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