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Classical Monte-Carlo simulationsGraduate Summer Institute „Complex Plasmas“
at the Stevens Insitute of Technology
Henning Baumgartner,A. Filinov, H. Kählert, P. Ludwig and M. Bonitz
Christian-Albrechts-University of Kiel
Hoboken, NJ, 4th August 2008
Ф Simulation technique
Ф Monte-Carlo method
Ф Metropolis algorithm
Ф Examples in dusty plasma research
Ф Yukawa balls
Ф Ground states
Ф Radial potential barriers
Ф Probability of Configurations
Ф Conclusions
Picture of the Casino in Monte-Carlo
Outline
full description of the physical system
Simulation basics
Sir Isaac Newton(1643-1724)
Ludwig Boltzmann(1844-1906)
F=d pdt
r it 0
v it 0
(Molecular dynamics simulations)
Newton's second law of motion
+ initial conditionspositions
velocities
L. Boltzmann (1887):A trajectory should pass every point in phase space, which is consistent with external constraints.
„Ergodic hypothesis“
A=⟨ A ⟩ time average = statistical average
(Monte Carlo simulations)
⇒
Monte-Carlo methodIn general, every method that utilizes random numbers
Picture of the Casino in Monte-Carlo
like the randomness in the games in the Casino
used for e.g. calculation of calculation of integrals
Points insidePoints total
=14
with r=1
M2 0.00 3.1296 0.07 3.1406 3.14153 0.00 4.2071 0.09 4.1907 4.18874 0.00 4.9657 0.12 4.9268 4.93485 0.03 5.2863 0.14 5.2710 5.26376 0.62 5.2012 0.17 5.1721 5.16777 14.90 4.7650 0.19 4.7182 4.72478 369.00 4.0919 0.22 4.0724 4.0587
Quadr. Time Result MC time Result Correct val.
Volume of a sphere in M dimensions
Ideal method to compute high dimensional integrals
Monte-Carlo sampling
f x
a bx0
straightforward sampling➢ random points are choosen uniformly
big error in integral estimate
importance sampling
error ∝/NMC
reduce 2 by
➢ random points are choosen by a distribution
Error reduction by importance sampling without increase of the sample size N
MC
p1x =1/b−a
p2x=exp− x−x0
2/ 2 2
2 2
I=∫a
bf x dx
I=∫a
b f xp x
p x dx
≈b−a N MC
∑i=1
N MC f x i
xi
x i
≈ 1N MC
∑i=1
N MC f x ip x i
⇒
Monte-Carlo integration in statistical physics
consider a canonical [N,V,T] ensemble
observables in thermodynamics:
where R={r1, r 2, ... , rN } particle coordinates
U N R potential energy of all N interacting particles
=1/k BT inverse temperature
and the partition function
Z N ,V ,T = 1N !D N∫V
...∫Vexp −U N Rd
N R
=2ℏ2/mk BT 1/2with and D dimensionality
Equilibrium distribution of states is given by the Boltzmann factor
AN ,V ,T = 1Z N ,V ,T ∫V
...∫V AN Rexp −U N RdN R
pB R=exp −U N RZ N ,V ,T
Metropolis algorithm - 1How to create independent microstates ?A Ri „Markov chain“
from initial state R0
generate all further states with some transition probability Ri , Ri1some restrictions on to ensure that the states Ri , Ri1 A Riare distributed according to the equilibrium probability of states pB RThe condition
p Ri Ri , Ri1= p Ri1 Ri1 , Ri „detailed balance“
applied to the stationary solution of dp R/dt=0„Master equation“dp Ri
dt=−∑ R i1
Ri , Ri1 p Ri∑ Ri1 Ri1 , Ri p Ri1
guarantees that one correctly generates the Markov chain with states distributed by ⇒ pB R
Metropolis algorithm - 2
the calculation of the observables reduces from
Nicholas Constantine Metropolis(1915-1999)
Ri , Ri1 ={exp −U N R, if U N R≥01 , otherwise
How to choose the transition probability ? Ri , Ri1N. Metropolis[1]:
AN ,V ,T =∫∫V
AN Rexp −U N RdN R
∫∫Vexp−U N Rd
N R
=∫∫V
AN Rexp−U N R / p RdN R
∫∫Vexp −U N R / p Rd
N Rimportance
sampling
=∫∫V
AN RdN R
∫∫V1d N R
≈1N MC
∑i=1
N MC AN Ri
detailed balance + Metropolis transition prob.
observables can be calculated by simple averages
[1] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Journal of Chem. Phys., 21(1087), 1953
Ф Simulation technique
Ф Monte-Carlo method
Ф Metropolis algorithm
Ф Examples in dusty plasma research
Ф Yukawa balls
Ф Ground states
Ф Radial potential barriers
Ф Probability of Configurations
Ф Conclusions
Picture of the Casino in Monte-Carlo
Outline
j
✔
✔
✔
The experiment
CC
D-c
amer
a
outer region central region
[2] O.Arp, D. Block, A. Piel und A. Melzer, PRL 93 (1165004), 2004
setup[2]
pictures from the CCD-camera
⇒ all positions
and velocties
r iv i
⇒ easy comparisonexperiment vs. simulation
The model
F G=−GmM r−R
∣r−R∣3
F th= rd2/ vth kB ∇ T
F E=q E
forces on the dust particlesgravitation:
thermophoretic force:
external electrostatic force:
interaction: F Y=q1q2 r1−r2
40∣r1−r2∣3exp−∣r1−r 2∣
} confinement
measured experimental confinement[3]
⇒
spherical symmetric harmonic confinement
H=∑i=1
N p2
2m∑
i=1
N 2ri
2∑i=1
N
∑j=i1
N q2
∣r i−r j∣e−∣r i−r j∣
[3] O. Arp, D. Block, M. Klindworth, and A. Piel, PoP 12, 122102 (2005)
⇒
Model Hamiltonian
Monte-Carlo simulation of Yukawa balls
1. place particles {r 1 , r 2 , ... , rN }2. calculate the energy of the particles by
0. initialize the system Nset fixed ,set fixed T
H=∑i=1
N 2ri
2∑i j
q2
∣ri−r j∣e−∣r i−r j∣
3. displace a randomly choosen particle by some distance di4. calculate the energy of the energy difference to the old stateE5. compare a random number with the Metropolis function
exp − E
*
* repetition of this procedure times, we define as 1 Monte-Carlo stepN
accept/ reject the new state∈[0,1 ]
E 0 [E i , E i1 ]
?
1.-2. 3.-5. (1) [E i1 , E i2 ]3.5. (2)
...?
typically 107 MC steps for one set
6. calculate the averages of the observables, e.g. ⟨E ⟩
[N ,T ]
Ground states of Yukawa balls
stable statesT 0
Ground states of Yukawa balls
NMC MC
2 2 2 0.5953 0.595286 6 6 2.10651 2.10651
12 12 12 3.84069 3.8406930 26,4 26,4 7.8092 7.8091931 27,4 27,4 8.0001 8.0001132 28,4 28,4 8.1899 8.1899440 34,6 34,6 9.6436 9.64361
Configuration Energy [E/N]MD[4] MD[4]
cooling down
experiment
0.0 (115;56;18;1) 36.3570.2 (114;57;18;1) 23.7290.4 (110;58;20;2) 17.6080.6 (107;60;21;2) 14.0280.8 (105;60;22;3) 11.6721.0 (102;60;24;4) 9.998
E /Nsimulation
screeningchanges theground stateconfiguration
MC simulation
MD simulation⇔
[4] P. Ludwig, S. Kosse and M. Bonitz, Phys. Rev. E 71 (046403), 2005
stable statesT 0T 1
T 2
T 0T 1T 2
Metastable states of Yukawa balls
N=31 =0.800 =1.500N=31
(28;3)(27;4)(26;5)(25;6)(24;7)
p(conf) – probability of occurance during simulation (107 MC steps)
Radial potential barriers
N=31
(25;6)
1
0
2
3
4
5
(26;5)
(27;4)
=0.0
=1.585 =1.635
inward barriers
estimation of radial stability / melting temperatures
=1 /k BT(5;26)
SummaryWhen/Why should one (not) use Monte-Carlo?
It is easy➢ to implement➢ to run a fast code➢ to access equilibrium properties
downside➢ no non-equilibrium properties➢ no time dynamics
requirements➢ good pseudo-random-number generator,
e.g. Mersenne Twister (period of 219937 − 1 )➢ good error estimation
further details and more examplesat posters➢ Yukawa tubes (done by K. Tierney, Boston College during stay
in the RISE program of the DAAD)➢ Yukawa balls
but➢ Dynamic Monte Carlo➢ Kinetic Monte Carlo
Thank you!
Picture of the windjammer on the Kieler Woche 2008
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