RANDOM RANDOM WALKWALK

the real basicthe real basicHEO TAE MINHEO TAE MIN

PHYSICSPHYSICS

contentscontents Brownian motion & polymerBrownian motion & polymer Random walk Random walk Self-avoiding walkSelf-avoiding walk Critical exponentCritical exponent Octupus walkOctupus walk

Brownian motionBrownian motion

Scottish botanist Robert brown(1820s) Scottish botanist Robert brown(1820s) A tiny pollen grainA tiny pollen grain

-- brown believed active moleculesbrown believed active molecules Random collisionRandom collision Einstein in 1905Einstein in 1905 Full mathematical theoryFull mathematical theory

PolymerPolymer

An indispensable part of technologyAn indispensable part of technology

Elementary mathematical models of Elementary mathematical models of Brownian motion and polymers can be Brownian motion and polymers can be produced from so – called random walksproduced from so – called random walks

A linear polymerA linear polymer

So what is random walks?So what is random walks?

discrete random motion in which a particle repeatedly discrete random motion in which a particle repeatedly moves a fixed distance up,down,east,west,north or somoves a fixed distance up,down,east,west,north or southuth

useful in applied fields, useful in applied fields, including ..including ..

Qunatum physics Statistical mechanics Electrical networks

Financial mathematics

As well as polymer theory

Mathematical description of a linear polymer Mathematical description of a linear polymer

•monomer -> pointlike dotsmonomer -> pointlike dots

•monomer-monomer bonds -> steps of random wamonomer-monomer bonds -> steps of random walk lk

•angle between bonds remain constantangle between bonds remain constant

ThereThere is an important is an important flaw….flaw….

volume effectvolume effect

We need a realistic We need a realistic modelmodel

Coin TossingCoin Tossing

It’s a random walkIt’s a random walk head(+1) or tail(-1)head(+1) or tail(-1) with more tosses, it looks like a with more tosses, it looks like a

one-dim Brownian motionone-dim Brownian motion position of the walk isposition of the walk is

1 2 ...n nS X X X n Look for Reif

• Likewise ,for Brownian motion … 2( )B t t

Random walk of high Random walk of high dim.dim.

Dimensions of Dimensions of 4 and higher cannot 4 and higher cannot visualizedvisualized

In five dimensions, a random walk In five dimensions, a random walk could be (0,0,0,0,0)could be (0,0,0,0,0) (0,0,0,1,0)(0,0,0,1,0)

(0,-1,0,1,0)(0,-1,0,1,0)

1

2dThe probability of making a specific step in dimension d

Returns to the OriginReturns to the Origin Polymer structure can be modeled by self-avoiding walksPolymer structure can be modeled by self-avoiding walks Understanding S.A.W. depends on determining the odds that a random walk Understanding S.A.W. depends on determining the odds that a random walk

will intersect itselfwill intersect itself One way is by returning to its starting pointOne way is by returning to its starting point

The minimum probability is 1/2dThe minimum probability is 1/2d In 1921, George Polya proved that one- and two- dimensional random walks In 1921, George Polya proved that one- and two- dimensional random walks

eventually return to the origineventually return to the origin In higher dimension, In higher dimension,

random walk with no self-interactions

DimensioDimensionn

probabiliprobabilityty

33 0.340.34

44 0.190.19

55 0.1350.135

Returns to the originReturns to the origin

In one & two dimensions, there will In one & two dimensions, there will be infinitely many returnsbe infinitely many returns

Once a walk returns to the origin, it is Once a walk returns to the origin, it is like starting againlike starting again

A random walk in 1 or 2 dimension A random walk in 1 or 2 dimension will visit every point infinitely many will visit every point infinitely many times, but not higher.times, but not higher.

Odds of IntersectingOdds of Intersecting

A related but easier modelA related but easier modelTwo independent d-dimensional random walks Two independent d-dimensional random walks that start at the origin and take some large that start at the origin and take some large number of stepsnumber of steps

• There is a positive probability of collision

• 1/2d is the prob of collision after their first step

• If they do not collide after their first steps, however , will they meet eventually?

Yes! Isn’t it trivialYes! Isn’t it trivial?? It is possible for two It is possible for two paths never to paths never to intersect in more than intersect in more than four dimensionsfour dimensions

Then, just think of the cases for dimensions 1 Then, just think of the cases for dimensions 1 through 4through 4

NNnn(d) is the probability of two w(d) is the probability of two walks of length n have no intersealks of length n have no intersection except the originction except the origin

We already know NWe already know Nnn(d) (d) goes to zerogoes to zero

How fast?How fast?

Three basic categories Three basic categories

(go to zero as n goes to infinity)(go to zero as n goes to infinity)

• exponetially

• power law

• logarithmically

slower slower in this in this orderorder

• NNnn(d) can be power law like a form of (d) can be power law like a form of

• In one dimension, NIn one dimension, Nnn(1) can be 1/n(1) can be 1/n

• Finding NFinding Nnn(2) and N(2) and Nnn(3) is rather difficult(3) is rather difficult

• Krzysztof Burdzy and Gregory Lawler showed that NKrzysztof Burdzy and Gregory Lawler showed that Nnn(2) and N(2) and Nnn(3) (3)

go to zero via a power law go to zero via a power law for some positive p for some positive p

1/ pn

1/ pn

1 1 32

2 8 41 1

34 2

p for d

p for d

The power does get smaller as the dimension The power does get smaller as the dimension gets biggergets bigger

• The correct power in two dimensions is p The correct power in two dimensions is p = 0.625= 0.625

• No similar prediction for three No similar prediction for three dimensions, dimensions,

But computer simulations give p = 0.29But computer simulations give p = 0.29

• Four dimensions presents a border line Four dimensions presents a border line casecase

For Nn(4) goes to zero as For Nn(4) goes to zero as

1/ log n

An exponent p can be defined for theAn exponent p can be defined for the

nonintersection of two long Browniannonintersection of two long Brownian

motion paths motion paths

Three WalksThree Walks

RandoRandom Walkm Walk

Dense region where most points Dense region where most points visited at oncevisited at once Looks like a map of towns and cities Looks like a map of towns and cities along a riveralong a river the mean-squared displacement is nthe mean-squared displacement is n

NonreverNonreversing walksing walk

similar with a random walksimilar with a random walk suggests a more open landscapesuggests a more open landscape 2n2n

Self-Self-avoidinavoidin

gg

walkwalk

Looks not like cities along a river but Looks not like cities along a river but like the river itself, or like a coast linelike the river itself, or like a coast line n^(3/2)n^(3/2)

Random walk can go on forever, but a self-Random walk can go on forever, but a self-avoiding walk can get trapped at a lattice site avoiding walk can get trapped at a lattice site

where none of the neighbors are unvisitedwhere none of the neighbors are unvisited

THE PROBABILITY OF GETTING TRAPPED THE PROBABILITY OF GETTING TRAPPED IS SMALL- a little less 1% – BUT IF YOU IS SMALL- a little less 1% – BUT IF YOU GO AHEAD INDEFINITELY, YOUR DESTINY GO AHEAD INDEFINITELY, YOUR DESTINY IS…IS…

For low n, the probability t(n) forFor low n, the probability t(n) for self-trapping self-trapping toto occur after n occur after n steps in two dim latticessteps in two dim lattices

3/5 / 40

(1) ... (6) 0

2(7)

7295 31

(8) ; (9)2187 6561

( ) ( 1) n

t t

t

t t

t n n e

The average length of the self-avoiding The average length of the self-avoiding walks iswalks is

70.7 0.2n

Polymer ConfigurationPolymer Configuration

What is the number of possible What is the number of possible configurations that a long polymer configurations that a long polymer chain can adopt?chain can adopt?

What is the typical distance What is the typical distance separating polymer’s endpoints?separating polymer’s endpoints?

CCnn : number of n-step walk S.A.W. : number of n-step walk S.A.W.

In two dimensionsIn two dimensions

4 one-step walks4 one-step walks

12 two-step walks 12 two-step walks

36 three-step walks 36 three-step walks

From now, we needmore careFrom now, we needmore care

Careful counting Careful counting reveals 100 four-step reveals 100 four-step

walks..walks..In 1954, CIn 1954, Cnn in 2 dim was showed in 2 dim was showed

to grow exponentially..to grow exponentially..

nnC

Connective constantConnective constant

For large n in two For large n in two dimdim

2.620 2.696

For n<40 ,For n<40 ,

2.6382.638

A measurement of A measurement of the average the average

number of possible number of possible next stepsnext steps

It measures how It measures how easy it is for a long easy it is for a long S.A.W. to connect S.A.W. to connect up with a up with a subsequent stepsubsequent step

Critical ExponentsCritical Exponents

Chemists and physics have not waited for Chemists and physics have not waited for mathematicians to prove theorems to advance mathematicians to prove theorems to advance their understanding of polymerstheir understanding of polymers

The consensus is that there are universal criticThe consensus is that there are universal critical exponents that govern the values of Cal exponents that govern the values of Cnn a and the mean-squared displacementnd the mean-squared displacement

One critical One critical exponent alpha exponent alpha arises in a arises in a refinement of the refinement of the connective connective constantconstant

n nn nC to C A n

2Dn

1 1/ 2

1/ 2 1 2,3,4 . . .

D and for random walks

for d S AW

There is a critical exponent nu and a There is a critical exponent nu and a

positive constant D such that the positive constant D such that the

average of displacement is approximately average of displacement is approximately

Takashi proved for dimeTakashi proved for dimension 5 and higher that nsion 5 and higher that nu=1/2 and alpha = 0nu=1/2 and alpha = 0

These values These values are the same as are the same as for random for random walkswalks

Counting stepsCounting steps

For 15 steps, you just say For 15 steps, you just say

“ “ show me all the 14 steps” show me all the 14 steps” Regress continues back to the 0 step walkRegress continues back to the 0 step walk

=> arduous work…=> arduous work… For n=15, the number of walks is For n=15, the number of walks is

6,416,5966,416,596 Counting the walks of n+1 steps takes Counting the walks of n+1 steps takes

longer than counting all the walks from 1 longer than counting all the walks from 1 through nthrough n

There are There are as much asas much as walks of 51 walks of 51 stepssteps

An more sophiscated algorithm was rAn more sophiscated algorithm was required for this calculation, as well aequired for this calculation, as well as an Intel Paragon supercomputer ths an Intel Paragon supercomputer that dedicated 1,024 processors and at dedicated 1,024 processors and 10 gigabytes of memory to the task10 gigabytes of memory to the task

14,059,415,980,606,050,644,814,059,415,980,606,050,644,84444

Repetition of S.A.WRepetition of S.A.W

pivopivott

To eliminat the To eliminat the rotational symmetryrotational symmetry of the lattic of the lattice, only walks that start with a step in east and e, only walks that start with a step in east and make the first turn to north! ( except straight wmake the first turn to north! ( except straight walks )alks )

Rooted walks : starting point is planted in the Rooted walks : starting point is planted in the groundgroundViewing S.A.W as unrooted , the number of distinct Viewing S.A.W as unrooted , the number of distinct

walks is reduced by additional 2 walks is reduced by additional 2

For unrooted,For unrooted,

the two walks at left ..the two walks at left ..

the walk at right ..the walk at right ..

Enumerating unrooted S.A.WEnumerating unrooted S.A.W

What turns to take What turns to take at every at every

intersection?intersection?

L-F-R-F-FL-F-R-F-F

F-F-L-F-RF-F-L-F-R

retroreflectionretroreflection

Ternary Ternary number number saves saves memorymemory

forward:forward: 00

left left : : 11

right right : : -1 -1

The octupus walkerThe octupus walker

8

1

( ) 1

(1) (3) (5) (7)

1 4 (3)(2) (4) (6) (8)

4

I

p I

p p p p

pp p p p

number of stepsnumber of steps

pro

bab

ilip

rob

ab

ilityty

Probability distribuion of S.A.W.Probability distribuion of S.A.W.

Max probability Max probability =0.0055=0.0055

Mean value of steps Mean value of steps 213.9 about twice of 213.9 about twice of max (107.5)max (107.5)

10000 walkers10000 walkers

where p(I)=0.125where p(I)=0.125

algorithmsalgorithms

Dimerization exploits a divide-and-conquer straDimerization exploits a divide-and-conquer strategy familiar in many other areas of computer tegy familiar in many other areas of computer sciencescience

Pivot algorithmPivot algorithm takes a walk and transforms it i takes a walk and transforms it into another ; randomly choose a pivot point sonto another ; randomly choose a pivot point somewhere along the walk, and then rotate or refmewhere along the walk, and then rotate or reflect or reverse the segment on one side of the lect or reverse the segment on one side of the pivotpivot

ApplicationApplication

S.A.W.’s have been used for modeling the lS.A.W.’s have been used for modeling the large scale properties of long-flexible macroarge scale properties of long-flexible macromolecules in solutionmolecules in solution

The study of polymers trapped in porous mThe study of polymers trapped in porous media, gel electrophoresis, and size exclusioedia, gel electrophoresis, and size exclusion chromatography, which deal with the trann chromatography, which deal with the transport of polymers through membranes with sport of polymers through membranes with small pores small pores

Application(2)Application(2)

Employed to characterize complex Employed to characterize complex crystal structures and to analyze crystal structures and to analyze critical phenomena in lattice modelscritical phenomena in lattice models

ConclusionConclusion

SAW is used to analyze the real SAW is used to analyze the real linear polymer model which has a linear polymer model which has a volumevolume

It is a hard work to count the number It is a hard work to count the number of SAW walks because there are of SAW walks because there are many constraintsmany constraints

There are universal critical There are universal critical exponents that govern connective exponents that govern connective constant and mean-squared constant and mean-squared displacementdisplacement

referencereference Gordon Slade.1996.Random Walk.American Scientist v.8Gordon Slade.1996.Random Walk.American Scientist v.8

4 no.2 4 no.2 Brian Hayes.1998.How to avoid yourself.American scientiBrian Hayes.1998.How to avoid yourself.American scienti

st v.86st v.86 Horacio A.Caruso and Sebastian M.Marotta.2000.Horacio A.Caruso and Sebastian M.Marotta.2000.

Self-avoiding random walkers with steps along eight possSelf-avoiding random walkers with steps along eight possible directions(octupus walkers)ible directions(octupus walkers)

Carlos P.Herrero and Martha Saboya.2003.connective coCarlos P.Herrero and Martha Saboya.2003.connective constants in small-world networksnstants in small-world networks

Harvey Gould, Jan Tobochnik, and Wolfgang Christian.20Harvey Gould, Jan Tobochnik, and Wolfgang Christian.2001.Random walks(chapter 12)01.Random walks(chapter 12)

reference(2)reference(2) S.Hemmer and P.C.Hemmer.1984.An average self-avoidiS.Hemmer and P.C.Hemmer.1984.An average self-avoidi

ng random walk on the square lattice lasts 71 steps.J.Chng random walk on the square lattice lasts 71 steps.J.Chem.Phys.81(1)em.Phys.81(1)

Recommendation for more readingRecommendation for more reading

Madras,Neal and Gordon Slade. 1993. Madras,Neal and Gordon Slade. 1993. The Self-Avoiding Walk.(Book)The Self-Avoiding Walk.(Book)