Monte Carlo: Naïve Lecture #19 Rigoberto Hernandez. Monte Carlo Methods 3

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1 Major Concepts Naïve Monte Carlo E.g., calculating π" Metropolis Monte Carlo Detailed Balance Improved Sampling Techniques Importance Sampling Umbrella Sampling Torrie and Valleau (1977) The Wolff Algorithm PRL 62, 361 (1989) Parallel Tempering & Replica Exchange Swendsen & Wang (1986) Wang-Landau (PRL 2001) Monte Carlo of Polymers Monte Carlo Methods 1

2 Monte Carlo: Naïve Monte Carlo Methods 3

Naïve MC: Calculating π" Numerics of π: N circ, N sq, 4*N

164 (2 digits) 78520 100000 3.1408 785322 1000000 3.

3 Naïve MC: Calculating π" Numerics of π: N circ, N sq, 4*N circ /N sq (1 digit) (2 digits) (3 digits) (4 digits) ,, (8 digits) Monte Carlo Methods 4

4 Monte Carlo: Detailed Balance Metropolis Monte Carlo: x x P(x) = Probability that trajectory resides in a given state W(x x ) = Probability per unit time that if the system is in state x, it will make a transition to state x ( ΔE x x' = E(x') E(x) ) Let Monte Carlo Methods 6

5 Monte Carlo: Detailed Balance Metropolis Monte Carlo: x x P(x) = Probability that trajectory resides in a given state W(x x ) = Probability per unit time that if the system is in state x, it will make a transition to state x N. Metropolis, A. Rosenbluth, M.Rosenbluth, A. Teller and E. Teller, J. Chem. Phys. 21, 1087 (1953). Importance Sampling (using loaded dice): Choose a good distribution from which to simulate the random variables of choice. Monte Carlo Methods 7

6 Monte Carlo: Metropolis N. Metropolis, A. Rosenbluth, M.Rosenbluth, A. Teller and E. Teller, J. Chem. Phys. 21, 1087 (1953) Monte Carlo Methods 8

7 Monte Carlo A sample of representative states or configurations is created by performing a random walk through configuration space. A Monte Carlo trajectory in the Ising magnet model would involve picking a random spin in the system and flipping the direction of the spin. The energy of the new configuration is calculated and compared to the energy of the old configuration: If the new energy is lower, the move is accepted. If the energy is higher, a random number is generated and compared to the Boltzmann distribution. It is rejected if the random number is greater than the Boltzmann distribution Monte Carlo Methods 9

8 E(x) Monte Carlo: Umbrella Sampling? x Note hyperdynamics... Monte Carlo Methods 12

9 Monte Carlo: Umbrella Sampling What if E(x) x OK. Let s write: Monte Carlo Methods 13

10 Monte Carlo: Umbrella Sampling Thus, < f > can be calculated from another distribution which corresponds to E 1 (x): as if P(x) = P 0 (x) P 1 (x). Monte Carlo Methods 14

11 Monte Carlo: Umbrella Sampling Monte Carlo Methods 15

12 Parallel Tempering & Replica Exchange Problem: How do you sample all of the space on a corrugated energy landscape? (barriers greater than target T.) Answer: run N many replicas using MC at N different temperatures, T i, in parallel Every so often, attempt exchanges between replicas according to probability = min( 1, exp([1/kt i - 1/kT j ][E i - E j ])) Satisfies detailed balance RH Swendsen and JS Wang, Replica Monte Carlo simulation of spin glasses, Phys. Rev. Lett. 57, (1986) " C. J. Geyer, in Computing Science and Statistics Proceedings of the 23rd Symposium on the Interface, American Statistical Association, New York, 1991, p Monte Carlo Methods 16

13 1-phase system Wang-Landau 2-phase system E E If we know ρ(e), then we know p(e) at any T! If we know p(e), then we know all the thermodynamics! Monte Carlo Methods 17

14 Wang-Landau A Monte Carlo approach for obtaining the density of states, ρ(e), directly Rather than computing g(r) or other observables A random walk through energy space! Algorithm: Initialize ρ(e)=1 for all E Accept moves according to: probability = min[ 1, ρ(e initial )/ρ(e trial move ) ] Let E' be the energy of the state at the end of the trial Update ρ(e') = f ρ(e') & the histogram H( Set(E') )+=1 Stop once the histogram is flat, [error in ρ(e) = O{ln(f)} ] Iterate, now using smaller f F. Wang & D.P. Landau, Phys. Rev. Lett. 86, 205 (2001). [original version for spin Hamiltonian] Monte Carlo Methods 18

15 Monte Carlo of Polymers Example: Metropolis Monte Carlo with the following types of moves: monomer diffusion: randomly selected monomer is displaced it a random small distance; reptation: randomly selected monomer from either end of the chain is reattached to the opposite end of the chain; crank-shaft: the portion of the chain between two randomly selected non-bonded monomers is rotated by a small random angle; random pivot: randomly selects a single monomer and the portion of the connected chain on either side of the monomer. A random direction is then chosen and the selected portion of the chain is then rotated around this axis by a small random angle. D. T. Seaton, S. J. Mitchell, and D. P. Landau Brazilian Journal of Physics, vol. 36, no. 3A, September, 2006, p. 623 Monte Carlo Methods 19

Monte Carlo: Naïve Lecture #14 Rigoberto Hernandez. Monte Carlo Methods 3

Monte Carlo: Naïve Lecture #14 Rigoberto Hernandez. Monte Carlo Methods 3 Major Concepts Naïve Monte Carlo E.g., calculating π" Metropolis Monte Carlo Detailed Balance Improved Sampling Techniques Importance Sampling Umbrella Sampling Torrie and Valleau (1977) The Wolff Algorithm