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 a complex system 

• Brownian motion and random walk 

• Dimensions, fractal, power laws & self-similarity 

• Self-organized criticality 

• Earthquake and its modeling 

• Some current research topics

What is a complex system ()? 

 

來說 

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

Brownian motion random walk 

Its trajectory () does look random

Random walk of a drunkard ( ) 

He moves one step at a time, 

equal chance to the left and right 

Let us label the steps of the drunkard by 

x 1 for the first step, x 2 for the second step, 

and so on. Each of them equals ∆x or – 

∆x. 

time t 

Let n=total number of steps 

∆t=time between steps (e.g. 1 sec) 

t=total time=n ∆t 

The total distance of the walk is 

X = x 1 + x 2 + x 3 +…+ x n 

∆x 

Note the averages: =0, =(∆x) 2

Let us do some statistics: 

Although each walk looks random, after 

many such walks, their averages are no 

longer random: 

∆x 

X = x 1 + x 2 + x 3 +…+ x n 

= n =0 

X 2 = (x 1 + x 2 + x 3 +…+ x n ) (x 1 + x 2 + x 3 +…+ x n ) 

= 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 

= + + +…+ + 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 2, 3,… dimensions. 

C=diffusion constant 

~ 10 -9 cm 2 /sec 

for 2µm particle in water

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 in 力 

v=g t 

x 

= 

vt 

∝t 

v=constant 

2 

x ≈ C t 

Random walk 

v=??

Exercises: 

Can we talk about the speed of diffusion? 

What is it? [Hint: how do you define speed?] 

Related: 

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

eating a candy? 

How long does it take to spread a rumour (or 

information, or disease) in a class?

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 

1 

2 

4 

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

N ~ 64/l D

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 is a fractal that you can make

Sierpinski gasket 

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

to this picture in the program called Photoshop) 

l=1=2 0 l=1/2=2 -1 l=1/4=2 -2 l=1/8=2 -3 

N=3 N=9=3 2 N=27=3 3 N=81=3 4 

So l=(1/2) n =2 -n 

N=3 n

l = (1/2) n =2 -n 

N = 3 n 

we want to know how N changes with l. 

we must eliminate n. Log is the way to do it: 

log l = - n log 2 

n = - log l / log 2 

Basic formula of logarithm: 

For x=a y , we have 

N = 3 n = 3 

-log l /log 2 

y=log a 

x=log x / log a, 

= 3 -log 3 l /log 3 2 

= (3 log 3 l ) -1/log 3 2 

a log a x =a y =x 

= l -1 /log 3 2 

= l 

–log 3/log 2 

= (1/l) D D = log3/log2 =1.585 

log a 

x / log a 

y=log b 

x / log b 

y, for any a, b.

Exercise: 

What is the average density of fractal? 

What is the difference between that and the density 

of normal 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 a 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 ≈ m L 2 /C ~ L 2 

D=2 in any dimension

Self-similarity 

切 

 

Sierpinski gasket

切 

It looks random, but 

it also looks “self similar”

: Now you’ve learnt self-similarity, let us apply it!

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 

Current through any surface at 

radius r must be constant: 

absorber 

density ρ=0 

R 

I = constant ~ C ρ 0 R 

by dimensional analysis ( 量 ). 

But I =J * area = J 4π r 2 

density 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


• What is complex system 


• 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

Note that we have only talked about statistics, 

nothing about prediction. 

Sadly, nobody is able to predict when, where 

and how an earthquake happens. 

Earthquake remains a difficult problem.







• Some current research topics 

- crack pattern formation 

- water striders 

-…..

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.

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, so you’d better 

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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