17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner | Lecture: Phase Transitions And Renormalization Group Presenter: Theresa Lindner Self-Similarity And The Random Walk 1 (1)
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Lecture: Phase Transitions And Renormalization Group Presenter: Theresa Lindner
Self-Similarity And The Random Walk
1
(1)
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Agenda
• Motivation • Definitions • Cantor sets
• One or more scaling factors • Fractals in higher dimensions • The Random Walk
• Fractal dimension • Gaussian distribution • Drift term
2
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Agenda
• Motivation • Definitions • Cantor sets
• One or more scaling factors • Fractals in higher dimensions • The Random Walk
• Fractal dimension • Gaussian distribution • Drift term
3
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Motivation
• Scale invariance • Magnification by suitable scale factor • Identical object • Self-similarity on all scales
• Critical behaviour • Loss of scale • Divergence of correlation length • Perturbation expansion fails
4
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Agenda
• Motivation • Definitions • Cantor sets
• One or more scaling factors • Fractals in higher dimensions • The Random Walk
• Fractal dimension • Gaussian distribution • Drift term
5
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Definitions
• Euclidian dimension • Dimension of observed space
• Topological dimension • Dimension of observed object
• Fractal (Hausdorff-) dimension • Cover object in d-dimensional balls of radius • •
6
d
dT
a ! 0
N(a) ⇠ a�D
dT D d
a
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Agenda
• Motivation • Definitions • Cantor sets
• One or more scaling factors • Fractals in higher dimensions • The Random Walk
• Fractal dimension • Gaussian distribution • Drift term
7
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Cantor Sets - One Scaling Factor
• Divide each line segment into three of . • Cover object in line segments of length .
• Approach 1: in step n
• Approach 2:
8
a0a03
a
d = 1, dT = 0
a = a03�n $ n =
ln (a/a0)
ln 3
N(a) = 2n =
✓a
a0
◆�ln2/ln3
⇠ a�DD =
ln2
ln3⇡ 0.6309
N(a) = 2N(3a) = 4N(9a) = ...
a�D = 2 · 3�D · a�D D =ln2
ln3⇡ 0.6309
(2)
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Cantor Sets - Two Scaling Factors
• Generator with more than one element and different scaling factors ( )
9
riX
ri < 1
r1 r2
rescale with rescale with 1
r1
1
r2
N(a) = N(a
r1) +N(
a
r2) ! 1 =
NX
j=1
rDj
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Agenda
• Motivation • Definitions • Cantor sets
• One or more scaling factors • Fractals in higher dimensions • The Random Walk
• Fractal dimension • Gaussian distribution • Drift term
10
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Koch Islands - Fractals In Higher Dimensions
11
!N(a) = 4N(3a) D =
ln4
ln3⇡ 1.262
!N(a) = 8N(4a) D =
ln8
ln4=
3
2 (4)(5)
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Agenda
• Motivation • Definitions • Cantor sets
• One or more scaling factors • Fractals in higher dimensions • The Random Walk
• Fractal dimension • Gaussian distribution • Drift term
12
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
The Random Walk
• Brownian motion • Diffusion • Conformation of linear chain molecules
• End point:
13
~R =MX
i=1
~ri
h~Ri = 0 h|~R|2i = Ma20 R =pMa0
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
The Random Walk - Fractal Dimension
• Divide RW in subwalks of steps
• for each subwalk • Coarse grained RW with steps of length
• Assumption:
14
M
nn
r(n) =pna0
a =pna0
M
n
N(pna0) ⇠
M
n⇠ 1
nN(a0) $ nN(
pna) = N(a0)
N(a0) ⇠ M N(pna0) ⇠
M
n⇠ 1
nN(a0)!
N(a) ⇠ a�2 d � 2for
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
p(~r) = (2⇡�20)
�d/2exp
� |~r|2
2�20
�
The Random Walk - Gaussian Distribution
• constant Gaussian distribution
• Coarse graining • Rescaling
15
a0 !
~r ! ~r 0 ! ~r 00
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
= (2⇡n�20)
�d/2exp
� |~r|2
2n�20
�
The Random Walk - Gaussian Distribution
• Coarse graining
16
~r ! ~r 0
~r 0 =nX
i=1
~ri
P (~r 0) =
Zd~r1...d~rn�
~r 0 �
nX
i=1
~ri
!p(~r1)...p(~rn)
=
Zddk
(2⇡)dexp
�n|~k|2�
20
2
� i~k · ~r 0�
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
P (~r 00) = (2⇡�2
0)�d/2
exp
� |~r 00|2
2�20
�
The Random Walk - Gaussian Distribution
• Rescaling
17
~r 0 ! ~r 00
~r 0 =pn~r 00
P (~r 0)ddr0 = P (~r 00)ddr00
P (~r 0)nd/2ddr00 = P (~r 00)ddr00
�(R) = �0
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
The Random Walk - Gaussian Distribution
• Consider RMS distance
• Dimensional analysis:
• Coarse graining:
Scaling exponent
18
R(M)
R(M,�) ⇠ �M⌫
�M⌫ =pn�(
M
n)⌫ ⌫ =
1
2
⌫ =1
D
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
p(~r) = (2⇡�2)
�d/2exp
� |~r � ~r0|2
2�2
�
The Random Walk - Adding A Drift
• Add a drift term
• Coarse graining:
• Rescaling:
• Fixed points: and
19
~r0
�0 =pn�
~r00 = n~r0
�(R) = �
~r0(R) =
pn~r0
(�,~r0 = 0) (�,~r0 ! 1)
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
The Random Walk
20
Without drift: Self-similarity With drift: Appearence of 1D-walk
(6)
17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |
Sources
1. https://bodyofsunshine.files.wordpress.com/2015/02/romanesco.jpg 2. https://en.wikipedia.org/wiki/Cantor_set 3. http://langferd.wikispaces.com/file/view/kochprog440.jpg/
236961482/204x324/kochprog440.jpg 4. https://en.wikipedia.org/wiki/Koch_snowflake 5. https://upload.wikimedia.org/wikipedia/commons/1/1b/
Quadratic_Koch.png 6. Creswick, Farach, Poole: Introduction to renormalization group
methods in physics
21