introduction to complex systems...

Introduction to Complex Systems複雜系統之簡介

梁鈞泰中央研究院物理所

Kwan-tai LeungInstitute 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

• Power laws & self-similarity

• Self-organized criticality

• Earthquake and its modeling

• Some current research topics - crack pattern formation- water striders- …..

What is a complex system (複雜系統)?

複雜系統是一個由多個簡單單元所組成的結構。一般來講,複雜系統最有趣的地方在於所組成的單元經由非線性交互作用，會產生集体的行為。這些集体行為可在空間或時間中表現為圖形化、新層次之結構。

這類系統往往受外界影響，而在熱力、噪音的作用下運作。要瞭解這些系統的行為，我們需要運用熱力學和統計力學的概念和方法。

Brownian motion & random walk布朗運動與隨機行走

Let us start with a simple system:2 µm polysteryne spheres

in waterdoing random walks

布朗先生在顯微鏡下看到水中的花粉不斷進行

不規則的運動

The random walker’s trajectory (軌跡)

It does look random

The motion of a random walker is better described byan equation known as Langevin equation:

ξµ += fdtdx

dxdUf −=

f 是外力，µ 是 mobility，而 ξ 則是噪音項

假如流體中有不只一顆而是很多顆粒子，我們需要知道的不是每顆粒子的位移，而是粒子的密度場。密度場遵守擴散方程式:

),( txρ.

2

2

xD

t ∂∂

=∂∂ ρρ

統計規律

tDx ≈2 D=diffusion constant~ 10-9 cm2/sec for 2µm particle in water

Power laws 指數定律When two physical variables are related, the simplest relationship is a power law. E.g.

g Free fall22

21 tgtx ∝=

Constant speedtvtx ∝=

Random walktDx ≈2

Self-similarity 自我相似性

cut and blow up

It looks random, butit also looks “self similar”

Self-similarity in fractal (碎形)

Sierpinski gasket

cut

Blow up

A practical example of self-similarity

Self-similarity and power laws can be found in phase transitions

Phase transitions 相變

Common material’s 3 phases Magnetic material

Power laws in thermodynamic quantites at phase transition

Magnetization 磁化率

Specific heat 比熱

Lattice-gas model of phase transitions~ many random walkers with weak attraction

T=Tc T=1.05 Tc T=2Tc

Scale invariance or self-similarity at Tc

After the operation, the left pictureIs reduced into this box

A self-similar trajectory lacks characteristic length andtime scale. If you do a Fourier analysis, there is nocharacteristic frequency – the power spectrum is alsoa power law:

αffP 1)( ≈ α=2

αffP 1)( ≈

α=0White noise

1/f noise

Brownian noise

α=1

α=2

Self-Organized Criticality (SOC)自組臨界性

對於任一時間序列，我們總可以計算它對應的功率譜。人們逐漸發現世界上有很多人為或自然產生的時間序列，它們對應的功率譜都沒有明顯的特徵頻率，而且經常具有這特性的現象有很多，包括河流的水位、恆星的亮度、地震的訊號、股票指數的起伏、甚至古典音樂的音量等。如何去合理解釋這普適性便成為理論物理學家的挑戰。

於是，Per Bak等人在1987年提出即使在缺乏明顯可調控的參數下也能產生指數定律的機制，並稱之為「自組臨界性」(self-organized criticality)。他們更同時提出一個圖像化而直覺的「沙堆模型」(sandpile model)來闡釋該概念。

Bak & Chen, Sc. Am. 1991

Sandpile as a paradigm of SOC

• “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 開放的邊界條件

are important to ensure self-organized criticality

One major success of SOC is in earthquake modeling

Gutenberg-Richter law for earthquake magnitudeC

umul

ativ

e di

strib

ut’n

Earthquake magnitude m

Slope 1.62 (compared to 1.8 of model)

N(>m) per yrData from sesmicity catalog 1973-1997

self-organized critical earthquake model

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

fault

Slip-size distribution in earthquake model

P(S)~S -1.8Latest results: Lise & Paczuski PRE 2001

One slip initiates an avalanche of ultimately S slips

S

P(S)

S

Outline of talk

• What is complex system

• Brownian motion and random walk

• Power laws & self-similarity

• Self-organized criticality

• Earthquake and its modeling

• Some current research topics - crack pattern formation- water striders- …..

The scales of cracks

50µm

monolayer of microspheres

20cmdried clay

1m

dried lake

fissures in Black Rock Desert, Nevada

100m

earthquake faults

100 km

monolayer of polystyrene spheres (µm size)

Early stage

Skjeltorp & Meakin

0.5 mm

0.05 mm

In presence of dissipation (e.g. friction), crack growthis subcritical, and speed of crack tip

What happens at crack tips?

Tensile stress in a block

0σ

0σσ >>yy

0σ

Stress concentration at tip

Stress field Crack path

xxσ

0=yyσ

A crack modifies the stress field around it, which in turn dictates its subsequent propagation.

crack nucleated from boundary

crack nucleated in bulk, thenpropagates toward boundary

Inexpensive experiments

slipperysubstrate

frictional substrate

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

Leung & Neda, Phys. Rev. Lett. 2000

thick layerthin layer

• starch-water mixture

with 120o joint

without 120o joint

thickness2

fragment area

Length scale

fraction of 120o joint

Groisman & Kaplan, Europhys Lett. 1994

Morphology

from coffee-water mixture expt

Approach: simplify the problem as much as possible—one step from being trivial. Try to capture the essentialphysics, then test the results against real, complex situations.

Simulating cracks: a spring-block model

cracking

0=τcrackF>τtension

},{ ii yx

kH

slipi FF >force

stick-slip

0=iF localequilibrium

slippings

crackings

…

Some typical evolution of cracks by simulations

Simulation using spring-block modelstrain=0.5 κ=1 thickness H=2

Close-up of crack propagation and branching

Simulation using spring-block modelstrain=0.3 κ=0.5 thickness H=3

Diffusive cracks

real cracks on desiccatingpaint (Summer Palace, Beijing)

Simulation using spring-block modelstrain=0.5 κ=0.5 thickness H=3Connected (or percolating) cracks

Simulation using spring-block modelstrain=0.1 κ=0.5 thickness H=9

Straight cracks, then diffusive

生物物理 (Biology-Inspired Physics )

生命體中有許多巨份子不斷在進行極為複雜的物理、生化相互作用。因此,從物理學的觀點看來,生命體無疑是最難被理解的物理系統。在探索生物的過程中會產生大量數據,物理學家希望能從中尋找一些基本原理。

由於生物中的巨份子一直處於一個熱力環境中,統計物理自然是研究生物不可或缺的工具。在這個基礎上,物理學家可以研究: ‧生理訊號(如心跳)的數據分析, ‧生物巨分子(如DNA、RNA和蛋白質)的模擬﹔‧發展最佳化的數值方法在蛋白質結構的預測﹔‧份子馬達的物理機制﹔‧神經網絡的同步發火現象的實驗。

Current research reported in first-class scientific journalon a daily-life type of problem

The water strider’s leg

Locomotion and the water-repellent legsof 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 cmWeight w ~10 dyn ( m ~ 0.01 gm)

1 cm

水蠅@南港胡適公園

Excerpt fromMicrocosmos -- 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 legs2. How it walks on water hydrodynamics

Two recent papers address those problems:

1. Structure of legs

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 closelySEM 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 µm200 nm

Trapping of air by setae and nanogrooves provides cushion for the leg from getting wet, and enables the insect to float.

Legs filled with air

2. Mechanism of locomotion

To move, one must push on something backward, something that carriesthe 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 striderscan’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,sendingvortices 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 R3/3, 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

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 more than proteins and DNA.

Conclusion

Two 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.

• Interesting problems are around you, as long as you keep an opened eye. And they can be inexpensive.

Introduction to Complex Systems複雜系統之簡介

introduction to complex systems...

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