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

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

Monte Carlo Sample Mean Integration To Solve: We Write: Then: When on Each Trial We Randomly Choose  from 

Monte Carlo Sample Mean Integration: Uniform Sampling to Estimate  To Estimate Using a Uniform Distribution Generate  tot Uniform, Random Numbers

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) Integration Points!!!! L L L Uniform Sample Mean Integration Generate 300 uniform random coordinates in (-L/2,L/2) Calculate U Repeat  tot times…

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

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

Importance Sampling for Ensemble Averages If We Want to Estimate an Ensemble Average Efficiently… We Just Need to Sample It With    NVT !!

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

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

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

Metropolis Monte Carlo Detailed Balance is Satisfied: Use:

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

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

Nearest-Neighbor, Square Lattice Gas A B Interactions  AA  BB  AB kT kT 0.0

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

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

When Is Enough Enough? Make a Picture!

When Is Enough Enough? Try Different Initial Conditions!

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

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

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 Overlap Volume Depletion Zones Ordering Can Increase Entropy! Hard Systems: It’s All About Entropy

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

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

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

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

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

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

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

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

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

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