self avoiding walk & spacetime random walk 20030384 이 승 주 computational physics ㅡ...
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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.