Advanced Monte Carlo algorithms
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1 Advanced Monte Carlo algorithms Bad Honnef Physics School Computational Physics of Complex and Disordered Systemss Werner Krauth Département de Physique Ecole Normale Supérieure, Paris, France 23 September 2015 Monte Carlo algorithms - basic notions. Hard disks: From detailed balance to global balance and to cluster algorithms. Integration and Sampling: From Gaussians to Maxwell and Boltzmann. Cluster algorithms for spin models: Ising, XY, Spin glasses.
2 Breaking the rules
3 MOOC Statistical Mechanics: Algorithms and Computations see WK Coming home from a MOOC (CSE 2015)
4 Newton: event-driven molecular dynamics Next event at t = Molecular dynamics (Alder & Wainwright 1957)
5 Event-disks.py (1/2)
6 Event-disks.py (2/2)
7 Boltzmann: Equiprobability
8 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
9 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
10 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
11 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
12 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
13 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
14 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
15 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
16 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
17 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
18 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
19 Boltzmann: Equiprobability Random positions x i, y i (σ < x i, y i < 1 σ) If illegal: Tabula rasa! Else: OK!
20 Direct-disks.py
21 Newton vs. Boltzmann Equiprobability: π Newton (C) = π Boltzmann (C) = { constant 0 else for legal C Rigorous proofs: Sinai (1970), Simanyi (2003).
22 Newton vs. Newton (Chaos) initial position t = 0 of the green disks on the left (x, y) green = (0.25, 0.25) initial position t = 0 of the green disks on the right (x, y) boule verte = ( , 0.25)
23 Asakura-Oosawa Depletion interaction (5th force in nature) The probability π(x) is non-uniform (Asakura-Oosawa 1954).
24 Asakura-Oosawa Depletion interaction (5th force in nature) probability π(x) non-uniform: Disks are attracted to the walls Disks are attracted to each other
25 Asakura-Oosawa Depletion interaction (5th force in nature) Animations by Maxim Berman (for MOOC)
26 Configurations at density η = 0.42 All configurations equally likely
27 Liquid-solid transition of hard disks Described in 1964 using MD (Alder & Wainwright) Understood in 2011 (Bernard & Krauth)
28 Markov-chain Monte Carlo ( Boltzmann ) Local Markov-chain Monte Carlo for hard disks: i = 1 (rej.) i = 2 i = 3 i = 4 (rej.) i = 5 i = 6 i = 7 i = 8 (rej.) i = 9 (rej.) i = 10 i = 11 i = 12 (rej.) Metropolis et al. (1953) (see lecture 1). Exponential convergence (see lecture 1).
29 Markov-disks.py Metropolis et al. (1953).
30 Correlation time in larger simulations disk k same disk t = 0... t = τ exists, but it is large (τ ).
31 Detailed balance and global balance f a c f a c e d e d global balance detailed balance g b f a c e d maximal global balance
32 Lifting - two hard spheres hard wall hard wall d + a + b + c + d a b c x d 1 a 1 b 1 c 1 a 1 (transl.) c 2 (transl.) b 2 a 2 (transl.) d 2 (transl.) t t+1 t+2 t+3 t+4 t+5 Following: Diaconis, Holmes, Neal (2000)
33 Lift-two-disks-discrete.py Break detailed balance (move up or right)
34 Lift-two-disks-discrete.py (output) Pair distances ( x, y) shown. Break detailed balance (move up or right)
35 Lifting impossible (?) - three hard spheres c 2 a 2 d 2 a 1 b 1 e 3 a 3 f 3 t t+1 t+2
36 Event-chain algorithm Bernard, Krauth, Wilson PRE (2009) (hard-sphere potentials). Rejection-free, fixed total length. Global balance OK (!) (moving right and up).
37 event-chain.py
38 Pressure formula of Event-chain algorithm Excess distance pressure.
39 Hard-disk configuration hard disks circular color code for orientational order Bernard, Krauth (PRL 2011)
40 Event-chain for continuous potentials Event-chain algorithm for continuous potentials: Exact sampling of canonical partition function. Event-driven: time-discretization, energy drift, thermostat, potential cutoff (!).
41 Event-chain for continuous potentials Event-chain algorithm for continuous potentials: Exact sampling of canonical partition function. Event-driven: time-discretization, energy drift, thermostat, potential cutoff (!).
42 Conclusions Concepts considered: Molecular dynamics (Event-driven) Monte Carlo, equiprobability, tabula rasa algorithm Asakura-Oosawa depletion interaction Hard-disk transition Lifting Algorithms considered: event-disks.py direct-disks.py markov-disks.py event-chain.py
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