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An Introduction to Monte Carlo Methods in Statistical Physics

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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
- 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?

Monte Carlo Sample Mean Integration

To Solve:

We Write:

Then:

When on Each Trial

We Randomly

Choose x from r

Monte Carlo Sample Mean Integration:

Uniform Sampling to Estimate p

To Estimate

Using a Uniform Distribution

Generate ttot Uniform, Random Numbers

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…

L

L

L

Problems with Uniform Sampling…

Too Many Configurations Where

Especially for a Dense

Fluid!!

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).

Importance Sampling for Ensemble Averages

If We Want to Estimate

an Ensemble Average

Efficiently…

We Just Need to

Sample It With

r=rNVT !!

: 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

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

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).

Metropolis Monte Carlo

Use:

Detailed Balance is Satisfied:

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

L

d

L

Periodic Boundary Conditions

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

eAA

eBB

eAB

0.0 -1.0kT

0.0 0.0

-1.0kT 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

TMV and spheres

Nature 393, 349 (1998).

E. coli

Electronic circuits

Bottom-up assembly of spheres

Lamellar

Nematic

Smectic

Miscible

Nematic

Isotropic

Miscible

Isotropic

Use MC Simulation to Understand

the Phase Behavior of

Hard Rod and Disk Systems

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!

Metropolis Monte Carlo

Old Configuration

Perform Move at Random

New Configuration

Ouch!

Small Moves or…

A Lot of Infeasible Trials!

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

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

Properties of Interest

Orientational Correlation

Functions

Nematic Order Parameter

Radial Distribution Function

Snapshots

1257 rods

ρ = 5.5 L-2

800 rods

ρ = 3.5 L-2

Snapshots

6213 rods

ρ = 6.75 L-2

8055 rods

ρ = 8.75 L-2

Energy

x

Accelerating Monte Carlo Sampling

How Can We Overcome the High

Free-Energy Barriers to Sample Everything?

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

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).

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).