Scaling and Dimensionalityin Statistical Physics
Hans FogedbyAarhus University
Lecture at the Niels Bohr Institute, April 26, 2006
There are more things in heaven and earth,Horatio,Than are dreamt of in your philosophy.
Hamlet
Hamlet Horatio
Sir Isaac Newton(1642-1727)
Galileo Galilei(1564-1642)
James Clerk Maxwell(1831-1879)
Josiah Willard Gibbs(1839-1903)
Ludwig Boltzmann(1844-1906)
Classical Physics
Outline
Statistical physicsScalingRandom walkPolymersCritical phenomenaFractalsGrowthSOC and Turbulence
Mathematical fractal
Statistical PhysicsStatistical physics enjoys a very special position among the subfields of physics. Itsprinciples hold whenever there are many degrees of freedom interacting with eachother to make a ”complex" system. Its subject vary from microsopic and mesoscopicscales to macroscopic and cosmic scales
1. Thermal equilibrium (the Boltzmann-Gibbs scheme):
2. Driven system out of equilibrium (e.g. Langevin equation of motion):
3. Numerical simulations universal tool, e.g. Monte Carlo simulation of systems in equilibrium and out of equilibrium
BEntropy: logstatistical weight
S k WW
= BProbability: exp( / ) energy, temperature
P E k TE T
∝ −
Langevin equation:
variable, potential, noise
dx dFdt dx
x F
ξ
ξ
= − +
ScalingWhat do
Phase transitions – turbulence – random walkPolymers – onset of chaos –growing interfaces Coast of Norway – stock market pricesDistribution of galaxies – networks – fractals ….
have in common?Answer:
SCALE INVARIANCE
Self Similarity
Self Similarity
Jonathan Swift(1667-1745)
The coast of Norway
Turbulence
Leonardo da Vinci (1452- 1519)
Fractals
Mandelbrot Fractal
Diffusion Limited Aggregation(Witten-Sander Model)
Mathematical fractal without scale
Scale invariant mathematical fractal with coloring
Palazzo Cavalli Franchetti with scale (and style)
Palazzo Cavalli Franchetti (Istituto Veneto di Scienze, Lettere ed Arti)
Scale invariance in general
Scale invariance means no scale in space or/and in timePhenomena look the same on many scalesPhenomena show self similarityNo characteristic time constant or spatial range
CommentComment::
We here discuss scaling in space or/and time (or frequendcy and wave number)We are mainly concerned with classical physicsIn quantum mechanics frequency is also energy and we can have scaling in energy
Scale invariance encountered in
Nonlinear systems, e.g. onset of chaosSystems with many degrees of freedom, e.g. turbulenceSystems in equilibrium, e.g. second order phase transitions Systems far from equilibrium, e.g. growing interfacesComplex systems, e.g. self-organized critical systems, scale-free networks
Scale invariance is an emergent phenomenon
Scale invariance emerges when many degrees of freedominteract forming a complex system
Power law behaviour in space R or/and in time TScaling exponents independent of microscopic detailsScaling exponents take universal valuesSystems fall in universality classes
Scaling exponent
( )
(
:
Scaling expone t
)n :
F R R
G T T
αβ
β
α≈
≈
Scale invariance characterized by power laws
log - log plot
Scaling
Random walk
Robert Brown observed ”dance” of pollen in 1827Albert Einstein explained ”Brownian motion” in 1905Jean Perrin demonstrated that atoms are real in 1908Paul Langevin also studied Brownian motion
Robert Brown (1773-1858)
Albert Einstein(1879-1955)
Jean Perrin(1870-1942)
Paul Langevin(1872-1946)
Robert Brown
Leading Scottish botanistExplored the coasts of Australiaand TasmaniaAdviced Darwin before journeyof the Beagle (1831)Identified the nucleus of the cell
Robert Brown (1773-1858)
Brownian Motion explained in 1905 by Albert Einstein
Title : “On the movement of small particles suspended in a stationary fluid as demanded by the laws of kinetic theory”
Motivation: To justify the kinetic theory of atoms and molecules – and make quantitative predictions
“In this paper it will be shown that according to the laws of molecular-kinetic theory of heat, bodies of a microscopically visible size suspended in a liquid must as a result of thermal molecular motions, perform motion visible under a microscope.”
Albert Einstein (1879-1955)Nobel prize 1921
Jean Perrin
Jean Perrin observed Brownian motion experimentally and verified Einstein’sprediction
In his letter to Einstein:“I did not believe it waspossible to study Brownian motion with such precision”
Accurate calculation ofAvogadro number (number ofmolecules in 18 grams of water ~6x1023)
Jean Perrin (1870-1942)Nobel prize 1926
Perrin’s record (paper and pencil)
Real Brownian Movement
Schematic Brownian movement
2D computer simulationBlue particle representspollen grainRed particles representmolecules bombarding thepollenPollen grain performs a random walk
Random walk on latticeComputer simulation
Random walk is scale invariance
N: Number of steps R: Size of RW clusterT: T time of RW (one step pr unit time)
Scale invariance
Power law behaviour
R
Scaling exponent
1 = for RW2
R N N
R T T
υ
υ
υ
υ
=
=
Further properties of RW
12
2 2
22
RW end-to-end distance
step
RW mean square distance , 1/ 2 RW Gaussian distribution
RW equation of motion
,
<
( , ) exp
( ), ( )
N
i ii
i j ij
d
R r r
rr r
R N r
RP R N NN
dR t tdt
ξ
υ
δ
ξ
=
−
=
< >=< >
>= < >
⎛ ⎞∝ −⎜ ⎟
⎝ ⎠
= <
=
∑
(0) ( ) tξ δ>∝
2
RW in d=2 is recurrent RW in d=3 is transientMany walkers with density move according to the diffusion equation
is the diffusion coeff
Eiicient
nstei
Dt
D
ρ
ρ
ρ∂= ∇
∂
B
B
RW of particle of size in medium with viscosity at temperature ( is Boltzmann's constant)
n relation
6
k Tk
a
aT
D
η
πη=
Gaussian distribution
Central limit theorem
Sum of N random variables withvariance <x2> approach Gaussian distribution withvariance N<x2 > for large NRW scale invariance is a consequence of the central limit theorem
Polymers
Macromolecules composed of monomersCrucial importance in soft matter and biologyMean field theory by Paul FloryScaling theory by Pierre-Giles deGennes
Paul Flory(1910-1985)
Nobel prize 1974
PG deGennesNobel prize 1991
Motor protein
Polymer modeled by random walk
Excluded volume effectModel polymer by self-avoiding random walk (SAW)
Simulation of polymer made of 500 monomers
Self-avoiding path with 20.000 steps
2D random walk
Mean field theory for polymersFlory theory
2
2
2 2
Mean field theory derivation: RW distribution:
Free energy for RW:
Excluded volume:
(equilib
( ) exp( / )
log
, 0 rium)
, sca
RW
EXC d d
SAWSAW RW EXC
P R R NRF PN
N NF dVR R
dFF F FdR
R Nυ υ
≈ −
≈ − ≈
⎛ ⎞≈ ≈⎜ ⎟⎝ ⎠
= + =
≈
∫
ling exponent
Flory's result for
, for 3 4 2
dd
υ
υ = ≤+
1 1 ( ) 2 3 / 4 (exact) 3 3 / 5 (approx. 0.586) 4 1/ 2 (RW)
d R Nddd
υυυυ
= = ∝= == == =
υ depends on dimension dυ > ½, polymer coil expansowing to self-avoidanceυ = ½ above d=4d=4 is critical dimensionself-avoidance irrelevant above d=4
Scaling exponent
Self-avoidance
Critical phenomena
Liquid and gas phases identicalDensity fluctuations on all scalesCorrelation length divergesCritical scattering of light Compressibility divergesPower law behaviorScaling exponents
Critical point Phase diagram for fluid
One of the triumphs of equilibrium statistical mechanics is the completeunderstanding of 2nd order phase transitions – critical phenomena
Basic properties (fluid)
Critical temperature TcOrder parameter Δρ= ρLiquid- ρGasCorrelation length ξ (size of densityfluctuations)Response (compressibility) κ~ dρ/dp
Power law behavior
( )
( )
( )
c
c
c
T T
T T
T T
βρυξγκ
Δ ≈ −
−≈ −
−≈ −
Scaling exponents 1.3-1.5 0.6 1.3
βυγ
≈≈≈
Tc depends on fluid, interaction, etc.Scaling exponents only depend on: 1. Dimension (symmetry) of order
parameter n (n=1 for fluid)2. Dimension of space d
Order parameter Correlation length Response (compressibility)
Ising Model
Paradigm in statistical physicsUsed to model magnetism, alloys, lattice gasses, neural networks, flocking birds, etcSolved by Ising in 1D (1924)Solved by Onsager in 2D (1944)
i Local degree of freedom , lattice point Hamiltonian:
coupling constant, magnetic field nearest neighbor coupl
1
< > ing Partition funct
i j i
i
H J hij i
J hij
σ
σ σ σ
= ±
⎛ ⎞⎜ ⎟= − −⎜ ⎟< > ⎝ ⎠
∑ ∑
1
0
ion: , temperature
Magnetization (order parameter):
Susceptibility (response):
Thermodynamics (Free energy):
exp( / ){ }
exp( / ){ }
log ,
i i
h
Z H T
i
m Z H T
idmdh
F T Z d
T
F SdT
σ
σ σσ
χ
−
=
= −
=< >= −
=
= − = − −
∑
∑
mdh
Ernest Ising(1900-1998)
Lars Onsager(1903-1976)
Nobel Prize 1968
The Ising model
2D Ising lattice
Basic properties (Ising)
2nd order phase transition at TcSpontaneous magnetization below TcBroken symmetry – long range spin orderParamagnetic for T>Tc,Ferromagnetic for T<TcCritical fluctuations at TcCorrelation length measures size ofdomainsCorrelation length diverges at TcSusceptibility (response) diverges at TcSpecific heat diverges at Tc
Monte Carlo simulation Ising-Model
Order parameter Correlation length
Susceptibility
Ising model at Tc. Fluctuations on all length scales
Onsager’s solution(for the afficionado)
[ ]π/2 2 2
N0
2
1+ 1-q sin θF 1 Lim = T log(2cosh(2J/T)) - dθlog N π 2
2sinh(2J/T) q = elliptic integralcosh (2J/T)
log(
Free energy pr spin (Onsager 1944):
SpecificHeat:
C T
→∞
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
≈ −
∫
2 4
4
), 0 , ( ( ) )
1/ 8(1 tanh ( / ) 1 , 1/ 8
16 tanh ( / )
Spontaneous magnetization (Yang 1952):
c cT C T T
J Tmj T
αα
β
−= ≈ −
⎡ ⎤−= − =⎢ ⎥⎣ ⎦
2D Ising model for h=0 on square lattice
Lars Onsager
Tour de force in mathematical physicsFirst exact many body calculation in statistical mechanicsSolution yields spontaneousmagnetization - phase transition
Brief history of critical phenomena
Second order phase transitions (critical phenomena) ubiquitousMean field theory (MFT) developed early for fluids and magnetsMFT codified by Landau (1937), MFT scaling exponentsExact solution by Onsager of 2D Ising model (1944)Experiments and high-T expansions yield exponents (1960 - 70)Exponents disagree with MFT exponentsUniversality and scaling laws (Widom, Fisher, Josephson)Kadanoff construction (1967)Renormalization group (RG) ideas by Wilson 1963 - 1971RG calculation, critical phenomena ”in the bag” 1971 (Wilson)RG ε-expansion (ε=4 - d) 1972 (Wilson and Fisher)Scaling and RG methods and ideas pervade modern theoretical physics, statistical physics and ”complexity theory”
Landau theory (MFT)Order parameter central ideaCoarse grained (hydrodynamical) Expansion of free energy near Tc
MFT
MF
2 4
T
Landau Theory (MFT) Order parameter Free energy expansion for small and near
Scaling exponents
, 1
( )
/ 2,
,
, 0
( ) ( )
( )
(
) 1
c
c
c
c
MM
TdFF B T M UMdM
B T T T
M T T
T T
β
γ
β
γχ
ξ
= + =
∝ −
≈
− =
−
−≈
=
MFT
Onsager Onsager Onsager
, 1/ 2 Onsager's results for 2D Ising model
(
=1/ 8, =7/4 =
)
, 1
cT T υ
β γ υ
υ−− =≈
Free energy
Order parameter
Lev Landau(1908-1968)
Nobel prize 1962
Phase diagram
Scaling laws
Critical exponents related by four scaling lawsBased on mathematical homogeneity assumptionSupported by experiments and numerics
Scaling laws: 2 2 (Rushbrooke)
( -1)= (Widom) (2 ) (Fisher) 2 (Josephson)
is spatial dimensiondd
α β γβ δ λγ η υυ α
+ + =
= −= −
Ben Widom
Michael Fisher Brian JosephsonNobel prize 1973
Kadanoff construction I
c
* *
Block construction and coarse graining Coupling K = J/T Renormalization group (RG) transformation: K K' K'' fixed point K RG equation K' = F(K), scale factor 2 Fixed point equation K = F(K ) Corre
→ → →
*
*
*c
lation length (K') = 2 (K) a) Stable fixed point at K = 0, paramagnetic b) Stable fixed point at K = , ferromagnetic c) Unstable fixed point at K = K , critical point correlation length Kada
ξ ξ
ξ
∞
= ∞noff construction
Yields scaling laws - notqualit
exponative
ents
Kadanoff construction (1967)
as T Tc For T Tc, is the For T=Tc,
Correlation length , fundamental concept
only length scaleno length s,
System becomcale
scale invaries an at Tc Flu
t ctuati
ξξ
ξ
ξ→∞ →
→=∞
allons le (doma ngth sins cal) on es
Leo Kadanoff
Kadanoff construction II
T=0.99 Tc T=Tc T=1.22 Tc
The Renormalization Group
1 1 (4 ), 2 6
11 (4 ),6
1 1 (4 )2 12
d
d
d
β
γ
υ
= − −
= + −
= + −
,,The renormalization group (RG) is a method to extract scaling propertiesfrom a partition functionConstitutes an extension of Kadanoffbloch constructionQuantitative coarse graining and construction of effective HamiltonianThe RG method is a general schemeto construct theories for scaleinvariant systemsThe RG addresses the statisticalcontinuum limit where degrees offreedom interact on many scales
Scale dependent coupling constantsNonlinear RG equations and flow in RG spaceFixed points and relevant and irrelevant scaling fieldsFixed points determine effective”scaling” HamiltoniansVicinity of fixed points yields scalingexponentsCritical dimension d=4Landau theory correct for d>4Correction to Landau theory for d<4Expansion about critical dimension
Scaling exponents to first order in (4-d)
Ken WilsonNobel prize 1982
MFT
MFT
MFT
12
112
β
γ
υ
=
=
=
”Poor man”s RG I
,0 1/
,0 1/ ,1/ 1/
Hamiltonian ( ), order parameter, wavenumber Partition function
exp[ ( ) / ]
Kadanoff construction, is box size
exp[ ( ) / ]
exp(
k
k k
k k
kM k a
kM k L M L k a
H M M k
Z H M T
L
Z H M T
< <
< < < <
= −
⎧ ⎫⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭
−
∑
∑ ∑
( )
1/,1/ 1/
,0 1/
2 2 4
({ } ) exp[ ( ) / ]
exp[ ( )]
Free energy on scale
( )
Sum over configurations between 1/ and 1/( ) exp( ( ))
k
k
L k k L kM L k a
L LM k a
dL L L L L L L
L L L L
F M H M T
Z F M
L
F M d x M R M U M
L L LF Mδ δ
δ
<< <
< <
+ +
= −
= −
⎡ ⎤= ∇ + +⎣ ⎦+
− =
∑
∑
∫
,1/( ) 1/exp( ( ))
k
L LM L L k L
F Mδ+ < <
−∑
Coarse grained order parameter
Momentum shell integration
Coarse grained order parameter
”Poor man”s RG II
1
2
2 3
2 2
2 / 2 1
31
9 (
Renormalization group equations for couplings and
D1 )
,imensionless form
RG equations for and
, log
3 21
dL L
Ld
L L
L
d
L L
L L
dR U LdL R L
d
R U
r
U U LdL R L
r L R u L U t L
dr u
u
u
rdt rd
−
−
−
=+
= −+
= = =
= ++
2
2
9(1 )
4
uudt r
d
ε
ε
= −+
= −
( , ) (0,0)(
RG equations define flow in ( , ) space MFT fixed point: New fixed point: ****************************
For d>4 MFT fixed point independent of , (
, ) ( / 6, / 9)
stable)L L c
r ur u
r u
U L R T T
ε ε
∗ ∗
∗ ∗
≈ −
=
= −
/ 3
unstabl
Landau theory valid, MFT exponents*********************************
For d<4 MFT fixed point New fixed point
Non-classical critic
e stable
, ( )1 1
al expo
, , /
n
L L cU L R T T L
M R URR
ε ε
ξ ξξξ
ξ χ
− −≈ ≈ −
≈ ≈ ≈ −
1/ 2 /12, en
1 / 6,ts
1 / 6 υ ε γ ε β ε= + = + = −
For d>4
For d<4
Fractals
Mandelbrot and Nature"Clouds are not spheres, mountains are not cones,coastlines are not circles, and bark is not smooth,nor does lightning travel in a straight line.“
Mandelbrot 1983
New geometrical description of scale invariant objectsin natural sciences and mathematics
Fractals characterized by non integer dimension
Early history of self similarity
Leibnitz: Recursive self similarityWeierstrass function, ”Monster function”Hilbert and Peano, ”Monster curves”Koch’s curveCantor’s fractal setsPoincare, Klein, Fatou, Julia, Mandelbrot: Iterated functions in the complex planeRichardson: The length of a coast line
The Weierstrass function
Karl Weierstrass(1815-1887)
1
1( ) sin( )aa a
k
F x k xk
ππ
∞
=
= ∑Weierstrass’ ”Monster function”Sum of waves with increasing wavenumber and decreasing amplitudeContinuous but nowhere differentiableFractal dimension
von Koch’s curve
Helge von Koch(1870-1924)
Hilbert and Peano
Hilbert Curve Generator
The space-filling Peano curve upset traditional mathematicsVilenkin: "Everything has come unstrung! It's difficult to put into words the effect that Peano's result had on the mathematical world. It seemed that everything was in ruins, that all the basic mathematical concepts had lost their meaning.“
Hilbert curve deterministic RW
David Hilbert(1862-1943)
Giuseppe Peano(1858-1932)
Georg Cantor
Georg Cantor (1845 - 1918)
Founded set theoryIntroduced the concept of infinite numbersRevolutionized mathematics
I place myself in a certain opposition to widespread views on the nature of themathematical infinite (Cantor)
From his paradise no oneshall ever evict us (Hilbert)
”Cantorism” is a disease from whichmathematics would have to recover (Poincare)
Fog on Fog (Weyl)
Some quotations:
The Cantor set
Cantor set is uncountableCanter set has no lengthCantor set is a fractal
What is removed:
L = 1
Julia sets
21
Quadratic Julia set , complex "Filled-in" Julia set is the set of points which do not approach infinity under iteration
n nz z c z
z
+ = +
Gaston Julia (1893 - 1978)
Mandelbrot set
Mandelbrot clickZoomable clickAutomatic click
Blue metal
Medusa
Grapes
Fractals in the arts
!!
Lewis Fry Richardson
• Mathematician, physicist and psychologist
• Weather forecasting• Mathematical analysis of war,
”Statistics of Deadly Quarrels”• Turbulence theory• Research on length of coastlines
and bordersLewis Fry Richardson
(1881-1953)
The coast of Britain
Richardson posed the question: How long is the coast of Britain
SLength (Scale)Log(Length) S Log(Scale)S is the scaling exponent
×
S=-0.25 for the west coast of BritainS=-0.15 for the land frontier of GermanyS=-0.14 for the land frontier of PortugalS=-0.02 for the South African coastGoogle Earth
Mandelbrot
“Clouds are not spheres, mountains are not cones,coastlines are not circles, and bark is not smooth,nor does lightning travel in a straight line”
Benoit Mandelbrot
Coined the word ”fractal” Demonstrated the application offractal geometry in mathematics and physics“The Fractal Geometry of Nature”
The fractal dimension D
Cover object with N boxes of size aN(a) will depend on a as a power
log log
DN aN D a
−== − × D is the fractal dimension
The coast of Norway
Jens Feder posed the question: How long is the coast of Norway
Outline traced from atlasSquare grid spacing 50 km
From ”Fractals” by Jens FederNorway is scale invariantFractal dimension DNorway ~1.52
Jens Feder
Koch curve
N(a)=a-D, D fractal dimension
N(1) = 1, N(1/3) = 4, N(1/9)=16, ..N(1/3n)=4n
DKoch =log4/log3 ~ 1.261< DKoch <2
Koch curve between line and areaKoch curve ”wollen”Koch curve has infinite lengthRepresents ”mathematical” coastline
Cantor set
N(a)=a-D, D fractal dimension
N(1) = 1, N(1/3) = 2, N(1/9)=4, ..N(1/3n)=2n
DCantor =log2/log3 ~ 0.630< DCantor <1
Cantor set between point and lineCantor set is a ”dust” (Mandelbrot)Cantor set has no length
Sierpinski Gasket
N(a)=a-D, D fractal dimension
N(1) = 1, N(1/2) = 3, N(1/4)=9, ..N(1/2n)=3n
DSier =log3/log2 ~ 1.581< DSier <2
Sierpinski gasket between line and areaSierpinski gasket diluteSierpinski gasket has no area
Waclaw Sierpinsky(1882-1969)
Menger Sponge
Fractal dimension D=log 20/log 3 = 2.73Dimension between 2 and 3Area but no volume
Karl Menger(1902-1985)
3D Sierpinski gasket
Fractal dimension D=log4/log2 = 22D fractal in 3D embedding spaceCompact area but no volume
Waclaw Sierpinsky(1882-1969)
Hilbert curve
Hilbert curve has fractal dimension D=2Hilbert curve is plane filling
David Hilbert(1862-1943)
Growth
Diffusion Limited Aggregation (DLA)
Computer simulation model of growthProposed by Witten and Sander (1981)
Add seed at centerParticle diffuse in from perimeterParticle sticks to seedIntricate branch structure (diffusion limited)Aggregate has fractal dimension
Diffusion limited aggregation (DLA)
DLA fractal dimension
Log-log plot of N(R) versus RSlope yields fractal dimension DFractal dimension of DLA
D ≈ 1.7
Morphology of DLA cluster analysed in terms of the fractal dimension D
R size of cluster (radius of gyration)N(R) # of aggregated particles within radius RD = 2 for compact growthD < 2 for fractal growth
( ) DN R R≈
Bacterial colony
Fractal growth of bacillus subtilisExperiment by Matsushita and FujikawaFractal growth observedDLA-like morhologiesFractal dimension D ~1.72
Bacterial colony
Computer generated fractal
The self-similar rippling of leafedges and torn plastic sheets
Self-similar Koch curve
torn plastic sheet
rippled leaf edge
Discharges (Lichtenberg figures)
Surface leader discharge Discharge pattern in plastic block. Charged with 2 MeV electron beam
Gold colloid and ramified electrode
Viscous fingering
Coloured water injected into wet clay Squeezing oil from bituminous rock
Flame front and Snow deposition
Propagation of flame frontDeposition of snow
Computer simulation
Corals and Cauliflower
Scale invariant cauliflower Scale invariant coral growth
SOC and Turbulence
Self-Organized Criticality (SOC I)
One of the founders and most influential contributors to the study of complex systems. Per Bak made many contributions to science, but the best known was a general theory ofself-organization, which he called, "self-organized criticality". His ideas and discoverieshave had an influence over how people think about a broad range of phenomena, from physics to biology, neurosciences, cosmology, earth sciences, economics and beyond.
Per Bak (1947-2002)
SOC II
SOC introduced by Per Bak, Chan Tang, and Kurt Wisenfeld in 1987Computer model - the ”sand pile” paradigmOpen driven systems self-organize to a critical statePhysics driven by avalanches (domains) on all scalesRobust power law scaling – no fine tuning to critical pointPurported to be underlying mechanism for (aspects) of earth quakes(Gutenberg-Richter law), stock prices, fluctuating river levels, traffic jams, quasar signals, flicker noise, punctuated equilibrium, etc
The ”sand pile” paradigm
Probability P(s) vs avalanche size s
Earth quakes
Energy release versus time (1995)Number of earth quakes (N)versus size (S) (log-log scale)
Gutenberg-Richter law
2/3( )N S S −∼
Turbulence
I am an old man now and when I die and go to Heaven there are two matters on which I hope for enlightenment. One is quantum electro-dynamics, and the other is turbulent motion of fluids. And about the former I am really rather optimistic.
Sir Horace Lamb (1932)
Big whorls have little whorls, which feed on their velocity, And little whorls have lesser whorls, And so on to viscosity.
L. F. Richardson
The most important unsolved problem ofclassical physics.
R. Feynmann
Turbulence
( ) 2u u u p ut
ν∂+ ⋅∇ = −∇ + ∇
∂
Andrei Kolmogorov(1903-1987)
Lewis Fry Richardson(1881-1953)
3/5)( −≈ kkE
Navier Stokes equation
Universal feature:The Kolmogorov5/3 law
Observed power spectrum E(k) in turbulent flows
Nonlinear Dissipative
The end