these scales can be approached by different disciplines. Research in emergent phenomena is based on a translational process from a lower level of description (world/scale) to a higher level of description or vice-versa, i.e. understanding emergent phenomena or emergent theories requires establishing a dictionary between a theory and its emergent phenomena or between two theories, one of which emerges from the other. 

Emergence, however, is a quite universal notion that extends beyond the realm of physics, mathematics and chemistry, reaching many other disciplines such as social sciences, information/computer science, as well as humanities. Systems that are defined, or whose existence is determined, by a collection of interacting components at some scale can in principle exhibit emergent behaviour at a larger scale. Examples include traffic jams (whose components are individual vehicles), emergent collective behaviour in networks such as communication networks or the emergence of communication networks itself, the collective behaviour of flocks of birds and schools of fish or large scale crowd behaviour, the collective behaviour (rational/irrational) in social media such as the spread of misinformation, the emergence of consciousness from complex networks of individual neurons or the behaviour of financial markets as self-organising networks of agents.  

Cross-fertilisation of similarly novel concepts across the different disciplines will indubitably propel science forward in the 21th century. In general, emergence establishes a relationship between theories or between effective descriptions at different levels of reality. When two descriptions meet, expertise of the two descriptions is required in order to properly understand emergent phenomena. This is why DIEP is a broad interdisciplinary research centre spanning several of the fundamental sciences.

The abstract definition above requires concrete examples. Below we have collected a few and brief examples, organised into classes.

History and philosophy of emergence

The modern usage of the term "emergence" was first introduced by the philosopher G.W. Lewes in the context of philosophy of mind. The term quickly spread to other parts of philosophy and science and expressed the belief of many that sciences such as chemistry, biology or psychology were described by fundamental laws and properties widely different from those of their supposed small constituents as studied by physics. C.D. Broad, one of the so called "British emergentists" in the early 1900's, was particularly interested in the laws of chemistry and how they greatly differed from the laws of physics. He believed that the combination of certain chemical substances required the introduction of specific laws beyond the general laws of combination applicable to all substances. However, with increasing scientific advances, the examples studied by Broad were in fact shown to be reducible to the laws of physics. With the success of reductionism (that higher level theories are ultimately reducible to lower level ones such as physics), the role of emergence was continuously pushed to higher level sciences (such as psychology) until the emergentist movement died, so to speak. 
The debate on emergence and emergent properties in the philosophical/scientific context was revived by the 1972 paper of Nobel prize winner and condensed matter physicist Philip W. Anderson, More is different. His opinion is largely shared by several others of his peers, including Robert Laughlin, David Pines and Piers Coleman. In this paper, Anderson argues that the laws of condensed matter systems are as fundamental as the laws of high energy physics (such as those governing the standard model) though still agreeing with the doctrine of microphysicalism, by which all such systems are made of microscopic constituents governed by microscopic laws. However, he argues that even though all matter is reducible to microscopic building blocks, it does not follow from there that one can derive all the workings of the universe. It may be theoretically possible to derive macroscopic laws from microscopic ones though perhaps not practically possible. Behind the curtain, Anderson's reasoning stems from examples in condensed matter physics where large aggregates of microscopic building blocks do not exhibit the same symmetries and laws as those of the underlying microscopic theory. In most cases, spontaneous symmetry breaking (as in solid crystals) occurs and the laws governing those are difficult, if even possible, to understand in terms of the natural variables/parameters used to describe the microscopic theory. 
This revival of "emergent philosophy" led to many developments and refinements of the concept of "emergence" itself. Some of these philosophical explorations led to metaphysical considerations of the notion of emergence, novelty and reducibility. Jaegwon Kim, in the context of philosophy of mind, believed that emergence involved higher levels of complexity, unpredictability and irreducibility. Batterman, in the context of philosophy of science, believed that emergence required a limiting and singular behaviour, such as the appearance of divergences in the free energy when the limit of large number of particles (thermodynamic limit) is taken in statistical mechanics systems. Butterfield, in turn, claims that emergence and reduction (interpreted as deduction) are mutually independent and that emergence can occur in non-singular limiting behaviour. The notion of emergence is still currently being refined and tested against many new contexts within science. As such, many of the examples given in the sections below are putative examples for many philosophers, whose work focuses on qualifying the degree of emergence existent in each example (strong/weak emergence, ontological/epistemic emergence, etc).
G.W. Lewes: The Physical Basis of Mind, Problems of Life and Mind, Trübner, 1877
P.W. Anderson: More is Different, Science, 177:393-396, 1972
J. Kim: Making Sense of Emergence, Philosophical Studies, 95: 3–36, 1999
R.W. Batterman, The Devil in the Details, Oxford University Press, 2002a
P. Mainwood: Is More Different? Emergent Properties in Physics, PhD thesis, 2006
J. Butterfield: Less is Different: Emergence and Reduction Reconciled, Foundations of Physics, Vol 41, Issue 6, 2011
T. O'Connor, H. Y. Wong: Emergent Properties, Stanford Encyclopedia of Philosophy, 2015

Emergence in string theory

String theory is a theory that combines quantum theory with gravity and is seen by the majority of the string community as a collection of tools, methods and approximation schemes for constructing quantum states that extends the framework of conventional quantum field theory, allowing to describe, in particular, theories of quantum gravity. String theory provides many examples of emergent behaviour as will be described next but perhaps the most surprising case of emergence is that of the emergence of gravity itself, and thus of spacetime. Despite being a very active research field, and hence of being in a continuous process of development, there are multiple indications within the framework of string theory that gravity is an emergent phenomenon, such as mirror symmetry, topology change transitions and many non-perturbative dualities.
One of most well studied examples of these dualities is the so-called AdS/CFT correspondence which is a holographic duality relating a conventional quantum field theory (without gravity) living in the boundary of Anti-de Sitter space to a theory of quantum gravity living in one higher dimension in the bulk of that same space. In this context, gravity and spacetime are thought to emerge from the local degrees of freedom that characterise the conventional boundary quantum field theory. There are several examples where the emergence of space can be made explicit. In particular, the renormalisation group flows of the quantum field theory (determining the physics at a given energy scale) can be interpreted as the emergence of the bulk holographic coordinate, in turn responsible for the non-trivial curvature of the bulk spacetime. Recently, the Einstein equations for the bulk theory of gravity were derived based on the evolution of entanglement entropy of the quantum field theory while notions of bulk locality were show to be expressible in terms of the language of quantum error correction in the dual quantum field theory. In fact, connections between this duality and generic (quantum) information theory have led string theorists to apply similar ideas to biophysics and neuroscience.
Holographic dualities in string theory have also proven to be powerful tools to study emergent phenomena in different areas of physics. For instance, strongly coupled phenomena occurring in quantum field theory can be tackled using classical gravity due to the strong/weak coupling nature of the holographic duality. This has furthered the study of toy models for condensed matter systems where rich emergent phenomena takes place (see the brief notes on condensed matter), such as certain scaling behaviour in strange metals or the emergence of superconducting behaviour. Hydrodynamics itself emerges in this context and has been key in the modelling of the quark gluon plasma generated at heavy ion collisions. The emergence of macroscopic thermodynamic properties for objects such as certain classes of black holes has also shown to be deducible by applying similar techniques as in statistical physics, allowing to derive the Hawking-Bekenstein area law for black holes.
Ideas that originated in the study of non-perturbative dualities have been applied to other contexts. The Kerr black hole, which is the standard theoretical model for describing astrophysical black holes, is characterised by an emergent conformal symmetry when it is spinning very fast. This behaviour has led string theorists to postulate a Kerr/CFT correspondence, which has recently proven to be quite useful for analytically understanding gravitational wave signatures of rapidly rotating astrophysical black holes.
Dualities and holography are a playground for testing philosophical notions of emergence.
J. Maldacena, The Large N limit of superconformal field theories and supergravity , Adv.Theor.Math.Phys., 1998 
J. de Boer, E. Verlinde, H. Verlinde: On the holographic renormalization group, JHEP 0008 003, 2000 
E. Witten: Emergent Phenomena in Condensed Matter and Particle Physics, talk given at SidneyFest 2005
S. BhattacharyyaV. E. HubenyS. MinwallaM. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 0802 045, 2008
M. GuicaT. HartmanW. SongA. Strominger: The Kerr/CFT Correspondence, Phys.Rev. D80 124008, 2009
S. A. HartnollJ. PolchinskiE. SilversteinD. Tong: Towards strange metallic holography, JHEP 1004 120, 2010 
J. BhattacharyaS. BhattacharyyaS. Minwalla, Dissipative superfluid dynamics from gravity, JHEP 1104 125, 2011
V. Balasubramanian: The Variational Universe: From Strings to Neurons, talk given at The Principles of Complexity: Life, Scale, and Civilization, 2012   
V. BalasubramanianB. D. ChowdhuryB. CzechJ. de BoerEntwinement and the emergence of spacetime, JHEP 1501 048, 2015 
A. AlmheiriX. DongD. Harlow: Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 1504 163, 2015
D. Dieks, J. van Dongen, S. de Haro: Emergence in holographic scenarios for gravityStudies in History and Philosophy of Science B, Vol 52, 2015
T. FaulknerF. M. HaehlE. HijanoO. ParrikarC. RabideauM. Van RaamsdonkNonlinear Gravity from Entanglement in Conformal Field Theories​JHEP 1708 057, 2017 
J. Armas, Conversations on Quantum Gravity, Cambridge University Press, to appear, 2018

Emergence in condensed matter

As mentioned in the brief notes on the history of emergence, condensed matter physicist Philip W. Anderson had a tremendous influence on the resurgence of emergent philosophy in science. One of the main reasons for this is that condensed matter systems are large enough systems (i.e. composed of a large collection of microscopic constituents) to exhibit a wide range of emergent collective behaviour, which is extremely difficult to understand by using the quantum mechanical theory of its microscopic constituents. As argued by Robert Laughlin and David Pines, solving the Schrödinger equation for more than 10 atoms accurately will never be possible with whatever computer one might make in the future due to the enormous amount of memory required (though this may be possible using a functioning quantum computer). In addition, many of the emergent phenomena encountered in condensed matter are very robust,  there is  little dependence on the microscopic details of the system. This has led the condensed matter community to consider the possibility of finding a general theory of emergence that could be rigorously tested in condensed matter systems.

For many years, emergence in condensed matter was focused on strongly correlated electron systems - systems in which the Coulomb interactions between electrons are always strong and important. Strongly interacting, or strongly correlated, electrons often lead to many almost degenerate (or at least rather finely balanced, energetically) ground states of the system, so that relatively small changes to the electronic environment  induce phase transitions into phases associated with the collective behaviour of the electron system. The small changes can be the application of magnetic or electric field, the application of pressure, or modification of the chemical composition of the material. Examples of these emergent phases include (unconventional) superconductivity, various novel magnetic phases, and strange metallic states that do not obey the standard Landau Fermi liquid theory. A key similarity between many of the  most studied emergent phases in correlated electron systems is that they develop at quantum phase transitions (zero temperature phase transitions driven by a non-thermal tuning parameter). Typical quantum phase transitions and emergent phases in strongly correlated electrons involve symmetry breaking, and the development of an order parameter which reflects the evolution of the phase.  Due to the difficulty in understanding from first principles the emergent phases and behaviour of correlated electron systems, progress is heavily based on experiments which attempt to identify the order parameters responsible for the onset of different phases, as well as to identify universality classes amongst different materials.

During the past few years, new types of emergent phases have arrived in condensed matter, in the form of topological phases. They are not entirely new, in that a few examples have existed for many years, but the interest in topological phases has dramatically increased as more examples have been discovered experimentally. A crucial difference between emergent topological phases and the emergent phases in correlated electron systems mentioned above is that topological order is non-symmetry breaking.  One well-established example of topological order is the fractional quantum Hall phase, whose appearance is not delineated by an order parameter, but by by the  development of long-range quantum entanglement. Gapped spin liquids and skyrmionic states are more recent  examples of topological phases, as are topological insulators. 

Emergent topological phases, like emergent phases in correlated electron systems, can often be described quite simply in terms of emergent quasiparticles. The "composite fermions" of the fractional quantum Hall effect are quasiparticles made up of a combination of electrons and flux quanta. They behave as ground state particles with integer charge, and can be described in the same way as electrons in the integer quantum Hall effect. The "heavy fermions" in lanthanide- and actinide-based correlated electron systems are quasiparticles made up of conduction electrons, collective interactions, and f-electrons, and they behave as ground state particles that obey Landau Fermi liquid theory. These emergent quasiparticles allow us to describe a very complicated system in terms of relatively simple excitations, and in a way that is insensitive to microscopic details. Spinons and skyrmions are more examples of emergent (topological) quasiparticles, which appear in the context of frustrated magnetism and other magnetic systems. The quasiparticle approach does not answer all the questions we have, but it has been important to much of the current understanding of emergent phases in condensed matter. It also offers a tantalising parallel to quasiparticle and coarse-graining approaches to emergence in other fields.

F. Steglich et al.: Superconductivity in the Presence of Strong Pauli Paramagnetism, Phys. Rev. Lett. 1979 
J. K. Jain: Composite-fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63, 199, 1989
N. D. Mathur et al: Magnetically mediated superconductivity in heavy fermion compounds, Nature 394, pages39–43, 1998
R. B. Laughlin, D. Pines: The Theory of Everything, PNAS January 4, 97 (1) 28-31, 2000
P. Coleman: Many Body Physics: Unfinished Revolution, Annales Henri Poincaré, Vol 4, pp 559-580, 2003
T. Senthil, M. Vojta, S. Sachdev: Weak magnetism and non-Fermi liquids near heavy-fermion critical points, Phys. Rev. B 69, 035111, 2004
S. Paschen et al.: Hall-effect evolution across a heavy-fermion quantum critical pointNature 432, 881-885, 2004
A. McCollam et al., Anomalous de Haas–van Alphen Oscillations in CeCoIn5, Phys. Rev. Lett. 2005
Y. Yang, D. Pines, Emergence of superconductivity in heavy-electron materials, PNAS December 23, 111 (51) 18178-18182, 2014
M. S. Golden, A. de Visser et al.: Low carrier concentration crystals of the topological insulator Bi2−xSbxTe3−ySey: a magnetotransport study, New Journal of Physics, 2014
Y. Yang, D. Pines, G. Lonzarich: Quantum critical scaling and fluctuations in Kondo lattice materials, PNAS June 13, 114 (24) 6250-6255, 2017
P. Coleman, Emergence and Reductionism: an awkward Baconian alliance, in Routledge Handbook of Philosophy of Emergence, 2017
S. Pezzini et al.: Unconventional mass enhancement around the Dirac nodal loop in ZrSiS, Nature Physics 14, 178-183, 2018   

Emergence in mathematical physics

Classical and quantum mechanical laws are time reversible. For instance, the collision of two particles look exactly the same if we move forward or backward in time. However, the world we see around us is obviously irreversible: there is an arrow of time. We see the sea waves approaching the shore and breaking but we do not all of a sudden see the waves rolling backwards and disappearing. Thermodynamically speaking, there is a law that tell us that the universe conspires in such a way that entropy always increases. How can macroscopic irreversibility be reconciled with microscopic reversibility? Boltzman himself, using probability theory, showed that irreversibility emerges for a gas of molecules by taking a singular limit where the number of particles in the system diverges and under appropriate initial conditions, i.e. only for microscopic trajectories that satisfy the molecular chaos hypothesis (that particles are uncorrelated before collision), which happen to be the majority of the trajectories.

Most of the materials and processes in everyday life appear smooth and continuous, even though they are made of many tiny particles. In their vast majority, they can be described by the laws of thermodynamics and the laws of hydrodynamics dictate their near equilibrium evolution. However, starting with molecular theory and obtaining the Navier-Stokes equations is a challenge that requires many footsteps and a multi-scale analysis. Obtaining the hydrodynamic limit from a collection of molecules gives rise to novel emergent behaviour: the emergence of an arrow of time. Interestingly, the emergence of hydrodynamics does not depend heavily on microscopic details (it is very robust), taking place for different classes of systems. In particular, hydrodynamics emerges as a macroscopic description of cellular automata, characterised by discrete probabilistic collision rules at the microscopic level. 

The world we see around us is also mostly classical, however the microscopic laws at the atomic scale are governed by quantum physics. Quantum physics has a certain degree of indeterminacy: a particle may be in a superposition of different states, which, by virtue of Heisenberg's Uncertainty Principle, does not allow us to extract both the position and velocity of the particle with equal precision. However, once billions of particles are put together and observed at macroscopic scales, classical physics emerges as well as deterministic measurements of the properties of macroscopic objects. Traditional methods for understanding this type of emergent phenomenon rely on the so called WKB approximation, however, it is only applicable to certain cases. A more powerful mathematical approach is (strict) deformation quantisation, which involves the machinery of C* algebras. Within this context, Planck's constant is a real number that can be formally varied and when ​taken to zero leads to the emergence of classical physics.

Mathematical physicists have developed some of the basic tools to deal with this kind of fundamental questions, such as partial differential equation theory, functional analysis, probability theory, symplectic geometry and operator algebras. These tools allow for understanding emergent behaviour in a vast realm of contexts within physics including the emergence of classical physics from quantum mechanics, the emergence of thermodynamics from statistical physics systems, the emergence of geometric optics from wave optics, the emergence of hydrodynamics from kinetic theory or molecular dynamics and the emergence of irreversibility at macroscopic scales.
R. Car: Unified approach for molecular dynamics and density-functional theoryPhys. Rev. Lett. 55(22), 2471, 1985
C. Cercignani et al.: The Mathematical Theory of Dilute Gases, Springer book, 1994
J. Fritz.: Introduction to Hydrodynamic Limits, 2001
M. V. Berry: Chaos and the semiclassical limit of quantum mechanics (is the moon there when somebody looks?), Vatican Observatory CTNS, 2011
A. N. Gorban et al.: Constructive methods of invariant manifolds for kinetic problems, Phys. Rep. Vol 396, 2004
F. Redig: Hydrodynamische limieten en de pijl van de tijd, Nieuw Archief voor Wiskunde, serie 5, 2011
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M. V. Berry: Classical limits, In The Theory of the Quantum World Proceedings of the 25th Solvay Conference on Physics, pp. 52, 2013
S. Chibbaro et al.: Reductionism, Emergence and Levels of Reality, Springer book, 2014
K. Landsman: Foundations of Quantum Theory, Springer book, 2017

Emergence in soft matter and chemistry

In 1929 after some of the great successes of formulating the theory of quantum mechanics, Paul Dirac wrote: The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solvedDirac suggested that deriving chemistry from quantum theory is only a practical matter. Up to this day, this still remains unfeasible. Soft matter and chemistry, similarly to condensed matter, involve systems of billions of billions of microscopic constituents that exhibit a wide variety of emergent behaviour occurring at multiple scales, in turn extremely difficult to track down in terms of the microscopic theory. A simple example is the difficulty in predicting the shape of different molecules using quantum mechanics, in particular pyramidal molecules, due to symmetry breaking. Small molecules, such as these, already require new emergent laws and principles to be discovered. The case of larger molecules such as polymers and biomolecules is even harder to predict.
Because of the difficulty in understanding the microscopics of systems such as colloids and polymers, which ultimately constitute the building blocks for biological life, active research in soft matter is constantly informed by heavy numerical simulations and experiments. To make theoretical progress in these areas, techniques for coarse-graining take a prominent role. The complicated interactions between building blocks can be replaced by effective interactions between larger assemblies e.g. nano particles, if all other degrees of freedom are unimportant. These forces can be seen as emergent. A classic example is the integration of short-range molecular van der Waals forces between colloids which ultimately allow geckos to climb smooth walls. In multi-scale numerical simulations, coarse-grained models are used to simulate regions of parameter space for which less detail is needed while other regions where finer detail is required make use of the microscopic model.
There are many other examples of emergent behaviour in soft matter and chemistry, such as metastability - the finite size version of thermodynamic phase transitions where interacting building blocks spontaneously order themselves. In this setting finite systems such as biomolecules and nanoparticles can undergo conformational changes such as nucleation, shape morphing and folding. Each of these states can have its own longevity. Another example is that of pattern formation where both in and out of equilibrium systems can form emergent patterns (such as dynamical travelling waves, oscillation and chaos), which are highly non-trivial given our current understanding of the underlying dynamical and interaction rules. 
A highly important example of emergence in soft matter and chemistry is hierarchical structural organisation and self-assembly, which has its large scale cosmological counterpart. It is well known that matter can arrange itself in hierarchal ways. Building blocks can spontaneously form larger entities that can then organise themselves in larger structures. The classic example is that of surfactants that organise themselves in bilayers that can stack into larger lamellar phases, or in micelles or vesicles that can organise themselves in higher order phases like crystals. Another example is that of bio-minerals, like bone or shells, that are build hierarchically. This is a very poorly understood emergent phenomenon that requires further investigation.
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Emergence in society

Many of the examples explained above show that often times emergent behaviour arises as collective behaviour due to the interaction of many fundamental building blocks. This type of behaviour is also observed in animal and human societies and can be approached using the same mathematical tools which are used in the different scientific disciplines. A curious and widely observed phenomenon is the organised behaviour of bird flocks, schools of fish and herds of sheep at macroscopic scales and the organised behaviour of bacteria and enzymes at mesoscopic scales. The emergence of collective behaviour in animal societies has direct consequences for the prediction of global catastrophic events such as earthquakes, volcanic explosions, disease spreading and sudden weather changes. This type of emergent phenomena requires the understanding of non-equilibrium statistical mechanics.

Several systems in human societies also exhibit self-organised emergent behaviour that can be modelled by phenomenological theories of critical phase transitions, allowing to predict the so called tipping points at which the system changes abruptly from one state to another. These systems include the emergence of traffic jams, the spontaneous systemic failures of the human body (asthma attacks, epileptic seizures), financial market crashes and abrupt changes in climate or of certain wildlife populations. The reason why such theories can be broadly applicable is because this type of emergent behaviour is robust and insensitive to many of the different properties of the microscopic constituents (i.e. to all the specific details of birds or human beings). 

Emergent self-organisation processes are also thought to occur in the architecture of urban networks, cities and economy cycles. Social interaction networks, human mobility patterns, power grids, ecosystems also display emergent behaviour, a lot of which can be studied using statistical mechanics techniques and the theory of complex adaptive systems.

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M. Scheffer et al.: Early-warning signals for critical transitions, Nature 461, 2009
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R. Ni, M. A. C. Stuart, P. G. Bolhuis: Tunable long range forces mediated by self-propelled colloidal hard spheres, Phys. Rev. Lett. 114, 018302, 2015
R. Kays, M. C. Crofoot, W. Jetz, M. Wikelski.: Terrestrial animal tracking as an eye on life and planet, Science 12 Jun, 2015


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