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Introduction to Computational Physics
Autumn term 2017
402-0809-00L
Tuesday 10.45 – 12.30 in HPT C 103
Exercises: Tuesday 8.45- 10.30 in HIT F21
Oral exams: end of January
www.ifb.ethz.ch/education/IntroductionComPhys
2
Who is your teacher?
Hans J. [email protected]
Institute of Building Materials (IfB)HIT G 23.1, Hönggerberg, ETH
Zürich
http://www.hans-herrmann.ethz.ch
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Studiengänge
• Mathematics, Computer Science (Bachelor)
• Mathematics, Computer Science (Master)
• Physics (Wahlfach)
• Material Science (Master)
• Civil Engineering (Master)
4
Plan of this course
• 19.09. Introduction, Random numbers (RN)• 26.09. Percolation• 03.10. Fractals, Cellular Automata• 10.10. Monte Carlo, Importance Sampling,
...........Metropolis • 17.10. Random Walks, Self-avoiding Walks• 24.10. Finite Size Effects, XY Model, ..........
...........first order transitions
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Plan of this course
• 31.10. Differential Eqs. (Euler, Runge Kutta..)• 07.11. Eqs. of Motion (Newton, Regula Falsi)• 14.11. Finite Difference Meth. Relaxation • 21.11. Multigrid, Finite Elements Method • 28.11. Gradient Methods • 05.12. Daniele Passerone• 12.12. Variational FEM, Crank-Nicholson
. Wave equation, Navier-Stokes eq. • 19.12. Giuseppe Carleo
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Spring term 2018
• Computational Statistical Physics (H.J.Herrmann)
• Computational Quantum Physics (G. Carleo, P. de Forcrand)
• Density Functional Theory (D. Passerone)
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Prerequisites
• Ability to work with UNIX
• Making of Graphical Plots
• Higher computer language (C++, FORTRAN..)
• Statistical Analysis (Averaging, Distributions)
• Linear Algebra, Analysis
• Classical Mechanics
• Basic Thermodynamics
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What is Computational Physics?
• Numerical solution of equations (since analytical solutions are rare)
• Simulation of many-particle systems (creation
of a virtual reality = 3rd branch of physics)
• Evaluation and visualization of large data sets (either experimental or numerical)
• Control of experiments (not treated in this course)
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Segregation under vibration
BrazilNut
Effect
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Mixing in a cylinder
hardspheres
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Sedimentation
comparing experiment and simulation
Glass beadsdescendingin silicon oil
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Motion of dunes
V. SCHWÄMMLE, H.J. HERRMANN, Nature 426, 619-620 (2003)
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Computer tools
• Object oriented programming
• Vector supercomputers
• Parallel computing (shared and distributed memory)
• Symbolic Algebra (Mathematica, Maple)
• Graphical animations
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Areas of computational physics
• CFD (Computational Fluid Dynamics)
• Classical Phase Transitions
• Solid State (quantum)
• High Energy Physics (Lattice QCD)
• Astrophysics
• Geophysics, Solid Mechanics
• Agent models (interdisciplinary)
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Literature
• H.Gould, J. Tobochnik and W. Christian: „Introduction to Computer Simulation Methods“ 3rd ed. (Wesley, 2006)
• D. Landau and K. Binder: „A Guide to Monte Carlo Simulations in Statistical Physics“ (Cambridge, 2000)
• D. Stauffer, F.W. Hehl, V. Winkelmann and J.G. Zabolitzky: „Computer Simulation and Computer Algebra“ 3rd ed. (Springer, 1993)
• K. Binder and D.W. Heermann: „Monte Carlo Simulation in Statistical Physics“ 4th ed. (Springer, 2002)
• N.J. Giordano: „Computational Physics“ (Wesley, 1996)• J.M. Thijssen: „Computational Physics“ (Cambridge, 1999)
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Book Series
• „Monte Carlo Method in Condensed Matter Physics“, ed. K. Binder (Springer Series)
• „Annual Reviews of Computational Physics“, ed. D. Stauffer (World Scientific)
• „Granada Lectures in Computational Physics“, ed. J.Marro (Springer Series)
• „Computer Simulations Studies in Condensed Matter Physics“, ed. D. Landau (Springer Series)
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Journals
• Journal of Computational Physics (Elsevier)
• Computer Physics Communications (Elsevier)
• International Journal of Modern Physics C (World Scientific)
CCP = Conference on Computational Physics
every year (2015 India, 2016 South Africa, 2017 Paris):
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Random numbers
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Why do we need Random numbers?
• Simulate experimental fluctuations (e.g. radioactive decay)
• Define temperature
• Complement lack of detailed knowledge (e.g. traffic or stock market simulations)
• Consider many degrees of freedom (e.g. Bownian motion)
• Test stability to perturbations
• Random sampling
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Literature to Random numbers
• Numerical Recipes
• D.E.Knuth: „ The Art of Programming: Seminumerical Algorithms“ 3rd ed. (Addison – Wesley, 1997) Vol. 2, Chapt. 3.3.1
• J.E. Gentle, „Random Number Generation and Monte Carlo Methods“ (Springer, 2003)
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Properties of Random numbers
• No correlations
• Fast implementation
• Reproductibility
• Long periods
• Follow well-defined distribution
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Distribution of random numbers
1 0and( ) ( )P x dx P x
( ) ( )x x
x
w x P x dx
examples: homogeneous, Gaussian, Poisson
Probability to find a random number
in the interval [ x, x + Δx ]:
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Random number generators (RNG)
• Congruential (Lehmer, 1948)
• Lagged-Fibonacci (Tausworthe,1965)
electrical flicker noise photon emission
from a semiconductor
Algorithms:
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Congruential generators
1 mod( ) , , ,i i ix c x p x c p Z
ii
xz
p
Fix two integers: c and p .
Start with a seed x0 .
Create new integers by iterating :
0 1,iz Make random numbers
through
Derrick Henry Lehmer
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Maximal period
Since all integers are less than p the sequence
must repeat after at least p -1 iterations, i.e.
the maximal period is p -1.
( x0 = 0 is a fixed point and cannot be used. )
R.D. Carmichael proved 1910 that
one gets the maximal period if p is
a Mersenne prime number and c the
smallest integer number for which1 1 modpc p
Robert D. Carmichael
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Mersenne prime numbers
2 1 qqM
43* 30,402,457 315416475…652943871 9,152,052December 15, 2005
GIMPS / Curtis & Steven Boone
44* 32,582,657 124575026…053967871 9,808,358September 4, 2006
GIMPS / Curtis & Steven Boone
Marin Mersenne, 1626
prime
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GIMPS news
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January 7, 2016
Curtis Cooper found Nr. 49
274,207,281 – 1 has 22,338,618 digits
Great Internet Mersenne Prime Search
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Example of congruential RNG
Park and Miller (1988):
const int p=2147483647;
const int c=16807;
int rnd=42; //seed
rnd = (c*rnd) % p;
print rnd;
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Square test
x
xi+1
Applet
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Theorem of Marsaglia
1 1 2 1 1 0mod , ..., : ( ... )n i i n i na a a x a x a x p
George Marsaglia (1968)For a congruential generator the
random numbers in an n-cube-test lie on
parallel n -1 dimensional hyperplanes.
1
11 0
at least one set
mod
, , , ..., :
...
n
nn
p c n a a
a c a c p
proof using:
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n-cube-test
1
4
np
One can also show that for congruential RNG
the distance between the planes must be larger
than
and that the maximum number of planes is
1np
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Lagged-Fibonacci RNG
• Initialization of b random bits xi
• Apply:
j jii xx 2mod)(
b,..,1Robert C. Tausworthe
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Lagged Fibbonacci RNG
2modi i a i b i a i bx x x x x
20 i i k ix x x k b
Typically one uses, since it is easy to implement:
Theorem of A. Compagner (1992) :If (a,b) Zierler trinomial then sequence has
maximal period 2b – 1 and :
a < b
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Zierler trinomials
(a, b)
(103, 250) (Kirkpatrick and Stoll, 1981)
(1689, 4187)
(54454, 132049) (J.R. Heringa et al., 1992)
(3037958, 6972592) (R.P.Brent et al., 2003)
ba xx 1 primitive on Z2[x]
(Neal Zierler, 1969)
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Making 64-bit integers
..…
zi = (01101….101110)
zi+1 = (10001….010111)
zi+2 = (00101….111001)
zi+3 = (01011….011100)
zi+4 = (00011….011011)
.....
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Tests for random numbers
• n-cube-test• Correlations should vanish• Average is 0.5• Average of each bit is 0.5• Check distribution• Spectral test: no peaks in Fourier transform• χ2 test: partial sums follow a Gaussian• Kolmogorov – Smirnov test
→ „Diehard battery“ of Marsaglia (1995)
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Diehard battery• Birthday Spacings: Choose random points on a large interval. The spacings between the points should be Poisson
distributed.• Overlapping Permutations: Analyze sequences of five consecutive random numbers. The 120 possible orderings
should occur with statistically equal probability.• Ranks of matrices: Select some number of bits from some number of random numbers to form a matrix over
{0,1}, then determine the rank of the matrix. Count the ranks. • Monkey Tests: Treat sequences of some number of bits as "words". Count the overlapping words in a stream.
The number of "words" that don't appear should follow a known distribution.• Count the 1s: Count the 1 bits in each of either successive or chosen bytes. Convert the counts to "letters", and
count the occurrences of five-letter "words". • Parking Lot Test: Randomly place unit circles in a 100 x 100 square. If the circle overlaps an existing one, try
again. After 12,000 tries, the number of successfully "parked" circles should follow a normal distribution.• Minimum Distance Test: Randomly place 8,000 points in a 10,000 x 10,000 square, then find the minimum
distance between the pairs. The square of this distance should be exponentially distributed.• Random Spheres Test: Randomly choose 4,000 points in a cube of edge 1,000. Center a sphere on each point,
whose radius is the minimum distance to another point. The smallest sphere's volume should be exponentially distributed with a certain mean.
• The Squeeze Test: Multiply 231 by random floats on [0,1) until you reach 1. Repeat this 100,000 times. The number of floats needed to reach 1 should follow a certain distribution.
• Overlapping Sums Test: Generate a long sequence of random floats on [0,1). Add sequences of 100 consecutive floats. The sums should be normally distributed with characteristic mean and sigma.
• Runs Test: Generate a long sequence of random floats on [0,1). Count ascending and descending runs. The counts should follow a certain distribution.
• The Craps Test: Play 200,000 games of craps, counting the wins and the number of throws per game and check the distribution.
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RN with other distributions
• Transformation method
• Rejection method
Poisson distribution
Gaussian distribution
(Box Muller, 1958)
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Transformation method
1 0 1
0
if
otherwise
,( )
zP z
0 0
' ' ' ' ( ) ( )yz
P z dz P y dy
We want random numbers y distributed as P(y).
Start with homogeneously
distributed numbers z:
P(y)
y z 0 0
' ' ' '( ) ( )yz
z P z dz P y dy
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Transformation method
( ) kyP y keexample:
generate Poisson distribution:
0
0
1
11
''
ln
[ ]y
ky y kykyz e
y zk
ke dy e
where z [0,1) are homogeneous random numbers.
This method only works if the integral can be
solved and the resulting function can be inverted.
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Box –Muller (1958)
21
( )y
P y e
1
2
1 2
1 2
2 221 2
1 2 1 2
0 0
2 2 21 2
1 2
0 0 0 0
2 21 21 1
1 1
2
1
21 1arctan
( ) ( )
y y
y y r
y yr
y y r
y
y
y y
z z dy dy
dy dy rdrd
e e
e e
e e
0
21
'
'y y
z dye
Gaussian distribution:
cannot be solved in closed form.
trick:2 2 2
1 2
1
2
1 2
tan
dy dy rdrd
r y y
y
y
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Box –Muller trick
2 21 2 2
1 11
2 1
1
22
2
ln
sintan
cos
y y z
y zz
y z
11 2
2
2 21 21
21
arctany y
yz z
ye
1 2 1
2 2 1
1 2
1 2
ln sin
ln cos
y z z
y z z
From two homogeneously distributed random
numbers z1 and z2 one gets two Gaussian
distributed random numbers y1 and y2 .
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Sample two homogeneously
distributed random numbers
z1 , z2 [0,1). If the point
(Bz1, Az2) lies above the curve
P (y), i.e. P (Bz1) < Az2 then
reject the attempt, otherwise y = Bz1 is retained as a
random number which is distributed according to P (y).
P
Rejection method
Generate random numbers y [0, B] sampled
according to a distribution P (y) with P (y) < A.
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Percolation
John M. Hammersley(1920 – 2004)
Broadbent and Hammersley
Proc. Cambridge Phil. Soc.
Vol. 53, p.629 (1957)
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References to percolation
• D. Stauffer: „Introduction to Percolation Theory“ (Taylor and Francis, 1985)
• D. Stauffer and A. Aharony: „Introduction to Percolation Theory, Revised Second Edition“ (Taylor and Francis, 1992)
• M. Sahimi: „Applications of Percolation Theory“ (Taylor and Francis, 1994)
• G. Grimmett: „Percolation“ (Springer, 1989)
• B.Bollobas and O.Riordan: „Percolation“ (Cambridge Univ. Press, 2006)
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Percolator
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Applications of percolation
• Porous media (oil production, pollution of soils)
• Sol-gel transition• Mixtures of conductors and insulators• Forest fires• Propagation of epidemics or computer virus • Crash of stock markets (Sornette)• Landslide election victories (Galam)• Recognition of antigens by T-cells (Perelson)
• …
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Gelatin formation
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Sol -gel transition
Shear modulus G vanishes
and viscosity η diverges
at tg as function of time t.
η •
G ◦
tg
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Percolation
bla
site percolation on square lattice
p is the probability to occupy a site.Neighboring occupied sites are „connected“
and belong to the same cluster.
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Burning method
2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 34 4 4 4
4 4 4 4 45 5 5 5
5 5 5 5
6
6 6 67
7 7
7
8
8
89
9
9
HH et al (1984)
10 11 12 13
14 15 16
17 18
19
20
21
22
23
24
L = 16
shortest
path
ts = 24
p = 0.59
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Probability to find a spanning cluster
pc = 0.592746…
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Percolation thresholds pc
lattice site bondcubic (body-centered) 0.246 0.1803
cubic (face-centered) 0.198 0.119
cubic (simple) 0.3116 0.2488
diamond 0.43 0.388
honeycomb 0.6962 0.65271*
4-hypercubic 0.197 0.1601
5-hypercubic 0.141 0.1182
6-hypercubic 0.107 0.0942
7-hypercubic 0.089 0.0787
square 0.592746 0.50000*
triangular 0.50000* 0.34729*
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Order parameter of percolation
P(p) = fraction of sites in the largest cluster
( ) ( )cP p p p = 5/36 (2d); ≈ 0.41 (3d)
pc
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Many clusters
bond
percolation
We have clusters
of different sizes s
and can study the
cluster size
distribution ns
ssn
N=
N
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Cluster size distribution
• N(i,j) {0,1}, 0 = empty, 1 = occupied
• Start: k = 2, N(first occupied site) = k, M(k) = 1
• If site top and left are empty: k = k + 1 and continue
• If one of them has value k0: N(i,j) = k0 , M(k0) = M(k0) + 1
• If both are occupied with k1 and k2: choose one, e.g. k1 , N(i,j) = k1, M(k1) = M(k1) + M(k2) + 1, M(k2) = - k1
• If any k has negative M(k): while(M(k)<0)k=-M(k)
• At end: for(k=2; k<=kmax; k++) n(M(k))=n(M(k))+1
Hoshen-Kopelman Algorithm (1976)Raoul Kopelman
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Evolution of N(i,j)
2 3 4 5 6 6 6 72 8 9 3 3 3 3 3 3 3 3 3 7
8 8 3 3 3 3 3 3 3 3 310 3 3 11 3 3 3 3 3 3 3
3 12 3
59
Cluster size distribution ns
ascs esppn )(
)/11(
)(dbs
cs eppn
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Cluster size distribution at pc
sn s
at pc
1872
912 18 3
52
2
in
in
.
τ
τ
d
d
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Scaling of cluster size distribution
±( ) [( ) ]s cτn p s p p s
±
D. Stauffer
(1978)
s = size of cluster
are scaling functions („+“ for p > pc and „–“ for p < pc)
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63
Second moment χ
3
cppC
3600
= 43/18 ≈ 2.39 (2d)
≈ 1.80 (3d)
' 2
s snsmeans that one excludes the largest cluster
'
'
64
Critical exponents
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65
Order parameter of percolation
P(p) = fraction of sites in the largest cluster
( ) ( )cP p p p = 5/36 (2d); ≈ 0.41 (3d)
pc
in infinite system
66
Size dependence of OP
fd d
fddPL s L df = 91/48 in d = 2
df 2.51 in d = 3
at pc:
We will
show later:
L is linear size
of the system
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67
Shortest path ts at pc
also called
„chemical distance“ℓ
minds Lt
site (upper) and bond (lower)
percolation in 4 dimensions
(Ziff, 2001)
dmin ≈ 1.13 (2d)
dmin ≈ 1.33 (3d)
dmin ≈ 1.61 (4d)
6868
Fractal dimension
• B.B.Mandelbrot, „Les Objets Fractals: Forme Hazard et Dimension“ (Flammarion, Paris,1975)
• J. Feder, „Fractals“ (Plenum Press, NY, 1988)• T. Vicsek, „Fractal Growth Phenomena“
(World Scientific, Singapore, 1989)• H.-O.Peitgen and P.H.Richter, „The Beauty
of Fractals“ (Springer, Berlin, 1986)• J.-F. Gouyet, „Physique et Structures
Fractales) (Masson, Paris, 1992)
Books:
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69
Self similarity
70
Self similarity
70
105 particles 106 particles
107 particlesDLA clusters
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7171
Fractal dimension
df = log(5)/log(3) 1.46
df = log(3)/log(2) 1.602
„box counting“ method:
fdM LSierpinski gasket
7272
Gold colloids
David Weitz, 1984
df = 1.70
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7373
Sand-box method
.
Forrest and Witten
(1979)
M(R) is the
number of
particles
in box
of size R .
7474
Sand-box method
slope = df
ln M
ln R
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7575
Correlation function method
)]()([2
)2/()(
12/rMrrM
rr
drc
dd
c(r) = <(0)(r)>
( ) fd dc r r
slope df - d
7676
Calculate c(r)
. r
r + Δr
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7777
Light scattering
( ) ( ) fqr d dI q c r e d r q
Intensity of scattered light with wavevector q:
silica gel
Schaefer
(1984)
7878
Many clusters
bond
percolation
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7979
Ensemble method
21
1( )
( ) ig ji jr
M MR r
Define „radius of gyration“ Rg:
Take cluster of M sites.
f
g
dM R
8080
Ensemble method
slope = df
g
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8383
Box-counting method
= grid spacing
N() = number
of occupied cells
8484
Box-counting method
slope = df
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8585
Box-counting method
= grid spacing
N() = number
of occupied cells
8686
Multifractality
i
qiq pM qd
qM L
0
1
1
lnlim lim
lnq
Nq
m
qd
Ni = number of points in box i
pi = Ni / total number of points
qqq MMm /1
0 )/(
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8787
Strange attractor
21
1
( 1)
n n n
n n
x y x
y x
a
b
Hénon map
Michel Hénon
8888
Strange attractor
self-similar
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90
Hénon Map
93
Percolation
bla
site percolation on square lattice
p is the probability to occupy a site.Neighboring occupied sites are „connected“
and belong to the same cluster.
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94
Order parameter of percolation
P(p) = fraction of sites in the largest cluster
( ) ( )cP p p p = 5/36 (2d); ≈ 0.41 (3d)
pc
95
Size dependence of OP
fd d
fddPL s L df = 91/48 in d = 2
df 2.51 in d = 3
at pc:
We will
show later:
L is linear size
of the system
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96
Volatile fractal
L3
L2
L1
9797
PercolationThe correlation function g(r) for percolation
describes the connectivity and is defined as the
probability that an occupied site is connected
to a site at distance r . This is equivalent to the
probability that the two sites belong to the same
cluster.
The correlation length ξ is the characteristic
length of the exponential decay of the
correlation function.
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9898
Calculate g(r)
. r
r + Δr
9999
Correlation length ξ
/2 1
( / 2)( ) ( ) ( )
2 d d
dg r M r r M r
r r
If one just analyses one cluster
connectivity correlation function g(r) = c(r)
For p < pc the correlation length is
proportional to the radius of a typical cluster.
0
with for( ) r
cg r C e C p p
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100
Correlation length ξ
cp p
pc
101101
Correlation length ξ
cpp
pc2( )( ) dg r r η = 5/24 2d
η ≈ -0.05 3d
at pc:
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102102
Finite size effectsproblem when:
system size L < correlation length i.e. close to the critical point:
Lcritical region
round-off
103103
Round-off in correlation length ξ
pc
3100
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104
Finite size effects
1 1( ) ( )cL p p p
1 2 1 cp p p p
1 2
1
p p L
L
p1 p2
1
1( ) ( )eff cp L p aL
size of
critical region:
105105
Apply finite size dependence
)1()(1
aLpLp ceff
Extrapolation to infinite size
-
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106106
L
χ L
Finite size scaling for χ
1
( , ) [( ) ]cp L L p p L
HH et al. (1982)
data collapse
LL )(max
at pc:
A.E.Ferdinand and M.E Fisher
(1967)
107
Finite size scaling of OP
])[(),(1
LppLLpP cP
)( cppP
fdM L
fdd d
M PL L L
LP
fd d
fraction of sites in spanning cluster (OP):
finite size scaling:
at pc:
fractal dimension:
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109
Continuum percolation
Continuum
Swiss cheese model
110
Bootstrap percolation
Start with p = 0.55 on square lattice.
Remove iteratively all sites that have less
than m =2 occupied neighbors: „culling“.
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111
Bootstrap percolation
triangular lattice, m = 3
112
Cellular Automata (CA)
John von Neuman and Stanislaw Ulam after 1940
discrete determinstic
dynamics
Boolean variables
discrete = on a lattice evolving
from t to t +1
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113
References to CA
• Stephen Wolfram: „Cellular Automata and Complexity“ (Perseus, 1994)
• S. Wolfram: „A New Kind of Science“ (Wolfram Media, 2002)
• A. Ilachinski: „Cellular Automata“ (World Scientific Publ., 2001)
• B. Chopard: „Cellular Automata Modelling of Physical Systems“ (Cambridge University Press, 2005)
114
Definition of CA
1 1 ( ) ( ), , ...,i i jt f t j k
σi binary variable on site i of a graph
rule:
k = number of inputs
There exist possible rules.22k
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115
Time evolution
time
„ rule 30“
entries: 111 110 101 100 011 010 001 000
f (n): 0 0 0 1 1 1 1 0
On every site of a
one-dimensional chain
we put the same rule f
with k = 3 inputs, namely
the site itself and its two
nearest neighbors and
put at t = 0 a random
configuration of bits.
example:
116
Classification of CA
12
0
)(2k
n
n nfc
entries: 111 110 101 100 011 010 001 000
f (n): 0 1 1 0 0 1 0 1
f (n) = 64 + 32 + 4 + 1 = 101
k = 3
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117
Examples for k = 3
applet
entries: 111 110 101 100 011 010 001 000
4 : 0 0 0 0 0 1 0 0
8 : 0 0 0 0 1 0 0 0
20: 0 0 0 1 0 1 0 0
28; 0 0 0 1 1 1 0 0
90: 0 1 0 1 1 0 1 0
118
Evolution of different rules
14 16 18
20 22 24k = 5
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119
Classes of Automata
class 2 class 3
class 1 class 4
(Wolfram)
121
The Game of Life
John Horton Conway (1970)
• n < 2 0
• n = 2 stay as before
• n = 3 1
• n > 3 0
Consider a square lattice.
Be n the number nearest
and next-nearest neighbors
that are „1“.
rule:
Game of Life
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122
The Game of Life
123
The Game of Life
gliders:
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124
The Game of Life
glider gun: