L échantillonnage parfait: un nouveau paradigme des calculs de Monte Carlo

Transcription

1 L échantillonnage parfait: un nouveau paradigme des calculs de Monte Carlo Séminaire général Département de Physique Ecole normale supérieure Werner Krauth Laboratoire de physique statistique Ecole normale supérieure 25 mars 2010

2 Table of contents Monte Carlo calculations in atomic physics... in physics of liquids... Markov chains (relation with perturbation theory) Convergence Perfect sampling Diffusion

3 Direct Monte Carlo sampling Monte Carlo sampling Computer program: print randint(1,6),randint(1,6)

4 Direct Monte Carlo sampling Monte Carlo sampling (Gaussians) print gauss(0,1),gauss(0,1)

5 Direct Monte Carlo sampling (Ideal bosons in trap) Ideal bosons in potential Program

6 Ideal cold atoms (3 d, simulation) (Path-integral representation) Holzmann, Krauth (1999)

7 Interacting cold atoms (3 d, simulation) Markov-chain algorithms allow us to incorporate interactions... relation with perturbation theory...

8 Interacting trapped bosons (experiment) 2-d cold-atom experiment (J. Dalibard + group)

9 Interacting trapped bosons (experiment) N 96, Rb atoms at T 132 nk (quasi-2d). interactions a 0 = 5.24nm

10 Comparison experiment - simulation Experimental and theoretical density profiles for identical traps, temperatures, number of bosons ( )... Monte Carlo simulation approximation-free... Optical density OD(r) r (µm) (a) r (µm) (b)... yet disagreement theory/experiment.. (Rath, Yefsah, Günter, Cheneau, Desbuquois, Holzmann, Krauth, Dalibard)

11 Classical two-d particles (Hard disks) η = 0.48 η = 0.72 Which phase transition between the two phases? how to generate these configurations?

12 Classical two-d particles (Hard disks) Metropolis (1953) algorithm: equilibrium for t.

13 Minimum running time of a Monte Carlo algorithm Monte Carlo algorithms approach thermal equilibrium (the stationary probability distribution) as exp[ t/τ]. No need to let simulations run until t. The condition t τ (for example, 10 τ) is enough. τ can only be estimated from simulations with effective running times t τ. This is a #1 problem,...

14 Longer hard-disk simulation disk k same disk t = 0... t = τ exists, but is large (τ ). this is still a very small system

15 Correlation time in systems of current interest back τ still exists...

16 Event-chain algorithm for hard spheres i f Bernard, Krauth, Wilson (2009) much faster than previous methods......allows to understand previous work...

17 Correlation time # of equiv. Metropolis moves 1e+13 1e+12 1e+11 1e+10 1e N η= SEC/h MD/h [17] Metropolis/h A faster algorithm allows to see that previous workers were too optimistic... config

18 One-d convergence t 0 t Markov-chain Monte Carlo algorithm on 5 sites converges as exp[ t/τ] with finite correlation time τ...

19 Transfer matrix transfer matrix: T mc = {p(i j)} = has largest eigenvalue 1, and a second eigenvalue......which fixes correlation time τ corr = 1/ log(λ mc 2 )....τ difficult to estimate in general. Empirically: longer, and longer simulations......get done later and later...

20 One-d calculation that finishes on time! t t=0...start earlier and earlier......get done on time......propp, Wilson (1995).

21 Correlations and coupling II (coupling) t 0 t 0 +t coup Markov-chain Monte Carlo algorithm......with random maps. This chain couples after 10 steps.

22 Transfer matrix II many-particle forward transfer matrix: T 1,1 T 2, T 2,2 T 3,2... T forward = 0 0 T 3,3..., T N,N...describes a physical system......coupling is also exponential...and always slower than convergence...

23 Correlations and coupling III (from the past) t t 0 t 0 +t fw coup -t bw coup t=0 Simulation starts really early (at time t )......At time t = 0, we are done infinite simulation.

24 Wide choice of coupled maps The independent choice of arrows is not the only possible Here s a better choice: 1/3 1/3 1/

25 Ising model, Ising spin glass spins {σ 1,...,σ N } = {±1,...,±1} E = J ij i,j σ iσ j (local energy) π({σ 1,...,σ N }) exp[ βe({σ 1,...,σ N })] one position (5-site problem) one such configuration (Ising)

26 Coupling from the past (alternative representation) Coupling from the past (Propp & Wilson (1995)) has raised great expectations. must prove that entire configuration space couples 2 N confs initial conf. perfect sample? t= t=t 0 t=t 0 +τ coup t=0

27 Updates on large lattices Ising spin glass has states. We must rigorously show that they all couple. No tricks known. Using patches k on the lattice, and sets of patches S k on patch k (k = 1,...,N), we define Ω = S 1 S 2 S N /(pairwise compat.). Ω is overcomplete, but storage requirement is only N 2 m2 /2

28 Patches and compatibilities in the Ising model key key k l patch k spin configs patch l eliminate all confs on l whose key is absent on k. eliminate all confs on k whose key is absent on l. repeatedly check all possible pairs of neighboring patches.

29 Dictionaries key value [,,,,, ] [,,,, ] [,, ] Dictionary (hash table) associating keys with values

30 Renormalization procedure number of configs. per patch sweeps 2d ±J spin glass at β = 0.5, works also in 3 dimensions

31 Birth and death for hard spheres I a move b Space of configurations infinite......achieving coupling is not hopeless......birth-and-death formulation for hard-spheres

32 Birth and death for hard spheres II t 0 t 0 +t sim t Hidden discrete structure in a continuous model.

33 Birth and death for hard spheres III L x 0 -T T start t=0 (now) Finite number of possible configurations at any given time.

34 Conclusion

35 References J. G. Propp and D. B. Wilson Exact sampling with coupled Markov chains and applications to statistical mechanics Random Structures & Algorithms 9, 223 (1995). W. Krauth Statistical Mechanics: Algorithms and Computations (Oxford University Press, 2006) Wiki site C. Chanal and W. Krauth Renormalization group approach to exact sampling PRL (2007), C. Chanal and W. Krauth Convergence and coupling for spin glasses and hard spheres PRE (2010) E. P. Bernard, W. Krauth, and D. B. Wilson Event-chain Monte Carlo algorithm for hard-sphere systems PRE (2010) S. P. Rath, T. Yefsah, K. J. Guenter, M. Cheneau, R. Desbuquois, M. Holzmann, W. Krauth, J. Dalibard The equilibrium state of a trapped two-dimensional Bose gas arxiv: