y - Institute for Building Materials - ETH Zürich

Autumn term 2008
I Introduction t d ti t to C Computational t ti l
Physics
401 401-0809 401 0809-00L 0809 00L
Friday 10.45 – 12.30 in HCI D8
Exercices: Friday 8.45 8.45- 10.30 in HCI D451
Oral exams: end of January
www.ifb.ethz.ch/education/IntroductionComPhys
www www.ifb.ethz.ch/education/IntroductionComPhys
ifb ethz ch/education/IntroductionComPhys
Studiengänge
g g
• Mathematics, Computer Science (Bachelor)
• Mathematics, Computer Science (Master)
• Physics (Wahlfach)
• Material Science (Master)
• Civil Engineering (Master)
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Who is your teacher?
Hans J. J JJ. Herrmann
Herrmann
hjherrmann@ethz.ch
since since April April 2006
2006
Institute of Building Materials (IfB)
HIF E12, Hönggerberg, ETH Zürich
http://comphys.ethz.ch/hans/
Spring term 2008
• Computational Statistical Physics
(H (H.J.Herrmann)
J Herrmann)
• Computational Quantum Physics
(P.Werner, P.de Forcrand, M.Troyer)
• Computational Polymer Physics
(M (M.Kröger) K ö )
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Plan of this course
• 19.09. Introduction, i random numbers (RN)
• 26.09. RN, , Percolation
• 03.10. Fractals, Cellular Automata
• 10 10.10. 10 Random Walks (Kröger)
• 17.10. Finite Size Effects, Monte Carlo
• 24.10. Importance Sampling, Metropolis
• 31 31.10. 10 Ising Model (Werner)
• 07.11. XY Model, first order transitions
Plan of this course
• 14.11. Differential Eqs. (Euler, Runge Kutta..)
• 21.11. Eqs. of Motion (Newton, Regula Falsi)
• 28.11. Finite Difference Meth. Relaxation
• 05.12. Gradient Methods
• 12 12.12. 12 Multigrid Multigrid, Finite Elements Method
• 19.12. Variational FEM, Crank-Nicholson
. Wave equation, Navier-Stokes
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Segregation under vibration
Mixing in a cylinder
Brazil
Nut
Effect
hard
spheres
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Sedimentation
Sedimentation
comparing experiment and simulation
Motion of dunes
V. SCHWÄMMLE, H.J. HERRMANN, Nature 426, 619-620 (2003)
Glass beads
ddescending di
in silicon oil
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Prerequisites
• Ability to work with UNIX
• Making of Graphical Plots
• Higher computer language (FORTRAN, C++..)
• Statistical Analysis (Averaging, Distributions)
• Linear Algebra Algebra, Analysis
• Classical Mechanics
• Basic Thermodynamics
Literature
• H.Gould, J. Tobochnik and W. Christian: „Introduction to
Computer Simulation Methods“ 3rd ed. (Wesley, 2006)
• DD. Landau and K. K Binder: „A A Guide to Monte Carlo
Simulations in Statistical Physics“ (Cambridge, 2000)
• D. Stauffer, , F.W. Hehl, , V. Winkelmann and J.G. Zabolitzky: y
„Computer Simulation and Computer Algebra“ 3rd ed.
(Springer, 1993)
• KBid K. Binder and dDWH D.W. Heermann: „Monte M Carlo C l Simulation Si l i in i
Statistical Physics“ 4th ed. (Springer, 2002)
• NN.J. J Giordano: „Computational Computational Physics Physics“ (Wesley (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“ Physics , ed ed. D. D Landau (Springer Series)
Journals
• Journal of Computational Physics (Elsevier)
• Computer Physics Communications
(Elsevier)
• IInternational i lJ Journal lof fM Modern d Physics Ph i C
(World Scientific)
every every year year (2007: (2007: Bruxelles, 2008: 2008: Brazil, Brazil, 2009 2009 Taiwan):
Taiwan):
CCP = Conference on Computational Physics
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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 p (not ( treated in this course) )
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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Computer tools
• Object oriented programming
• Vector supercomputers
• Parallel computing (shared and distributed
memory) )
• Symbolic Algebra (Mathematica, Maple)
• Graphical animations
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)
• CConsider id many ddegrees of fffreedom d (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 Se u e c Algorithms“ go s 3rd 3 d ed. ed ( (Addison dd so
– Wesley, 1997) Vol. 2, Chapt. 3.3.1
• JE J.E. Gentle, G tl „Random R d Number N b GGeneration ti and d
Monte Carlo Methods“ (Springer, 2003)
Properties p of Random numbers
• No correlations
• Long periods
• Follow well-defined distribution
• Fast implementation
• RReproductibility d tibilit
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Distribution of random numbers
+∞
∫
−∞∞
P( x) dx = 1 and P( x)
> 0
examples: homogeneous, Gaussian, Poisson
Probability to find a random number
i in the h i interval l[ [ [x, , x+ x+Δx]: Δ ] ]:
x +Δ x
wx ( ) = ∫ Pxdx ( )
Random number generators (RNG)
electrical flicker noise
Algorithms:
∫
x
photon emission
f from a semiconductor i d
• Congruential (Lehmer (Lehmer, 1948)
• Lagged-Fibonacci gg (Tausworth,1965)
( , )
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Congruential g generators g
Fix two integers: g c and p.
Start with a seed x0. Create new integers by iterating:
x = ( c⋅x ) p , x , c, p∈Z
i i− i
1 mod
Make random numbers
through
z =
i
x
i
p
Maximal period
z ∈
i
[ 01 , )
Since all all integers integers are are less less than than p the the sequence
sequence
must repeat after at least p-1 1 iterations, i.e.
the maximal period is p-1. 1.
(x (x0 = = 0 0 is is a a fixed fixed point and cannot cannot be used used.) )
R.D. Carmichael proved 1910 that one gets the
maximal period if p is a Mersenne prime number
and and the smallest smallest integer number for which
p−1
c mod p
= 1
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Mersenne prime p numbers
n
M = 2 −1
n
43 * December 15,
30,402,457 315416475…652943871 9,152,052
2005
44 * September 4,
32,582,657 124575026…053967871 9,808,358
2006
Example p of congruential g RNG
Park and Miller (1988):
const int p=2147483647;
const int c=16807;
int int rnd=42; rnd=42; //seed
rnd = (c*rnd) % p;
print rnd;
GIMPS / Curtis
& Steven Boone
GIMPS / Curtis
& Steven Boone
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Applet
x i+1
Square test
Theorem of Marsaglia g
x
(1968)
For a congruential generator the random
numbers b i in an n-cube cube-test b t ttest t li lie on parallel ll l
n-1 n 1 1 dimensional dimensional hyperplanes.
∃ a a a x + a x + + a x p =
1 1,..., , , n : ( 1 i 2 i + 1 ... n i + n n− 1 ) mod p 0
proof using:
∃∀ pcn , , at least one set
a a,..., a :
( 1
n )
1 ... ) n
ac + + a c mod p =
1 n p
1
0
n
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n-cube cube-test test
One One can can also sho show that that for for congruential congr ential RNG
RNG
the distance between the planes p must be larger g
than
p
n
and that the maximum number of planes is
p
1 n
Lagged Lagged-Fibonacci Fibonacci RNG
• Initialization ii i i of fbb random bits i xi • AApply: l
(∑ ( ∑ j∈ℑ
x = x ) mod 2
i
i i−
j
ℑ
⊂
[ 1,..,
,,
b ]
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Lagged gg Fibbonacci RNG
Typically one uses, since it is easy to implement:
x = x ⊕x ≡ x + x
( ) mod 2
i i−a i−b i−a i−b a < b
Theorem of fA A. C Compagner (1992):
If If ( (a,b ( (a,b a,b) ) Zierler trinomial then then sequence sequence has
has
maximal period 2 2b – 1 and :
2
x ⋅x − x = 0 ∀ k < b
i i−k i
Zierler trinomials
1+ x
a
+ x
b
primitive on Z Z2[x] (Zierler, 1969)
(a, ( , b) )
(103, 250) (Kirkpatrick and Stoll, 1981)
(1689, 4187)
(54454, 132049) (J.R. Heringa et al, 1992)
(3037958 (3037958, 6972592) (R.P.Brent et al., 2003)
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Making 64 64-bit bit integers
..…
z zi = (01101….101110)
(01101 101110)
z i+1 +1 = (10001….010111)
zi+2 i+2 2 = (00101….111001)
( )
zi+3 +3 = (01011….011100)
zi+4 +4 = (00011….011011)
.....
Tests for random numbers
• Check distribution
• Average is 0.5
• Average of each bit is 0.5
• n-cube-test n cube test
• Correlations should vanish
• SSpectral t lt test: t no peaks k iin Fourier F i transform t f
• χ2 χ test: partial p sums follow a Gaussian
• Kolmogorov – Smirnov test
→ „Diehard Di h d battery“ b tt “ of f M Marsaglia li (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 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: Tests: Treat sequences of some number of bits as "words" words . Count the overlapping words in a stream 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 square. If the circle overlaps an existing one 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 44,000 000 points in a cube of edge 11,000. 000 Center a sphere on each point, 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 u be oof floats oats needed eeded to reach eac 1 should s ou d follow o ow a certain ce ta distribution.
d st but o .
• 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.
RN with other distributions
• Transformation method
Poisson distribution
Gaussian distribution
(Box (B (Box Muller, M ll 1958)
1958)
• Rejection method
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Transformation method
We want random numbers y distributed as P(y). (y) ).
Start with homogeneously g y ⎧1 if z ∈ [ 0 , 1 ]
distributed numbers z:
P(y)
y z
⎪
P ( z ) = ⎨
⎪⎩
Transformation method
example:
p
generate Poisson distribution:
y
'
y y
−
' k y
ke k dy [ e ] 0 1
y
−
k
−
∫ =−
0
z = = −e
( 1 )
⇒ y =−kln −z
z
0 otherwise
y
' ' ' '
∫ ∫
z = P( z ) dz = P( y ) dy
0 0
y
P( y) ke k
−
=
where z ∈ [0,1) are homogeneous random numbers.
This method only works if the integral can be
solved l d and dth the resulting lti function f ti can be b inverted.
i t d
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Box –Muller Muller (1958)
Gaussian distribution:
distribution:
y y2
1 −
e dy σ
z e σ dy
∫
0
πσ
P( y) 1 −
y
e σ =
πσ
= ∫ cannot be solved in closed form.
trick:
r = y + y
y
tan tanϕϕ
=
y
2 2 2
1 2
1 2
1
2
dyy dyy = rdrdϕϕ
y 2 y 2
y y 1 1 y y
2
2
1 −
σ 1 −
σ
e e
∫ ∫
z ⋅ z = dy ⋅
dy
1 2 1 2
0 πσ 0 πσ
y y y2 2 2
1 + y2 ϕ r r
− −
σ σ
1 2
1 1
∫∫ ∫∫
e e
= dy dy = rdrdϕ
πσ πσ
1 2
0 0 0 0
2
ϕ σ
1 ⎛ y ⎞
= = ⎜ ⎟
πσ 2 2 2ππ ⎝ y2
⎠
Box –Muller Muller trick
1 ⎛ y ⎞ 1
z1⋅ z2
= ⎜ ⎟⋅
2π
⎝ y2
⎠
( 1 )
y + y =−σln −z
2 2
1 2 2
y sin 2π
z
= tan 2π
z =
y z
1 1
2
1
cos 2 2π
1
1
( 1−e σ ) arctan ( 1−e
σ )
2
2 2
1 + 2
r
y y
− −
2 2
1+ 2
y y
−
e σ
arctan ( 1−
)
⇒
( )
y = −σσ ln 1 1− z sin 2 2ππ
z
1 2 1
= − ( −
π ( )
y σ ln 1 z cos 2πz
2 2 1
From two homogeneously g y distributed random
numbers z1 and z 2 one gets two Gaussian
distributed random numbers y1 and y2 .
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Rejection method
Generate random numbers y ∈ [0, [, [0,B] ] sampled p
according to a distribution P(y) ) with P(y) < AA.
.
Sample two homogeneously
distributed random numbers
z 1 ,
, z 2 ∈ [0,1)
[0,1). . If the point
(B (Bz Bz1, , A Az Az2) )li ) lies above b the th curve
P(y), ), i.e. P(Bz Bz1) < Az Az2 then
reject the attempt, otherwise y = Bz Bz1 is retained as a
random number which which is is distributed according to P(y). ).
P
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