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

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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

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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

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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

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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

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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 • •

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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

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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:

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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

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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 ( )

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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

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17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |

Koch Islands - Fractals In Higher Dimensions

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!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

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17.07.2017 | Self-Similarity And The Random Walk | Theresa Lindner |

The Random Walk

• Brownian motion • Diffusion • Conformation of linear chain molecules

• End point:

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~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:

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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

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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

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~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

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~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

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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

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~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

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Without drift: Self-similarity With drift: Appearence of 1D-walk

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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

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