Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

1 of 27

Why Complexify?Principles of Complex Systems

CSYS/MATH 300, Fall, 2011

Prof. Peter Dodds

Department of Mathematics & Statistics | Center for Complex Systems |Vermont Advanced Computing Center | University of Vermont

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

2 of 27

Outline

Universality

Symmetry Breaking

The Big Theory

Final words

For your consideration

References

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

3 of 27

Limits to what’s possible:

Universality (�):I The property that the macroscopic aspects of a

system do not depend sensitively on the system’sdetails.

I Key figure: Leo Kadanoff (�).

Examples:I The Central Limit Theorem:

P(x ;µ, σ)dx =1√2πσ

e−(x−µ)2/2σ2dx .

I Navier Stokes equation for fluids.I Nature of phase transitions in statistical mechanics.

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

4 of 27

Universality

I Sometimes details don’t matter too much.I Many-to-one mapping from micro to macroI Suggests not all possible behaviors are available

at higher levels of complexity.

Large questions:I How universal is universality?I What are the possible of long-time states (attractors)

for a universe?

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

5 of 27

Fluids mechanics

I Fluid mechanics = One of the great successes ofunderstanding complex systems.

I Navier-Stokes equations: micro-macro systemevolution.

I The big three: Experiment + Theory + Simulations.I Works for many very different ‘fluids’:

I the atmosphere,I oceans,I blood,I galaxies,I the earth’s mantle...I and ball bearings on lattices...?

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

6 of 27

Lattice gas models

Collision rules in 2-d on a hexagonal lattice:

I Lattice matters...I No ‘good’ lattice in 3-d.I Upshot: play with ‘particles’ of a system to obtain

new or specific macro behaviours.

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

7 of 27

Hexagons—Honeycomb: (�)

I Orchestrated? Or an accident of bees working hard?I See “On Growth and Form” by D’Arcy Wentworth

Thompson (�). [4, 5]

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

8 of 27

Hexagons—Giant’s Causeway: (�)

http://newdesktopwallpapers.info

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

9 of 27

Hexagons—Giant’s Causeway: (�)

http://www.physics.utoronto.ca/

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

10 of 27

Hexagons run amok:

I Graphene (�): single layer ofcarbon molecules in a perfecthexagonal lattice (super strong).

I Chicken wire (�) . . .

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

11 of 27

Whimsical but great example of real science:

“How Cats Lap: Water Uptake by Felis catus” (�)Reis et al., Science, 2010.

Amusing interview here (�)

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

12 of 27

Symmetry Breaking

Philip Anderson (�)—“More is Different,” Science, 1972 [1]

I Argues against idea thatthe only real scientistsare those working onthe fundamental laws.

I Symmetry breaking→different laws/rules atdifferent scales...

2006 study→ “most creative physicist in the world” (�)

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

13 of 27

Symmetry Breaking

“Elementary entities of science X obey the laws ofscience Y”

I XI solid state or

many-body physicsI chemistry

I molecular biologyI cell biology...

I psychologyI social sciences

I YI elementary particle

physicsI solid state

many-body physicsI chemistryI molecular biology...

I physiologyI psychology

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

14 of 27

Symmetry Breaking

Anderson:I [the more we know about] “fundamental laws, the

less relevance they seem to have to the very realproblems of the rest of science.”

I Scale and complexity thwart the constructionisthypothesis.

I Accidents of history and path dependence (�)matter.

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

15 of 27

Symmetry Breaking

I Page 291–292 of Sornette [3]:Renormalization ≡ Anderson’s hierarchy.

I But Anderson’s hierarchy is not a simple one: therules change.

I Crucial dichotomy between evolving systemsfollowing stochastic paths that lead to(a) inevitable or (b) particular destinations (states).

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

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More is different:

http://xkcd.com/435/ (�)

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

17 of 27

A real science of complexity:

A real theory of everything anything:1. Is not just about the ridiculously small stuff...2. It’s about the increase of complexity

Symmetry breaking/Accidents of history

vs. Universality

I Second law of thermodynamics: we’re toast in thelong run.

I So how likely is the local complexification of structurewe enjoy?

I How likely are the Big Transitions?

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

18 of 27

Complexification—the Big Transitions:

I Big Bang.I Big Random-

ness.I Big Replicate.I Big Life.I Big Evolve.

I Big Word.I Big Story.I Big

Number.I Big God.I Big Make.

I Big Science.I Big Data.I Big Information.I Big Algorithm.I Big Connection.I Big Social.I Big Awareness.

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

19 of 27

Why complexify?

I “Why do things become more complex?” [2]

Brian ArthurScientific American, 268, 92, 1993.

I Complexification ≡ evolution of algorithms?I Differential equations and stories ⊂ Algorithms.I Life is a loaded word: The Search for Extraterrestrial

Algorithms (SETA)?

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

20 of 27

Why complexify?

Driving complexity’s trajectory:I Big BangI Randomness leads to replicating structures;I Biological evolution;I Sociocultural evolution;I Technological evolution;I Sociotechnological evolution.

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

21 of 27

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

22 of 27

Homo narrativus—What’s the Story?:

http://xkcd.com/904/ (�)

I Mechanisms =Evolution equations,algorithms, stories, ...

I Rollover zing: “Also, allfinancial analysis. And,more directly, D&D.”

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

23 of 27

(Sir Terry) Pratchett’s (�) Narrativium (�):

I “The most common element on thedisc, although not included in thelist of the standard five: earth, fire,air, water and surprise. It ensuresthat everything runs properly as astory.”

I “A little narrativium goes a longway: the simpler the story, thebetter you understand it.Storytelling is the opposite ofreductionism: 26 letters and somerules of grammar are no story atall.”

I “Heroes only win when outnumbered, and thingswhich have a one-in-a-million chance of succeedingoften do so.”

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

24 of 27

The absolute basics:

Science in three steps:1. Find interesting/meaningful/important phenomena

involving spectacular amounts of data.2. Describe what you see.3. Explain it.

Beware your assumptions:Don’t use tools/models because they’re there, or becauseeveryone else does...

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

25 of 27

Next:

Spring 2013: Complex Networks (CSYS/MATH 303)I Branching networks (rivers, cardiovascular systems)I Redistribution networks (airlines, post)I Structure detection for complex systemsI ContagionI Random networks-aramaI Distributed SearchI Organizational networksI Deeper investigations of scale-free networksI and more...

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

26 of 27

References I

[1] P. W. Anderson.More is different.Science, 177(4047):393–396, 1972. pdf (�)

[2] W. B. Arthur.Why do things become more complex?Scientific American, 268:92, 1993. pdf (�)

[3] D. Sornette.Critical Phenomena in Natural Sciences.Springer-Verlag, Berlin, 2nd edition, 2003.

[4] D. W. Thompson.On Growth and From.Cambridge University Pres, Great Britain, 2nd edition,1952.

Why Complexify?

Universality

SymmetryBreaking

The Big Theory

Final words

For yourconsideration

References

27 of 27

References II

[5] D. W. Thompson.On Growth and Form — Abridged Edition.Cambridge University Press, Great Britain, 1961.