There is more to emergence than novel properties, i.e., where a whole system has a property that the individual components of the system do not have. Here I focus on emergent properties, but in most cases “property” might be replaced with state, phenomenon, or entity. I now discuss ten characteristics often associated with emergence, beyond novelty. Some people include one or more of these characteristics in their definitions of emergence. However, I do not include them in my definition because as I explain some of the characteristics are contentious. Some may not be necessary or sufficient for novel system properties.
The first five characteristics discussed below might be classified as objective (i.e., observable properties of the system) and the second five as subjective (i.e., associated with how an investigator thinks about the system). In different words, the first five are mostly concerned with ontology (what is real) and the second five with epistemology (what we know). The first five characteristics concern discontinuities, universality, diversity, mesoscales, and modification of parts. The second five concern self-organisation, unpredictability, irreducibility, downward causation, and closure.
1. Discontinuities
Quantitative changes in the system can become qualitative changes in the system. For example, in condensed matter physics spontaneous symmetry breaking only occurs in the thermodynamic limit (i.e., when the number of particles of the system becomes infinite). More is different. Thus, as a quantitative change in the system size occurs the order parameter becomes non-zero. In a system that undergoes a phase transition at a non-zero temperature, a small change in temperature can lead to the appearance of order and to a new state of matter. For a first-order phase transition, there is discontinuity in properties such as the entropy and density. These discontinuities define a phase boundary in the pressure-temperature diagram. For continuous phase transitions the order parameter is a continuous function of temperature, becoming non-zero at the critical temperature. However the derivative with respect to temperature may be discontinuous and/or thermodynamic properties such as the specific heat and susceptibility associated with the order parameter may approach infinite as the critical temperature is approached.
Two different states of a system are said to be adiabatically connected if one can smoothly deform one state into the other and all the properties of the system also change smoothly. The case of the liquid-gas transition illustrates subtle issues about defining emergence. A discontinuity does not imply a qualitative difference (novelty). On the one hand, there is a discontinuity in the density and entropy of the system as the liquid-gas phase boundary is crossed in the pressure-temperature diagram. On the other hand, there is no qualitative difference between a gas and a liquid. There is only a quantitative difference: the density of the gas is less than the liquid. Albeit sometimes the difference is orders of magnitude. The liquid and gas state can be adiabatically connected. There is a path in the pressure-temperature phase diagram that can be followed to connect the liquid and gas states without any discontinuities in properties.
The ferromagnetic state also raises questions, as illustrated by a debate between Rudolf Peierls and Phil Anderson about whether ferromagnetism exhibits spontaneous symmetry breaking. Anderson argued that it did not as, in contrast to the antiferromagnetic state, a non-zero magnetisation (order parameter) occurs for finite systems and the magnetic order does not change the excitation spectrum, i.e., produce a Goldstone boson. On the other hand, singularities in properties at the Curie temperature (critical temperature for ferromagnetism) only exist in the thermodynamic limit. Also, a small change in the temperature, from just above the Curie temperature to below, can produce a qualitative change, a non-zero magnetisation.
2. Universality
Properties often referred to as emergent are universal in the sense that it is independent of many of the details of the parts of the system. There may be many different systems that can have a particular emergent property. For example, superconductivity is present in metals with a diverse range of crystal structures and chemical compositions.
Robustness is related to universality. If small changes are made to the composition of the system (for example replacing some of the atoms in the system with atoms of different chemical element) the novel property of the system is still present. In elementary superconductors, introducing non-magnetic impurity atoms has no effect on the superconductivity.
Universality is both a blessing and a curse for theory. Universality can make it easier to develop successful theories because it means that many details need not be included in a theory in order for it to successfully describe an emergent phenomenon. This is why effective theories and toy models can work even better than might be expected. Universality can make theories more powerful because they can describe a wider range of systems. For example, properties of elemental superconductors can be described by BCS theory and by Ginzburg-Landau theory, even though the materials are chemically and structurally diverse. The curse of universality for theory is that universality illustrates the problem of “under-determination of theory”, “over-fitting of data” and “sloppy theories” [Sethna et al.]. A theory can agree with the experiment even when the parameters used in the theory may be quite different from the actual ones. For example, the observed phase diagram of water can be reproduced, sometimes with impressive quantitative detail, by combining classical statistical mechanics with empirical force fields that assume water molecules can be treated purely being composed of point charges.
Suppose we start with a specific microscopic theory and calculate the macroscopic properties of the system, and they agree with experiment. It would then be tempting to think that we have the correct microscopic theory. However, universality suggests this may not be the case.
For example, consider the case of a gas of weakly interacting atoms or molecules. We can treat the gas particles as classical or quantum. Statistical mechanics gives exactly the same equation of state and specific heat capacity for both microscopic descriptions. The only difference may be the Gibbs paradox [the calculated entropy is not an extensive quantity] which is sensitive to whether or not the particles are treated as identical or not. Unlike the zeroth, first, and second law of thermodynamics, the third law does require that the microscopic theory be quantum. Laughlin discusses these issues in terms of “protectorates” that hide “ultimate causes” .
In some physical systems, universality can be defined in a rigorous technical sense, making use of the concepts and techniques of the renormalisation group and scaling. These techniques provide a method to perform coarse graining, to derive effective theories and effective interactions, and to define universality classes of systems. There are also questions of how universality is related to the robustness of strata, and the independence of effective theories from the coarse-graining procedure.
3. Diversity
Even when a system is composed of a small number of different components and interactions, the large number of possible stable states with qualitatively different properties that the system can have is amazing. Every snowflake is different. Water is found in 18 distinct solid states. All proteins are composed of linear chains of 20 different amino acids. Yet in the human body there are more than 100,000 different proteins and all perform specific biochemical functions. We encounter an incredible diversity of human personalities, cultures, and languages. A stunning case of diversity is life on earth. Billions of different plant and animal species are all an expression of different linear combinations of the four base pairs of DNA: A, G, T, and C.
This diversity is related to the idea that "simple models can describe complex behaviour". One example is Conway’s Game of Life. Another example is how simple Ising models with a few competing interactions can describe a devil's staircase of ground states or the multitude of different atomic orderings found in binary alloys.
Goldenfeld and Kadanoff defined complexity [emergence] as “structure with variations”. Holland (VSI) discusses “perpetual novelty” giving the example of the game of chess, where are typical game may involve the order of 1050 move sequences. “Motifs” are recurring patterns (sequences of moves) in games.
Condensed matter physics illustrates diversity with the many different states of matter that have been discovered. The underlying microscopics is “just” electrons and atomic nuclei interacting according to Coulomb’s law.
The significance of this diversity might be downplayed by saying that it is just a result of combinatorics. But such a claim overlooks the issue of the stability of the diverse states that are observed. In a system composed of many components each of which can take on a few states the number of possible states of the whole system grows exponentially with the number of components. For example, for a chain of ten amino acids there are 1013 different possible linear sequences. But this does not mean that all these sequences will produce a functional protein, i.e., a molecule that will fold rapidly (on the timescale of milliseconds) into a stable tertiary structure and perform a useful biochemical function such as catalysis of a specific chemical reaction or signal transduction.
4. Simple entities at the mesoscale
A key idea in condensed matter physics is that of quasi-particles. A system of strongly interacting particles may have excitations, seen in experiments such as inelastic neutron scattering and Angle Resolved PhotoElectron Spectroscopy (ARPES), that can be described as weakly interacting quasi-particles. These entities are composite particles, and have properties that are quantitatively different, and sometimes qualitatively different, from the microscopic particles. Sometimes this means that the scale (size) associated with the quasi-particles is intermediate between the micro- and the macro-scales, i.e., it is a mesoscale. The existence of quasi-particles leads naturally to the technique of constructing an effective Hamiltonian [effective theory] for the system where effective interactions describe the interactions between the quasi-particles.
The economist Herbert Simon argued that a characteristic of a complex system is that the system can be understood in terms of nearly decomposable units. Rosas et al., argue that emergence is associated with there being a scale at which the system is “strongly lumpable”. Denis Noble has highlighted how biological systems are modular, i.e., composed of simple interchangeable components.
5. Modification of parts and their relationships
Emergent properties are often associated with the state of the system exhibiting patterns, order, or structure, terms that may be used interchangeably. This reflects that there is a particular relationship (correlation) between the parts which is different to the relationships in a state without the emergent property. This relationship may also be reflected in a generalised rigidity. For example, in a solid applying a force on one surface results in all the atoms in the solid experiencing a force and moving together. The rigidity of the solid defines a particular relationship between the parts of the system.
Properties of the individual parts may also be different. For example, in a crystal single-atom properties such as electronic energy levels change quantitatively compared to their values for isolated atoms. Properties of finite subsystems are also modified, reflecting a change in interactions between the parts. For example, in a molecular crystal the frequencies associated with intramolecular atomic vibrations are different to their values for isolated molecules. However, emergence is a sufficient but not a necessary condition for these modifications. In gas and liquid states, novelty is not present but there are still such changes in the properties of the individual parts.
As stated at the beginning of this section the five characteristics above might be associated with ontology (what is real) and objective properties of the system that an investigator observes and depend less on what an observer thinks about the system. The next five characteristics might be considered to be more subjective, being concerned with epistemology (how we determine what is true). In making this dichotomy I do not want to gloss over the fuzziness of the distinction or of two thousand years of philosophical debates about the relationship between ontology and epistemology, or between reality and theory.
In the next post, I will discuss the remaining five characteristics: self-organisation, unpredictability, irreducibility, contextuality and downward causation, and intra-stratum closure.
Thanks for reading this far!
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