Classical Monte-Carlo simulations Graduate 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, J, 4th August 2008
Outline Ф Simulation technique Ф Monte-Carlo method Ф Metropolis algorithm Ф Examples in dusty plasma research Ф Yukawa balls Ф Ground states Ф adial potential barriers Ф Probability of Configurations Ф Conclusions Picture of the Casino in Monte-Carlo
Simulation basics ewton's second law of motion d p F= dt positions ri t 0 velocities vi t 0 + initial conditions full description of the physical system (Molecular dynamics simulations) Sir Isaac ewton (1643-1724) 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) Ludwig Boltzmann (1844-1906)
Monte-Carlo method In general, every method that utilizes random numbers like the randomness in the games in the Casino used for e.g. calculation of calculation of integrals Points inside 1 = Points total 4 with Picture of the Casino in Monte-Carlo r=1 Volume of a sphere in M dimensions M 2 3 4 5 6 7 8 Quadr. Time 0.00 0.00 0.00 0.03 0.62 14.90 369.00 esult MC time 3.1296 0.07 4.2071 0.09 4.9657 0.12 5.2863 0.14 5.2012 0.17 4.7650 0.19 4.0919 0.22 esult Correct val. 3.1406 3.1415 4.1907 4.1887 4.9268 4.9348 5.2710 5.2637 5.1721 5.1677 4.7182 4.7247 4.0724 4.0587 Ideal method to compute high dimensional integrals
Monte-Carlo sampling straightforward sampling random points x i are choosen uniformly f x b I = a f x dx b a i =1 f x i MC MC 2 p 2 x = / MC error 2 exp x x 0 / 2 2 2 big error in integral estimate 2 reduce by importance sampling random points x i are choosen by a distribution f x I = a p x dx px f x i 1 MC i =1 p x i p 1 x =1/b a b MC a x0 Error reduction by importance sampling without increase of the sample size MC b
Monte-Carlo integration in statistical physics consider a canonical [,V,T] ensemble observables in thermodynamics: 1 A, V, T =... A exp U d V V Z,V,T where ={r1, r2,..., r } particle coordinates potential energy of all interacting particles U inverse temperature =1/k B T and the partition function 1 Z, V,T =... V exp U d D V! with and =2 ℏ2 / mk B T 1/ 2 D dimensionality Equilibrium distribution of states is given by the Boltzmann factor exp U pb = Z,V, T
Metropolis algorithm - 1 How to create independent microstates from initial state i A? Markov chain 0 i, i 1 i, i 1 to ensure that the states A i some restrictions on are distributed according to the equilibrium probability of states p B generate all further states with some transition probability The condition i i, i 1 = p i 1 i 1, i p applied to the stationary solution dp / dt =0 detailed balance of Master equation i dp i, i 1 p i i 1, i p i 1 = dt i 1 i 1 guarantees that one correctly generates the Markov chain with states distributed by p B
Metropolis algorithm - 2 How to choose the transition probability. Metropolis[1]: i, i 1? { exp U, if U 0 i, i 1 = 1, otherwise the calculation of the observables reduces from A exp U d icholas Constantine Metropolis (1915-1999) [1]. Metropolis, A. W. osenbluth, M.. osenbluth, A. H. Teller and E. Teller, Journal of Chem. Phys., 21(1087), 1953 V A, V,T = exp U d V A exp U / p d importance V = sampling exp U / p d V A d V detailed balance + = Metropolis transition prob. 1 d V 1 A i i =1 MC MC observables can be calculated by simple averages
Outline Ф Simulation technique Ф Monte-Carlo method Ф Metropolis algorithm Ф Examples in dusty plasma research Ф Yukawa balls Ф Ground states Ф adial potential barriers Ф Probability of Configurations Ф Conclusions j Picture of the Casino in Monte-Carlo
The experiment CCD-camera setup[2] pictures from the CCD-camera outer region central region all positions and velocties ri vi easy comparison experiment vs. simulation [2] O.Arp, D. Block, A. Piel und A. Melzer, PL 93 (1165004), 2004
The model forces on the dust particles gravitation: thermophoretic force: G mm r F G= 3 r Fth= r2d / vth k B T E external electrostatic force: FE =q interaction: F Y= measured experimental confinement[3] q 1 q2 r1 r2 3 4 0 r 1 r 2 } confinement exp r 1 r 2 spherical symmetric harmonic confinement Model Hamiltonian H = i =1 p2 2 q2 r i e r r 2m i=1 2 i=1 j =i 1 r i r j [3] O. Arp, D. Block, M. Klindworth, and A. Piel, PoP 12, 122102 (2005) i j
1.-2. Monte-Carlo simulation of Yukawa balls 3.-5. (1) 3.5. (2) E, E E0 [ i i 1 ]?? 0. initialize the system set fixed 2. calculate the energy of the particles by 3. displace a randomly choosen particle *,set fixed {r1, r2,..., r } 1. place particles [ 0,1 ] T i i by some distance d E to the old state with the Metropolis function accept/ reject the new state 6. calculate the averages of the observables, e.g. * repetition of this procedure exp E E times, we define as 1 Monte-Carlo step typically 107 MC steps for one set... 2 q2 H = r i e r r i =1 2 i j r i r j 4. calculate the energy of the energy difference 5. compare a random number [ E i 1, E i 2 ] [,T ] j
Ground states of Yukawa balls stable states T0
Ground states of Yukawa balls stable states T0 T1 T2 2 6 12 30 31 32 40 experiment Configuration Energy [E/] [4] MD MD[4] MC MC 2 2 0.5953 0.59528 6 6 2.10651 2.10651 12 12 3.84069 3.84069 26,4 26,4 7.8092 7.80919 27,4 27,4 8.0001 8.00011 28,4 28,4 8.1899 8.18994 34,6 34,6 9.6436 9.64361 cooling down T 0 T 1 T 2 MC simulation MD simulation [4] P. Ludwig, S. Kosse and M. Bonitz, Phys. ev. E 71 (046403), 2005 simulation 0.0 0.2 0.4 0.6 0.8 1.0 (115;56;18;1) (114;57;18;1) (110;58;20;2) (107;60;21;2) (105;60;22;3) (102;60;24;4) E/ 36.357 23.729 17.608 14.028 11.672 9.998 screening changes the ground state configuration
Metastable states of Yukawa balls =31 =0.800 (28;3) (27;4) (26;5) (25;6) (24;7) =31 =1.500 p(conf) probability of occurance during simulation (107 MC steps)
adial potential barriers =0.0 inward barriers =31 5 =1 /k B T 4 (5;26) =1.585 =1.635 3 (25;6) 2 (26;5) 1 (27;4) 0 estimation of radial stability / melting temperatures
Summary When/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 but Dynamic Monte Carlo Kinetic Monte Carlo requirements good pseudo-random-number generator, e.g. Mersenne Twister (period of 219937 1 ) good error estimation further details and more examples at posters Yukawa tubes (done by K. Tierney, Boston College during stay in the ISE program of the DAAD) Yukawa balls
Thank you! Picture of the windjammer on the Kieler Woche 2008