Self Avoiding Walk &Spacetime Random Walk

20030384 이 승 주

Computational Physics ㅡ Project

Contents

1. Definition and Application of SAW2. Fundamental Problems on SAW3. Modified SAW (on a square lattice)

4. Random Walk in Spacetime5. Further Topic of interest6. Summary

1. Definition and Application of SAW (1/2)

Def -step SAW is a sequence of distinct vertices s.t. each vertex is a nearest neighbor of its predecessor

Example

n

# of steps : 200

Dimension : 2

on a Square Lattice

n

d

Homopolymer- Long molecules made of the same monomers bonded together in 1-dim chain

- Typically, neighboring monomers align at small angles

- Several monomers are needed to lose memory of the original angles

Lattice model of homopolymer- Uncorrelated SAW approximates a polymer structure

1. Definition and Application of SAW (2/2)

2. Fundamental Problems on SAW(1/9)

Number of n-step SAWs( )

- We want to know n-dependence of

- Conjecture : is called the connective constant

- Existence of connective constant was proved

- Exact value is known only on the hexagonal lattice( )

nc

1nnc A n

1/lim nn

nc

2 2 1.84776h

nc

2. Fundametal Problems on SAW(2/9)

End to end distance( )- We want to know n-dependence of - , is a distance exponent- Flory’s mean field theory

This is exact for d=1, 2, 4 and for d=3

nR

nR

3

2d

0.588

~nR n

2. Fundamental Problems on SAW(3/9)2.1. Calculation of lower bounds of connective constant

Bridge and Irreducible bridge- Bridge : a SAW s.t. for all j- Irreducible bridge : bridge that cannot be decomposed further

Examples of a Bridge(left) and

a Irreducible bridge (right)

- Three Connective constants are the same i.e. 1/ 1/ 1/lim lim limn n n

n n nx x xa b c

0 j nx x x

2. Fundamental Problems on SAW(4/9)

2.1. Calculation of lower bounds of connective constant

A lower bound

11( ) ( ) 1

1 ( )B x A

A x

1

-1

For ' with 0 ' , if ' 0

then

nn n n n c

c

a a a a x

x

N1

,n=1 1

If 1 then L

ncn l c

l

a x x

, : Irreducible bridges (length , height )n la n l

, : Bridges (length , height )n lb n l ,1

Take 'L

n n ll

a a

2. Fundamental Problems on SAW(5/9)

2.1. Calculation of lower bounds of connective constant

Generating functions

Alm, Janson theorem gives exact expressions for upto n=4

Coefficient table of upto n=4

1

1

( ) ( ) ( ) ( )l

l l l k kk

A x B x A x B x

( )nA x

( )nA xn 2 4 6 8 10 12 14 16 18 20 22 24

1 2 2 2 2 2 2 2 2 2 2 2 2

2         8 24 58 116 226 418 764 1368

3               40 248 956 2932 8158

4                     232 2208

2. Fundamental Problems on SAW(6/9)

2.1. Calculation of lower bounds of connective constant

Result

Note that the exact value is 1.84776h

N 1/Xc

2 1.41421

4 1.65289

6 1.7087

8 1.72464

10 1.74764

12 1.76395

14 1.7744

16 1.78215

18 1.78875

20 1.79417

22 1.79856

24 1.80229

2. Fundamental Problems on SAW(7/9)

2.2. End to end distance Result1) Square lattice

Note that the exact values are 0.75, 0.588 and

0.5 respectively

(d, n, e) Distance exp.

(d=2, n=30, e=300)

0.73 0.01

(d=3, n=30, e=300)

0.59 0.01

(d=4, n=30, e=100)

0.54 0.01

2. Fundamental Problems on SAW(8/9)

2.2. End to end distance2) Triangular- (d=2, n=35, e=150) : - Trajectory

0.71 0.01

2. Fundamental Problems on SAW(9/9)

2.2. End to end distance3) Hexagonal- (d=2, n=30, e=50) : - Trajectory

0.71 0.025

3. Modified SAW The exponent decreases to 0.5 as increases To distinguish SAWs from normal RWs, we define ununiform probability for each direction

On 2-dim square lattice-

-

Now it is natural to think of the effect of the curvature of spacetime! Consider the Schwarzschild spacetime!

d

0.25u d l rp p p p

0.4, 0.1u d l rp p p p n=30, e=200 =0.73 0.01

n=30, e=50 =0.68 0.02

4. Random Walk in Spacetime(1/4)

Metric & Christoffel symbols

Schwarzschild spacetime is described by

* These are the only nonzero elements of metric tensor and Christoffel symbols

00

111

222

2 233

(1 2 / )

(1 2 / )

sin

g M r

g M r

g r

g r

0 1 2 110 11

1 200

1 1 222 33

2 3 2 221 31 33

323

/ (1 2 / )

/ (1 2 / )

/ sin (1 2 / )

/ sin 1/

cot

M r M r

M r M r

r M r

r

4. Random Walk in Spacetime(2/4)

Description of RW1) Random velocity- Selection of a random 3-velocity vector - Normalize the 3-velocity to make its 3-norm be a random number between 0 and 1- Completion of 4-velocity vector by normalization

2) Geodesic along that direction

- Solve the above equation for geodesic- Motion along the geodesic for

2

20

d x dx dx

d d d

1

4. Random Walk in Spacetime(3/4)

3) Trajectory- Repeat these processes for to get a

1000-step trajectory

Expectation- RW will move toward the center of the gravity

Result- As the mass of the star increases, the RW moves toward the center of the field (Figures on the next page)

- RW diverges when it touch some characteristic radius

1000

4. Random Walk in Spacetime(4/4) Sample trajectories for some values

of M* Mass of star : M

* Position of star : (0, 0, 0)

* # of steps : for 1000

* Initial position of particle

[1] 50( )

[2] / 2( )

[3] 0( )

x r

x

x

0 0

0 0

* RW diverged when M=15

* Rough estimation for this divergence :

Divergence occurs if 2M/r = 1 i.e. if r =30

Correponding / 3 17

See the picture for M=14.5 !

x r

5. Further Topic of Interest

Simulation of Brownian motion in a gravity1) 3-vector normalization - Change the code to normalize random 3-velocity not by uniform number in [0,1) but by Boltzmann distribution

2) Number of particles - Increase the number of particles

Description of Molecular Brownian motion in spacetime !

6. Summary1) We have found lower bounds of connective constant in 2-dim hexagonal lattice ( )

2) We have calculated distance exponents in square(for some dimensions), triangular and hexagonal lattices (in plane)

3) We have observed the 2-dim Modified Random SAW

4) We have defined RWs in spacetime and got some samples of them

at least 1.80229

Reference[Papers]- Jensen, I. (2004) Improved lower bounds on the connective constants for SAW. J.phys. A- Alm, S.E. and Parviainen, R. (2003) Bounds for the connective constant of the hexagonal lattice. J. phys. A- Conway, A.R. and Guttmann, A.J. (1992) Lower bound on the connective constant for square lattice SAWs. J.phys. A

[Books]- Bernard F. Schutz, A first course in general relativity. Cambridge, 1985.- Marion and Thornton, Classical dynamics of particles and systems(4th ed.). Saunders College Publishing, 1995.