Self Avoiding Walk & Spacetime Random Walk 이 승 주 Computational Physics ㅡ Project

Contents 1. Definition and Application of SAW 2. Fundamental Problems on SAW 3. Modified SAW (on a square lattice) 4. Random Walk in Spacetime 5. Further Topic of interest 6. 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

 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( )

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

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.

2. Fundamental Problems on SAW(4/9) 2.1. Calculation of lower bounds of connective constant  A lower bound

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 n

2. Fundamental Problems on SAW(6/9) 2.1. Calculation of lower bounds of connective constant  Result Note that the exact value is N1/Xc

2. Fundamental Problems on SAW(7/9) 2.2. End to end distance  Result 1) Square lattice Note that the exact values are 0.75, and 0.5 respectively (d, n, e)Distance exp. (d=2, n=30, e=300) (d=3, n=30, e=300) (d=4, n=30, e=100)

2. Fundamental Problems on SAW(8/9) 2.2. End to end distance 2) Triangular - (d=2, n=35, e=150) : - Trajectory

2. Fundamental Problems on SAW(9/9) 2.2. End to end distance 3) Hexagonal - (d=2, n=30, e=50) : - Trajectory

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!

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

4. Random Walk in Spacetime(2/4)  Description of RW 1) 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

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

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

5. Further Topic of Interest  Simulation of Brownian motion in a gravity 1) 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. Summary 1) 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

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, Marion and Thornton, Classical dynamics of particles and systems(4 th ed.). Saunders College Publishing, 1995.