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An Introduction to Monte Carlo Methods in Statistical Physics

An Introduction to Monte Carlo Methods in Statistical Physics. Kristen A. Fichthorn The Pennsylvania State University University Park, PA 16803. 1. C. B. Algorithm: Generate uniform, random x and y between 0 and 1 Calculate the distance d from the origin

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An Introduction to Monte Carlo Methods in Statistical Physics

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  1. An Introduction to Monte Carlo Methods in Statistical Physics Kristen A. Fichthorn The Pennsylvania State University University Park, PA 16803

  2. 1 C B • Algorithm: • Generate uniform, random • x and y between 0 and 1 • Calculate the distance d from • the origin • If d ≤ 1, thit=thit+ 1 • Repeat for ttot trials y A 1 0 x Monte Carlo Methods: A New Way to Solve Integrals (in the 1950’s) “Hit or Miss” Method: What is p?

  3. Monte Carlo Sample Mean Integration To Solve: We Write: Then: When on Each Trial We Randomly Choose x from r

  4. Monte Carlo Sample Mean Integration: Uniform Sampling to Estimate p To Estimate Using a Uniform Distribution Generate ttot Uniform, Random Numbers

  5. L L L Monte Carlo Sample Mean Integration in Statistical Physics: Uniform Sampling Quadrature e.g., with N=100 Molecules 3N=300 Coordinates 10 Points per Coordinate to Span (-L/2,L/2) 10300 Integration Points!!!! • Uniform Sample Mean Integration • Generate 300 uniform random • coordinates in (-L/2,L/2) • Calculate U • Repeat ttot times…

  6. L L L Problems with Uniform Sampling… Too Many Configurations Where Especially for a Dense Fluid!!

  7. What is the Average Depth of the Nile? Integration Using… Quadrature vs. Importance Sampling or Uniform Sampling Adapted from Frenkel and Smit, “Understanding Molecular Simulation”, Academic Press (2002).

  8. Importance Sampling for Ensemble Averages If We Want to Estimate an Ensemble Average Efficiently… We Just Need to Sample It With r=rNVT !!

  9. : Probability to be at State at Time t : Transition Probability per Unit Time from to Importance Sampling: Monte Carlo as a Solution to the Master Equation

  10. The Detailed Balance Criterion After a Long Time, the System Reaches Equilibrium At Equilibrium, We Have: This Will Occur if the Transition Probabilities p Satisfy Detailed Balance

  11. Metropolis Monte Carlo Let p Take the Form: • = Probability to Choose a Particular Move acc = Probability to Accept the Move Use: With: N. Metropolis et al. J. Chem. Phys. 21, 1087 (1953).

  12. Metropolis Monte Carlo Use: Detailed Balance is Satisfied:

  13. Initialize the Positions Calculate the Ensemble Average Select a Particle at Random, Calculate the Energy Give the Particle a Random Displacement, Calculate the New Energy Metropolis MC Algorithm Yes Finished ? No Accept the Move with

  14. L d L Periodic Boundary Conditions If d>L/2 then d=L-d It’s Like Doing a Simulation on a Torus!

  15. Nearest-Neighbor, Square Lattice Gas A B Interactions eAA eBB eAB 0.0 -1.0kT 0.0 0.0 -1.0kT 0.0

  16. When Is Enough Enough? Run it Long... …and Longer!

  17. When Is Enough Enough? Run it Big… …and Bigger! Estimate the Error

  18. When Is Enough Enough? Make a Picture!

  19. When Is Enough Enough? Try Different Initial Conditions!

  20. Phase Behavior in Two-Dimensional Rod and Disk Systems TMV and spheres Nature 393, 349 (1998). E. coli Electronic circuits Bottom-up assembly of spheres

  21. Lamellar Nematic Smectic Miscible Nematic Isotropic Miscible Isotropic Use MC Simulation to Understand the Phase Behavior of Hard Rod and Disk Systems

  22. Depletion Zones Overlap Volume Hard Systems: It’s All About Entropy A = U – TS Hard Core Interactions U = 0 if particles do not overlap U = ∞ if particles do overlap Maximize Entropy to Minimize Helmholtz Free Energy Ordering Can Increase Entropy!

  23. Metropolis Monte Carlo Old Configuration Perform Move at Random New Configuration Ouch! Small Moves or… A Lot of Infeasible Trials!

  24. Select a New Configuration with Accept the New Configuration with Configurational Bias Monte Carlo Rosenbluth & Rosenbluth, J. Chem. Phys. 23, 356 (1955). Old Move Center of Mass Randomly Generate k-1 New Orientations bj New Final

  25. Configurational Bias Monte Carlo and Detailed Balance Recall we Have p of the Form: The Probability a of Choosing a Move: The Acceptance Ratio: Detailed Balance

  26. Properties of Interest Orientational Correlation Functions Nematic Order Parameter Radial Distribution Function

  27. Snapshots 1257 rods ρ = 5.5 L-2 800 rods ρ = 3.5 L-2

  28. Snapshots 6213 rods ρ = 6.75 L-2 8055 rods ρ = 8.75 L-2

  29. Energy x Accelerating Monte Carlo Sampling How Can We Overcome the High Free-Energy Barriers to Sample Everything?

  30. System N at TN … System 3 at T3 System 2 at T2 System 1 at T1 Accelerating Monte Carlo Sampling: Parallel Tempering E. Marinari and and G. Parisi, Europhys. Lett. 19, 451 (1992). Metropolis Monte Carlo Trials Within Each System Swaps Between Systems i and j TN >…>T3 >T2 >T1

  31. System 3 at kT3=5.0 System 2 at kT2=0.5 System 1 at kT1=0.05 Parallel Tempering in a Model Potential 90% Move Attempts within Systems 10% Move Attempts are Swaps Adapted from: F. Falcioni and M. Deem, J. Chem. Phys. 110, 1754 (1999).

  32. Good Sources on Monte Carlo: D. Frenkel and B. Smit, “Understanding Molecular Simulation”, 2nd Ed., Academic Press (2002). M. Allen and D. J. Tildesley, “Computer Simulation of Liquids”, Oxford (1987).

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