複雜系統之簡介(梁鈞泰) - 中研院物理研究所- Academia Sinica

Introduction to Complex Systems 

 

梁 

理 

Kwan-tai Leung 

Institute of Physics, Academia Sinica, 

Taipei, 115, Taiwan 

http://www.sinica.edu.tw/~leungkt

Outline of talk 

• What is complex system 

• Brownian motion and random walk 

• Fractal, power laws & self-similarity 

• Self-organized criticality 

• Earthquake and its modeling 

• Some current research topics

What is a complex system ()? 

 

來 

行 

行 

 

類力 

行 

力力念

Brownian motion & random walk 

朗行 

Let us start with a simple system: 

Robert Brown (1773–1858) 

朗 

 

不行不 

 

10 -6 

2 µm polysteryne spheres 

in water 

doing random walks

The random walker’s trajectory () 

It does look random

Random walk of a drunkard 

∆x 

Move one step at each time, 

equal chance to the left or right 

Statistics: 

after many realizations, regularities emerge when 

we compute their averages: 

time t 

X(t) = x 1 + x 2 + x 3 +…+ x n 

= 0 

X 2 (t) = x 1 

2 

+ x 2 

2 

+ x 3 

2 

+…+ x n 

2 

+ x 1 x 2 + x 1 x 3 +…+ x 1 x n 

+ x 2 x 1 + x 2 x 3 +…+ x 2 x n +… 

+ x n x 1 + x n x 2 +…+ x n x n-1 

X 

= + + +…+ + 0 

= n = (∆x) 2 n 

= [ (∆x) 2 /∆t ] (n ∆t) 

= C t 

where C≣ (∆x) 2 /∆t is called the diffusion constant

律 

= 了 t 度 

x 

2 

≈ C t 

Still holds in higher dimensions. 

C=diffusion constant 

~ 10 -9 cm 2 /sec 

for 2µm particle in water

Exercise: Can we talk about the speed of diffusion? 

What is it? 

Related: how long does it take for a fly to know you are 

eating a cake? How long does it take to spread rumour 

or information in a class?

Power laws 數律 

When two physical variables are related, the simplest 

relationship is a power law. E.g. 

x 

= 

1 

2 

gt 

2 

∝ 

t 

2 

g 

Free fall 

x 

= 

vt 

∝t 

Constant speed 

2 

x ≈ C t 

Random walk

Dimension D ( 度 ) 

Two ways to define D: 

1. Mass ( 量 ) M~L D 

M~L D=1 

M~L 2 D=2 M~L3 D=3 

2. Stick length ( 度 ) l Coverage ( 數 ) N ~ 1/l D 數律 

l=1, N=64 l=2, N=16=64/2 2 l=4, N=4=16/2 2

Fractal dimension 度 

fractal B. Mandelbrot 

(The fractal geometry of Nature, 1983) 

Stick length ( 度 ) l 

Coverage ( 數 ) N ~ 1/l D 

l=200 km l=100 km l=50 km 

1 < D < 2 

Most coastlines are fractal with dimension between 1 & 2

Sierpinski gasket

Sierpinski gasket 

l is the stick length (=radius for applying gaussian blur 

to this picture in Photoshop) 

l=1 l=1/2 l=1/4 l=1/8 

N=3 N=9 N=27 N=81 

So for l=(1/2) n =2 -n we want to know how N changes with l: 

N=3 n = 3 -log l /log 2 = 3 -log l /log 2 

3 3 = (3 log l 

3 ) -1/log 2 

3 

= l -1 /log 2 

3 =l –log 3/log 2 = (1/l) D 

D=log3/log2=1.585 

because for x=a y , we have y=log a 

x, and a log a x =a y =x

Exercise: What is the average density of fractal? 

What is the difference between that and the density 

of non-fractal objects? Can you use it to tell whether 

an object is a fractal or not?

Koch curve () 

Exercise: Can you show that the fractal 

dimension of the Koch curve is log4/log3=1.26?

Fractal dimension of random walk 

L 

2 

= 

x 

2 

≈ 

C t 

L 

Let each step carry a unit mass m. 

After t steps, total mass M is 

M=m t ~ L 2 

D=2 in any dimension

Self-similarity 

cut 

Blow up 

Sierpinski gasket

cut and blow up 

It looks random, but 

it also looks “self similar”

A practical example of self-similarity

The motion of a random walker is described by 

an equation known as Langevin equation ( 朗 ): 

dx 

dt 

= µ f + ξ 

f 

= 

− 

dU 

dx 

力 µ mobility ξ 

 

流不粒不 

粒粒度 ρ( x, 

t) 

度 

: 

∂ρ 

= C 

∂t 

∂ 

∂ 

2 

ρ 

2 

x 

ρ 

x

Steady diffusion 

absorber 

ρ=0 

R 

Current through any surface at 

radius r must be constant: 

I = constant ~ C ρ 0 R 

by dimensional analysis ( 量 ). 

But I =J * area = J 4π r 2 

ρ 0 

 

So J = I/4πr 2 = C ρ 0 R/r 2 

is the current per unit area. 

Important to see that I ∝ R , not R 2 , because the current density J 

decreases as 1/R.

Similar result for a disk absorber: I = 4 C ρ 0 R. 

absorber 

R 

These results can all be derived by solving the diffusion equation 

(a differential equation): 

∂ρ 

= 

∂t 

C 

∂ 

∂ 

ρ 

= 

2 

x 

2 

0 

subject to boundary conditions ρ(R)=0, ρ()=ρ 0

If some sea animal is fed by taking small creatures or algae that 

happen to diffuse into its mouth, the sea animal cannot grow too big: 

3 

Food consumption rate ∝ R 

Food intake rate 

∝ 

R 

R

Diffusion Limited Aggregation (DLA) ~ crystal growth 

Start with a seed at center. Random particles released from boundary, 

doing random walk until it sticks to the center. The seed grows into a cluster

Phase transitions 

Critical temperature 

Self-similarity and power laws 

can be found in phase 

transitions 

Common material’s 3 phases 

Magnetic material

Power laws in thermodynamic 

quantites at phase transition 

Magnetization 率 

Specific heat

Lattice-gas model of phase transitions 

~ many random walkers with weak attraction 

T~0 

T=T c T=1.05 T c T=2T c 

Exercise: which pattern is more complex, contains more information?

Scale invariance or self-similarity at T c 

After the operation, the left picture 

Is reduced into this box

A self-similar trajectory lacks characteristic length and 

time scale. If you do a Fourier analysis, there is no 

characteristic frequency – the power spectrum () 

is also a power law: 

1 

P( f ) ≈ α=2 

α 

f

P( f ) ≈ 

1 

f 

α 

White noise 

α=0 

1/f noise 

α=1 

α=2 

Brownian noise




• Power laws & self-similarity 




Self-Organized Criticality (SOC) 

臨 

列率 

列 

率都率 

流亮度 

數樂量理 

便理論理 

Per Bak1987 年參 

數數律臨 

(self-organized criticality) 更 

(sandpile model) 來念

Sandpile as a paradigm of SOC 

Bak & Chen, Sc. Am. 1991 

• “Self-organized” means systems reaching critical states without tuning. 

• “Critical”: at critical slope (“angle of repose”), no characteristic avalanche 

size exists. It covers all possible values with power-law distribution P(s)~s -b .

Open-boundary conditions 

are important to ensure self-organized criticality 

Number of jobs thrown out of window obeys P(n)~n -b








One major success of SOC is in earthquake modeling 

(Note: not prediction)

Gutenberg-Richter law for earthquake magnitude 

N(>m) per yr 

Cumulative distribut’n 

Data from sesmicity catalog 1973-1997 

Slope 1.62 

(compared to 1.8 of model) 

Earthquake magnitude m

self-organized critical earthquake model 

fault 

Olami, Feder, and Christensen, Phys. Rev. Lett. 68, 1244 (1992)

Slip-size distribution in earthquake model 

S 

Latest results: Lise & Paczuski PRE 2001 

One slip initiates an avalanche of ultimately S slips 

P(S)~S -1.8 

P(S) 

S







• Some current research topics 

- crack pattern formation 

- water striders 

-…..

The scales of cracks 

monolayer of microspheres 

dried lake 

earthquake faults 

100 km 

50µm 

1m 

dried clay 

20cm 

100m 

fissures in Black Rock 

Desert, Nevada

monolayer of polystyrene spheres (µm size) 

Early stage 

0.5 mm 

0.05 mm 

Skjeltorp & Meakin

In presence of dissipation (e.g. friction), crack growth 

is subcritical, and speed of crack tip

What happens at crack tips? 

σ 0 

σ >> σ 

yy 0 

σ 0 

Tensile stress in a block 

Stress concentration at tip

σ xx 

Stress field Crack path 

A crack modifies the stress field around it, which 

in turn dictates its subsequent propagation. 

σ yy 

= 0 

crack nucleated 

from boundary 

crack nucleated in bulk, then 

propagates toward boundary

Inexpensive experiments 

• coffee-water mixture Groisman & Kaplan, Europhys Lett. 1994 

frictional 

substrate 

slippery 

substrate 

• starch-water mixture 

Leung & Neda, Phys. Rev. Lett. 2000 

thin layer 

thick layer

Length scale 

fragment 

area 

without 120 o joint 

with 120 o joint 

Morphology 

thickness 2 

fraction of 

120 o joint 

from coffee-water mixture expt 

Groisman & Kaplan, Europhys Lett. 1994

Approach: simplify the problem as much as possible— 

one step from being trivial. Try to capture the essential 

physics, then test the results against real, complex situations.

Simulating cracks: a spring-block model 

{ 

i 

x i 

, y } 

stick-slip 

kH 

force 

F 

i 

> F slip 

Fi 

= 0 

local 

equilibrium 

cracking 

slippings 

tension 

τ > 

F crack 

crackings 

… 

τ = 0

Some typical evolution of cracks by simulations 

Simulation using spring-block model 

strain=0.5 κ=1 thickness H=2 

Close-up of crack propagation and branching


strain=0.3 κ=0.5 thickness H=3 

Diffusive cracks 

real cracks on desiccating 

paint (Summer Palace, Beijing)



Connected (or percolating) cracks



Straight cracks, then diffusive

理 (Biology-Inspired Physics ) 

不行理 

理來 

理理索 

量數理理 

力 

理不 

理 

‧ 理數 

‧DNARNA 

‧ 數 

‧ 理 

‧ 神

Current research reported in first-class scientific journal 

on a daily-life type of problem 

The water strider’s leg

Locomotion and the water-repellent legs 

of water striders 

@ 

• Looks like a big mosquito, lives on surface 

of still water. 

• Sensitive to surface vibrations—detects 

presence of preys. 

• Eats living and dead insects on surface. 

• No wing. Usually in group. 

• Do not bite people. 

• Body length L ~ 1 cm 

• Weight w ~10 dyn ( m ~ 0.01 gm) 

1 cm

Excerpt from 

Microcosmos -- Claude Nuridsany and Marie Perennou, 1995

Focus on the legs 

• Short front legs for grabbing prey, middle legs 

for rowing, and the rear legs steer and balance. 

• Legs and lower body covered with tiny “hairs” 

to keep it from getting wet. 

• Walking speed: 1m/sec ~ 100 body lengths/sec 

Main things to understand: 

1. How it stays afloat structure of its legs 

2. How it walks on water hydrodynamics

Two recent papers address those problems:

1. Structure of legs 

Water dropet on a leg θ=168 

152 dyn 

The wax extracted from the leg of striders has a contact angle θ=105 . 

For length L=5mm, σ=70 dyn/cm: 

F= 2Lσ cosθ ~20 dyn.

Looking closely 

•SEM scans reveal fine structures of leg: 

oriented setae (needle-shaped hair) of diameter hundreds nm to 3 µm, 

length 50µm, at angle 20 from axis 

•Moreover, there are elaborate nanoscale grooves on a seta 

20 µm 

Trapping of air by setae and nanogrooves provides cushion for the leg from 

getting wet, and enables the insect to float. 

200 nm

Legs filled with air

2. Mechanism of locomotion 

To move, one must push on something backward, something that carries 

the momentum. It's the ground (earth) that we push when we walk, and 

vortices in water when we swim. How about for water striders? 

Long believed to be surface waves (capillary waves). 

But surface wave speed = (4 g σ/ρ) ¼ = 23 cm/s for water. 

A strider must beat its legs faster than this speed. 

No problem for adults, but measurements show that infant striders 

can’t beat that fast. 

Denny's paradox.

Hu et al videotaped striders at 500 fps, showing no substaintial surface 

waves, but there are vortices beneath the water surface. 

The vortex filament cannot start and end in bulk, it must be U-shaped. 

So, the legs stroke the water like the oars of a rowing-boat,sending 

vortices backward to propel itself forward.

The balance of momenta 

For dipolar vortices at wake of stroke: 

Speed V=4 cm/s, radius R=4mm, Mass M=2 π R 3 /3, MV=1 g cm/s, 

For water Strider: v=100cm/s, m=0.01g, mv=1 g cm/s 

Estimation of capillary wave packet momentum gives 0.05 g cm/s

Robo-strider 

1 cm

How about other creatures? 

All rely on vortices, but different 

topology due to different 

boundary conditions.

Ancient “water striders” 

Excerpt from

What do we learn from the water strider? 

A common subject (such as water strider) may contain interesting, 

potentially important physics waiting for you to discover. 

In biophysics, to solve a problem one often needs to look very closely— 

as close as down to nanometer scale. This requires nano-technology, 

state-of-the-art imaging techniques, etc. 

Biophysics is not confined to proteins and DNA.

Conclusion 

Main Ideas we want to get across: 

• We have shown that statistical physics methods are 

useful in understanding complex phenomena by means 

of simple models and rules. 

• Random walk and its generalizations occupy an 

important role. 

• Interesting problems are around you, as long as you 

keep an opened eye (be curious).

複雜系統之簡介(梁鈞泰) - 中研院物理研究所- Academia Sinica

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