Abstract
The paper begins with a generic discussion of modelling, focusing on some of its practices and problems. I then move on to a philosophical discussion about emergence and multi-scale modelling; more specifically, the reasons why what looks like a promising strategy for dealing with emergence is sometimes incapable of delivering interesting results. This becomes especially evident when we look more closely at turbulence and what I take to be the main ontological feature of emergent behavior—universality. Finally, I conclude by showing why, despite displaying multi-scale behaviour and some of the characteristics we identify with emergence, turbulence fails to fit neatly into the latter category and is not successfully captured using multi-scale modelling. The complex nature of turbulence illustrates the difficulties in characterizing emergence and why specific criteria are needed in order to prevent every complex behaviour we don’t understand being classified as emergent.
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Reproduced with permission from Warhaft (2002)
Reproduced with permission from Warhaft (2002)
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Notes
Turbulence is not always identified in the physics literature as an emergent phenomenon but then defining emergence is not a topic that preoccupies many physicists. There are, however, instances where it is classified as such; see the collection of essays edited by Echekki and Mastorakos (2011) as well as Kivelson and Kivelson (2016). In these cases it is typically assumed that the flow properties are emergent but Kivelson and Kivelson provide their own definition of emergence which is “An emergent behavior of a physical system is a qualitative property that can only occur in the limit where the number of microscopic constituents tends to infinity”. They go on to claim that because hydrodynamics is explicitly concerned with the behaviour of fluids at scales large compared with all microscopic scales, all distinct hydrodynamic behaviour, including laminar flow, turbulence and shock waves, are emergent. What is especially interesting is the distinction they draw between qualitative distinctions and quantitative ones. They latter are not considered emergent, as in the case of highly correlated electronic materials and their weakly correlated counterparts; a distinction which is, definitionally, quantitative. The notion of “tending” to infinity is somewhat problematic as a definition of emergence, as I discuss later in the paper.
See Batterman (2015) for an interesting discussion of homogenization techniques.
Humphreys also has an interesting discussion of the problems that surround undecidability of values for certain properties and the difficulties in associating micro properties with real macro ones. Although these issues are important for understanding the micro–macro relation in cases of emergence space constraints prevent me from addressing those arguments here.
Of course, we need to distinguish here between the use of smooth limits that can easily be approximated and singular limits where this is not the case. For a further discussion of this see Morrison (2015).
The symbols Zα and Mα are the atomic number and mass of the αth nucleus, Rα is the location of this nucleus, e and m are the electron charge and mass, rj is the location of the jth electron, and h is Planck's constant. The NIST (National Institute of Standards and Technology) reference on constants, units, and uncertainty. US NIST gives the CODATA recommended values for fundamental physical constants and provides an explanation of how the values are arrived at. In the case of Planck’s constant, although it is generally regarded as the starting point of quantum physics its physical nature has not been well understood. It was originally seen as a fitting constant to explain the black-body radiation but no satisfactory theoretical justification or derivation has been forthcoming. In a recent paper Chang (2017) uses Maxwell’s theory to directly calculate the energy and momentum of a radiation wave packet which was determined to be proportional to its oscillation frequency. That in turn enabled him to derive the value of the Planck’s constant. See arxiv.org/pdf/1706.04475. But here again, this is not really a derivation from first principles.
Localization involves the absence of diffusion of waves in a random medium caused by a high concentration of defects or disorder in crystals or solids. In the case of electric properties in disordered solids we get electron localization which turns good conductors into insulators (See Anderson 1958).
When I say “accurately accounted for” what I have in mind here is that without taking the thermodynamic limit we are unable to calculate specific values for the fixed points/critical indices which mark the phase transition. Of course we can get a reasonably accurate description of the behaviour of the system around critical point without the thermodynamic limit but this is simply a phenomenological description and is unable to explain other aspects of phase transitions such as universality.
An example of an order parameter is the net magnetization in a ferromagnetic system undergoing a phase transition. For liquid/gas transitions, the order parameter is the difference of the densities. From a theoretical perspective, order parameters arise from symmetry breaking. When this happens, one needs to introduce one or more extra variables to describe the state of the system. For example, in the ferromagnetic phase, one must provide the net magnetization, whose direction was spontaneously chosen when the system cooled below the Curie point. However, note that order parameters can also be defined for non-symmetry-breaking transitions. Some phase transitions, such as superconducting and ferromagnetic, can have order parameters for more than one degree of freedom. In such phases, the order parameter may take the form of a complex number, a vector, or even a tensor, the magnitude of which goes to zero at the phase transition.
The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. Phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the critical exponents of the system. Diverse systems with the same critical exponents—those that display identical scaling behaviour as they approach criticality—can be shown, via RG, to share the same fundamental dynamics.
Cooperative behaviour is produced when many interacting entities attempt to satisfy some optimal conditions different from what is produced via aggregation. The correlations in systems displaying cooperative behaviour result in universal features.
I am grateful to an anonymous referee for suggesting that I clarify this point.
See Lesne (1998).
This assumption is well supported by numerical and experimental results (Sreenivasan 1984).
This is especially true since many turbulent flows have in the order of 1018 excited degrees of freedom; some of which must be modelled statistically if the flow is to be computable.
This point was originally made in Falkovich and Sreenivasan (2006).
I would like to thank the Social Sciences and Humanities Research Council of Canada for research support and two anonymous referees for helpful comments.
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Morrison, M. Turbulence, emergence and multi-scale modelling. Synthese 198 (Suppl 24), 5963–5985 (2021). https://doi.org/10.1007/s11229-018-1825-5
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DOI: https://doi.org/10.1007/s11229-018-1825-5