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I'm researching many topics including emergence and chaos theory, and I cannot for the life of me find strictly mathematical treatments of the idea of emergence. Is there any form or field of mathematics that can predict the emergence of one equation from another, or from a set of equations? A simple analogy would be the "emergence" of a velocity equation by differentiating the position equation, and an acceleration equation from a velocity equation. More aptly, for example, is there any known way in which the Navier-Stokes equation can "emerge" from the equations of Schrödinger, Pauli or Dirac (or even the equations of QCD)? Some relatively "simple" transformation based upon, perhaps, a single parameter (ideally, maybe scale, energy, etc), that can change an equation from one integrative level to an equation from a higher/lower integrative level?

I realize this seems to be a hotly debated topic in some ways, but I cannot seem to find what I am looking for. My intuition for some reason says this may involve, among other ideas, fractional differential equations, Galois theory, fractal geometry, nested matrices, Fourier/Laplace transformations, that kind of thing. Deep down (despite my lack of formal mathematical education), I truly feel there HAS to be a relatively simple way in which equations can be transformed from small-scale dynamics to larger, emergent phenomena. Imagine having a transformation that could transform the Schrödinger equation smoothly through the Pauli equation, then the Dirac equation, on up through the Navier-Stokes equation (finally?) arriving at the Einstein Field Equations, all based upon a few (maybe even a single) parameter(s).

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    $\begingroup$ I would say that the most similar thing one can think of is Catastrophe theory. $\endgroup$
    – John B
    Commented Jan 15, 2016 at 0:06
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    $\begingroup$ Are you asking about deriving macroscopic equations from microscopic equations? This is a deep subject, but its most basic aspects are in statistical thermodynamics. Statistical mechanics is a more difficult subject with more problem-specific aspects. It is actually a bit surprising, coming from statistical thermodynamics, that the situation is as problem-specific as it is. $\endgroup$
    – Ian
    Commented Jan 15, 2016 at 0:08
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    $\begingroup$ For example, one would naively expect to always have some kind of Fokker-Planck equation for whatever macroscopic variable you want, even if the parameters are complicated...but in fact the procedure for deriving such a thing, called van Kampen system size expansion, requires somewhat stringent hypotheses. $\endgroup$
    – Ian
    Commented Jan 15, 2016 at 0:10
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    $\begingroup$ from the many versions of Euler-Lagrange equations emerge movement $\endgroup$
    – janmarqz
    Commented Jan 15, 2016 at 0:22

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There is no method (known) in full generality. The does exist a huge amount of literature, mainly in physics, on deriving the equations of continuous or "large" systems as a statistical mechanics limit of N-particle systems.

Emergence usually means something broader than reduction of macroscopic equations to microscopic ones. The idea more is different is that phenomena and quantities that describe a large system can be qualitatively different than those useful for the description of its microscopic building blocks, and knowing the lower level completely is not always sufficient to understand the higher level.

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    $\begingroup$ I think the "general" bifurcation diagram depicts many things. I feel that the bifurcation diagram for ax(1-x) depicts the universe quite well. The bifurcation of the Schrödinger equation (with a single complex answer) leads to the 2-component Pauli equation, which bifurcates again to the 4-component Dirac equation, which possibly bifurcates again to the (seemingly) eight-component function of QCD (and its corresponding eight gluon flavors) which infinitely bifurcates until it hits the critical threshold (one of the Feigenbaum constants) where many possibilities exist in the form of molecules. $\endgroup$
    – Garrett Miller
    Commented Jan 15, 2016 at 0:58
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There is no known way of deriving the Navier-Stokes equation from the Bolzmann equations.

There are attempts at putting emergence on a firm mathematical foundation in very wide generality. While the following introduces it only in the context of cellular automata, it generalises well to other domains:

Robert S. MacKay.Space-time phases, page 387–426. London Mathematical Society Lecture Note Series. Cambridge University Press, 2013.

See also this paper for another method for quantifying emergence using shannon entropy:

ROBIN C. BALL, MARINA DIAKONOVA, and ROBERT S. MACKAY.Quantifying emergence in terms of persistent mutual information.Advancesin Complex Systems, 13(03):327–338, 2010.

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  • $\begingroup$ Thank you Joe Colvin! Great resources! $\endgroup$
    – Garrett Miller
    Commented Jul 6, 2022 at 22:20
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I am reading https://arxiv.org/abs/1904.03424 "Emergence via non-existence of averages" and it mentions Berger's emergence and evelops a pointwise emergence and this article contains statement ([10] is reference to Berger article):

we note that “emergence” is one of the most important concepts in complexity science [34], but had no rigorous formulation before [10] appeared

that suggests that such mathematical theory of emergence has been made at last.

I will expand my answer a bit later after reading the article.

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  • $\begingroup$ Thanks TomR! I'm a bit late catching up on my SE stuff, but I will take a look at that ASAP. $\endgroup$
    – Garrett Miller
    Commented Jun 23, 2023 at 0:11

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