Modellen - KNAW...Newton Maxwell 1687 1865 Clausius Kelvin Boltzmann 1870 Darwin 1860 Einstein...
Transcript of Modellen - KNAW...Newton Maxwell 1687 1865 Clausius Kelvin Boltzmann 1870 Darwin 1860 Einstein...
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Macht & Onmacht van het modelma4g denken in de natuurwetenschappen
Sander Bais Ins4tuut voor Theore4sche Fysica
Universiteit van Amsterdam Santa Fe Ins4tute
A universe in search of itself
De kennismachine: een model van modellen Modelambi4es Een natuurlijke hiërarchie van structuren Een hiërarchie van krachten Evolu4e Fronten van kennis Een stapeling van modellen Vergelijkingen Oplossingen Collec4ef gedrag en emergen4e: 1+1=3 Universaliteit? Irreducibele complexiteit? (Low probability large impact: de prijs van onzekerheid)
Comenius
De kennismachine: een model van modellen
vragen meten weten maken
“ …of wonder is the seed of knowledge is the…”
“…of knowledge is the seed of technology is the…” Comenius
De wetenschappelijke methode
Falsificatie
Informatica Simulatie
Dataverwerking
Wiskunde Logische consistentie
Taal van de natuur
Kennis
Technologie Modelambities:• beschrijven• verklaren• voorspellen
- Wetten-vergelijkingen - Ruimte van oplossingen
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Aarde
Quarks Heelal
Atoom
Mens
DNA
Cel Natuur en milieu
Zonnestelsel
Melkweg Kern
Natuurlijke hierarchie van structuren Zwakke kernkracht
Electromagnetische kracht
Zwaartekracht Sterke Kernkracht
Een hierarchie van krachten
Comenius
Kosmische Evolutie Expansie van kennis op alle fronten
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Comenius
Grote afstanden Kleine afstanden
Complexiteit
Drie Fronten Wetenschappelijke keerpunten: de wetten van dynamica & structuur
Klassieke fysica
Evolutietheorie
Relativiteitstheorie
Quantumtheorie
DNA
Newton Maxwell 1687 1865
Clausius Kelvin Boltzmann
1870
Darwin
1860
Einstein
1905-1916
Planck Bohr
Schrodinger
Heisenberg Dirac
1925
Crick Watson
1953
Standaard model Feynman,….
1950-1975
Plaattectoniek
Wegener Holmes
1960
De NatuurweXen
mp mw
c h
GN e
N , k
GN
De NatuurweXen
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The many faces of a cube….
Newton Mechanica Newton Gravita4ewet GN
Schrodinger/Heisenberg: Quantummechanica
Einstein: Algemene rela4viteit
Standaard Model Quantumvelden
Einstein: Speciale rela4viteit
?? ħ
c
The many faces of a cube….
Newton Mechanica Newton Gravita4ewet GN
Schrodinger/Heisenberg: Quantummechanica
Einstein: Algemene rela4viteit
Standaard Model Quantumvelden
Einstein: Speciale rela4viteit
?? ħ
c
Materie
Ruimtetijd
The many faces of a cube….
Dynamica Gravita4ewet GN
Quantummechanica
Algemene rela4viteit
Quantumvelden
Speciale rela4viteit
?? ħ
c
diff.rekening funct. analyse vector algebra. determ chaos fractals…
(diff) meetkunde niet euclidisch topologie
operatorwaardig operator algebra’s hilbertruimtes groepentheorie statistiek padintegralen
De NatuurweXen
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Newton
Boltzmann
BoXom up: collec4ve behaviour & emergence Navier‐Stokes
Newton
Boltzmann
Navier‐Stokes Op4malisa4e paradigma: Maximum entropie beginsel
H(pi, !) = !N!
i
pi log pi ! !((!
i
pi)! 1)! µ((!
i
pii2)! "2)
#H
#pi= 0 (i = 1, ..., N)
" ! log pi ! 1! !! µi2 = 0
" pi = $e!ui2
#H
#!= 0
"!
i
pi ! 1 = 0
#H
#µ= 0
"!
i
pii2 ! "2 = 0
Boltzmannn Gibbs Shannon Jaynes
pi kansverdeling ensemble
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The Gaussian or normal distribu4on
f(x) =1!
2!"2e!x2/2!2
(with average <x>=0 and variance σ =1)
The Gaussian distribu4on Central limit theorem
I know of scarcely anything so apt to impress the imagina4on as the wonderful form of cosmic order expressed by the ‘law of frequency of error’ [the normal or Gaussian distribu4on]. Whenever a large sample of chao4c elements is taken in hand and marshaled in the order of their magnitude, this unexpected and most beau4ful form of regularity proves to have been latent all along. The law … reigns with serenity and complete self‐effacement amidst the wildest confusion. The larger the mob and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of unreason.
(Galton, 1889)
The Ising model and its descendants The 2‐d ferro magne4c ising model
The Ising model in an external field
1. Groundstate(s): In a typical configura4on, are most of the spins +1 or −1, or are they split equally?
2. Correla4ons: If a spin at any given posi4on is 1, what is the probability that the spin at posi4on j is also 1?
3. Phase transis4ons: If β is changed, is there a phase transi4on? 4. Interac4on types: If J is random, how many different configura4ons are there at
any given inverse temperature? 5. On a laqce, what is the fractal dimension of the shape of a large cluster of +1
spins?
Distribu4on
Numerical Simula4ons
The model prolifera4on
The study of non‐perturba4ve collec4ve behaviour: Generalized spin models ((groups and irreps) Generalized laqces ( triangula4ons, networks, grapphs) Generalized interac4ons (asymmetric rules) Laqce gas Height models Percola4on models Spin glasses Matrix models Laqce gauge theories (link and plaqueXe variables) Kitaev models for topological order Loop gasses Networks and adap4ve systems
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Phase transi4ons
“No” phase transi4ons in finite systems Cri4cality (fluctua4ons on all scales!) Scale (conformal) invariance: power law behaviour
Con4nuum limit can be defined Universality (cri4cal exponents) If d=2 Conformal Field Theory (= space4me coordinates on worldsheet of stringtheory)
Networks and adap4ve systems dynamics
Agent based modeling
Aim: Predic4ng complex collec4ve behaviour from simple (local) cons4tuent behaviour.
Agent based modeling and simula4on
Agent‐based modeling and simula4on (ABMS) is a new approach to modeling systems comprised of interac4ng autonomous agents. ABMS promises to have far‐reaching effects on the way that businesses use computers to support decision‐making and researchers use electronic laboratories to do research. Computa4onal advances make possible a growing number of agent‐based applica4ons across many fields. Applica4ons range from modeling agent behavior in the stock market and supply chains, to predic4ng the spread of epidemics and the threat of bio‐warfare, from modeling the growth and decline of ancient civiliza4ons to modeling the complexi4es of the human immune system, and many more.
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Boyd’s AB model for flocking
1. Cohesion: each agent steers toward the average posi4on of its nearby “flockmates,” (long range aXrac4on)
2. Separa4on: each agent steers to avoid crowding local flockmates, (short range repulsion)
3. Alignment: each agent steers towards the average heading of local flockmates. (coupling to background mean field)
http://www.youtube.com/watch?v=UM8SzF6_0sM
Frame dependence of Modelling
How a biologist fixes a radio How a biologist fixes a radio
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How a biologist fixes a radio Modellen
Cons4tuents/bouwstenen en roosters, netwerken, clusters (labels, toestands/configura4e ruimte)
Interac4es (krachten, koppelingen, control/parameter ruimte) Beperkende voorwaarden. (constraints) => Ac4e of energie func4e (measure on configura4on space) Complexiteit (landschap, metastabiliteit, computa4onal NP
problemen) Sta4onaire toestanden (evenwicht)
(op4malisa4e, minimale energie, maximale entropie etc) Ensembles, correla4es, orderparameters, faseovergangen, kri4ek
gedrag. Universaliteit, emergen4e, schalingsgedrag. Dynamica ( beginvoorwaarden etc)
Books
De natuurweXen/The equa4ons 2005 AUP/HUP Keerpunten/In Praise of Science 2009 AUP/MIT Press The Physics of Informa4on (with D. Farmer) in The Philosophy of Informa4on (Ed. Adriaans and Van Benthem ) 2008, Elsevier