複雜系統 - 中研院物理研究所- Academia Sinica
複雜系統 - 中研院物理研究所- Academia Sinica
複雜系統 - 中研院物理研究所- Academia Sinica
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Introduction to Complex Systems<br />
<br />
梁 <br />
理 <br />
Kwan-tai Leung<br />
Institute of Physics, <strong>Academia</strong> <strong>Sinica</strong>,<br />
Taipei, 115, Taiwan<br />
http://www.sinica.edu.tw/~leungkt
Outline of talk<br />
• What is complex system<br />
• Brownian motion and random walk<br />
• Power laws & self-similarity<br />
• Self-organized criticality<br />
• Earthquake and its modeling<br />
• Some current research topics<br />
- crack pattern formation<br />
- water striders<br />
-…..
What is a complex system ()?<br />
<br />
來 <br />
行 <br />
行 <br />
<br />
類 力 <br />
行 <br />
力 力 念
Brownian motion & random walk<br />
朗 行 <br />
Let us start with a simple system:<br />
朗 <br />
不 行<br />
不 <br />
2 µm polysteryne spheres<br />
in water<br />
doing random walks
The random walker’s trajectory ()<br />
It does look random
The motion of a random walker is better described by<br />
an equation known as Langevin equation:<br />
dx<br />
dt<br />
= µ f + ξ<br />
f<br />
=<br />
−<br />
dU<br />
dx<br />
力 µ mobility ξ <br />
<br />
流 不 粒 不<br />
粒 粒 度 ρ( x,<br />
t)<br />
度 <br />
:<br />
∂ρ<br />
= D<br />
∂t<br />
∂<br />
∂<br />
2<br />
ρ<br />
2<br />
x
律<br />
x ≈ Dt<br />
2 D=diffusion constant<br />
~ 10 -9 cm 2 /sec<br />
for 2µm particle in water
Power laws 數 律<br />
When two physical variables are related, the simplest<br />
relationship is a power law. E.g.<br />
x<br />
=<br />
1<br />
2<br />
gt<br />
2<br />
∝<br />
t<br />
2<br />
g<br />
Free fall<br />
x<br />
=<br />
vt<br />
∝t<br />
Constant speed<br />
2<br />
x ≈ Dt<br />
Random walk
Self-similarity <br />
cut and blow up<br />
It looks random, but<br />
it also looks “self similar”
Self-similarity in fractal ()<br />
cut<br />
Blow up<br />
Sierpinski gasket
A practical example of self-similarity
Phase transitions <br />
Self-similarity and power laws<br />
can be found in phase<br />
transitions<br />
Common material’s 3 phases<br />
Magnetic material
Power laws in thermodynamic<br />
quantites at phase transition<br />
Magnetization 率<br />
Specific heat
Lattice-gas model of phase transitions<br />
~ many random walkers with weak attraction<br />
T=T c<br />
T=1.05 T c T=2T c
Scale invariance or self-similarity at T c<br />
After the operation, the left picture<br />
Is reduced into this box
A self-similar trajectory lacks characteristic length and<br />
time scale. If you do a Fourier analysis, there is no<br />
characteristic frequency – the power spectrum is also<br />
a power law:<br />
1<br />
P( f ) ≈ α=2<br />
α<br />
f
P( f ) ≈<br />
1<br />
f<br />
α<br />
White noise<br />
α=0<br />
1/f noise<br />
α=1<br />
α=2<br />
Brownian noise
Self-Organized Criticality (SOC)<br />
臨 <br />
列 率 <br />
列 <br />
率 都 率 <br />
流 亮 度 <br />
數 樂 量 理 <br />
便 理 論 理 <br />
Per Bak1987 年 參<br />
數 數 律 臨 <br />
(self-organized criticality) 更 <br />
(sandpile model) 來 念
Sandpile as a paradigm of SOC<br />
Bak & Chen, Sc. Am. 1991<br />
• “Self-organized” means systems reaching critical states without tuning.<br />
• “Critical”: at critical slope (“angle of repose”), no characteristic avalanche<br />
size exists. It covers all possible values with power-law distribution P(s)~s -b .
Open-boundary conditions <br />
are important to ensure self-organized criticality
One major success of SOC is in earthquake modeling
Gutenberg-Richter law for earthquake magnitude<br />
N(>m) per yr<br />
Cumulative distribut’n<br />
Data from sesmicity catalog 1973-1997<br />
Slope 1.62<br />
(compared to 1.8 of model)<br />
Earthquake magnitude m
self-organized critical earthquake model<br />
fault<br />
Olami, Feder, and Christensen, Phys. Rev. Lett. 68, 1244 (1992)
Slip-size distribution in earthquake model<br />
S<br />
Latest results: Lise & Paczuski PRE 2001<br />
One slip initiates an avalanche of ultimately S slips<br />
P(S)~S -1.8<br />
P(S)<br />
S
Outline of talk<br />
• What is complex system<br />
• Brownian motion and random walk<br />
• Power laws & self-similarity<br />
• Self-organized criticality<br />
• Earthquake and its modeling<br />
• Some current research topics<br />
- crack pattern formation<br />
- water striders<br />
-…..
The scales of cracks<br />
monolayer of microspheres<br />
dried lake<br />
earthquake faults<br />
100 km<br />
50µm<br />
1m<br />
dried clay<br />
20cm<br />
100m<br />
fissures in Black Rock<br />
Desert, Nevada
monolayer of polystyrene spheres (µm size)<br />
Early stage<br />
0.5 mm<br />
0.05 mm<br />
Skjeltorp & Meakin
In presence of dissipation (e.g. friction), crack growth<br />
is subcritical, and speed of crack tip
What happens at crack tips?<br />
σ 0<br />
σ >> σ<br />
yy 0<br />
σ 0<br />
Tensile stress in a block<br />
Stress concentration at tip
σ xx<br />
Stress field Crack path<br />
A crack modifies the stress field around it, which<br />
in turn dictates its subsequent propagation.<br />
σ yy<br />
= 0<br />
crack nucleated<br />
from boundary<br />
crack nucleated in bulk, then<br />
propagates toward boundary
Inexpensive experiments<br />
• coffee-water mixture Groisman & Kaplan, Europhys Lett. 1994<br />
frictional<br />
substrate<br />
slippery<br />
substrate<br />
• starch-water mixture<br />
Leung & Neda, Phys. Rev. Lett. 2000<br />
thin layer<br />
thick layer
Length scale<br />
fragment<br />
area<br />
without 120 o joint<br />
with 120 o joint<br />
Morphology<br />
thickness 2<br />
fraction of<br />
120 o joint<br />
from coffee-water mixture expt<br />
Groisman & Kaplan, Europhys Lett. 1994
Approach: simplify the problem as much as possible—<br />
one step from being trivial. Try to capture the essential<br />
physics, then test the results against real, complex situations.
Simulating cracks: a spring-block model<br />
{<br />
i<br />
x i<br />
, y }<br />
stick-slip<br />
kH<br />
force<br />
F<br />
i<br />
> F slip<br />
Fi<br />
= 0<br />
local<br />
equilibrium<br />
cracking<br />
slippings<br />
tension<br />
τ ><br />
F crack<br />
crackings<br />
…<br />
τ = 0
Some typical evolution of cracks by simulations<br />
Simulation using spring-block model<br />
strain=0.5 κ=1 thickness H=2<br />
Close-up of crack propagation and branching
Simulation using spring-block model<br />
strain=0.3 κ=0.5 thickness H=3<br />
Diffusive cracks<br />
real cracks on desiccating<br />
paint (Summer Palace, Beijing)
Simulation using spring-block model<br />
strain=0.5 κ=0.5 thickness H=3<br />
Connected (or percolating) cracks
Simulation using spring-block model<br />
strain=0.1 κ=0.5 thickness H=9<br />
Straight cracks, then diffusive
理 (Biology-Inspired Physics )<br />
不 行 理 <br />
理 來 <br />
理 理 索 <br />
量 數 理 理 <br />
力 <br />
理 不 <br />
理 <br />
‧ 理 數 <br />
‧DNARNA<br />
‧ 數 <br />
‧ 理 <br />
‧ 神
Current research reported in first-class scientific journal<br />
on a daily-life type of problem<br />
The water strider’s leg
Locomotion and the water-repellent legs<br />
of water striders<br />
@<br />
• Looks like a big mosquito, lives on surface<br />
of still water.<br />
• Sensitive to surface vibrations—detects<br />
presence of preys.<br />
• Eats living and dead insects on surface.<br />
• No wing. Usually in group.<br />
• Do not bite people.<br />
• Body length L ~ 1 cm<br />
• Weight w ~10 dyn ( m ~ 0.01 gm)<br />
1 cm
Excerpt from<br />
Microcosmos -- Claude Nuridsany and Marie Perennou, 1995
Focus on the legs<br />
• Short front legs for grabbing prey, middle legs<br />
for rowing, and the rear legs steer and balance.<br />
• Legs and lower body covered with tiny “hairs”<br />
to keep it from getting wet.<br />
• Walking speed: 1m/sec ~ 100 body lengths/sec<br />
Main things to understand:<br />
1. How it stays afloat structure of its legs<br />
2. How it walks on water hydrodynamics
Two recent papers address those problems:
1. Structure of legs<br />
Water dropet on a leg θ=168<br />
152 dyn<br />
The wax extracted from the leg of striders has a contact angle θ=105 .<br />
For length L=5mm, σ=70 dyn/cm:<br />
F= 2Lσ cosθ ~20 dyn.
Looking closely<br />
•SEM scans reveal fine structures of leg:<br />
oriented setae (needle-shaped hair) of diameter hundreds nm to 3 µm,<br />
length 50µm, at angle 20 from axis<br />
•Moreover, there are elaborate nanoscale grooves on a seta<br />
20 µm<br />
Trapping of air by setae and nanogrooves provides cushion for the leg from<br />
getting wet, and enables the insect to float.<br />
200 nm
Legs filled with air
2. Mechanism of locomotion<br />
To move, one must push on something backward, something that carries<br />
the momentum. It's the ground (earth) that we push when we walk, and<br />
vortices in water when we swim. How about for water striders?<br />
Long believed to be surface waves (capillary waves).<br />
But surface wave speed = (4 g σ/ρ) ¼ = 23 cm/s for water.<br />
A strider must beat its legs faster than this speed.<br />
No problem for adults, but measurements show that infant striders<br />
can’t beat that fast.<br />
Denny's paradox.
Hu et al videotaped striders at 500 fps, showing no substaintial surface<br />
waves, but there are vortices beneath the water surface.<br />
The vortex filament cannot start and end in bulk, it must be U-shaped.<br />
So, the legs stroke the water like the oars of a rowing-boat,sending<br />
vortices backward to propel itself forward.
The balance of momenta<br />
For dipolar vortices at wake of stroke:<br />
Speed V=4 cm/s, radius R=4mm, Mass M=2 π R 3 /3, MV=1 g cm/s,<br />
For water Strider: v=100cm/s, m=0.01g, mv=1 g cm/s<br />
Estimation of capillary wave packet momentum gives 0.05 g cm/s
Robo-strider<br />
1 cm
What do we learn from the water strider?<br />
A common subject (such as water strider) may contain interesting,<br />
potentially important physics waiting for you to discover.<br />
In biophysics, to solve a problem one often needs to look very closely—<br />
as close as down to nanometer scale. This requires nano-technology,<br />
state-of-the-art imaging techniques, etc.<br />
Biophysics is more than proteins and DNA.
Conclusion<br />
Two main Ideas we want to get across:<br />
• We have shown that statistical physics methods are<br />
useful in understanding complex phenomena by means<br />
of simple models and rules.<br />
• Interesting problems are around you, as long as you<br />
keep an opened eye. And they can be inexpensive.