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# An Introduction to Monte Carlo Methods in Statistical Physics - PowerPoint PPT Presentation

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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Monte Carlo Methods

in Statistical Physics

Kristen A. Fichthorn

The Pennsylvania State University

University Park, PA 16803

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?

To Solve:

We Write:

Then:

When on Each Trial

We Randomly

Choose x from r

Uniform Sampling to Estimate p

To Estimate

Using a Uniform Distribution

Generate ttot Uniform, Random Numbers

L

L

Monte Carlo Sample Mean Integration

in Statistical Physics: Uniform Sampling

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

Problems with Uniform Sampling…

Too Many Configurations Where

Especially for a Dense

Fluid!!

Integration Using…

or Uniform Sampling

“Understanding Molecular Simulation”,

If We Want to Estimate

an Ensemble Average

Efficiently…

We Just Need to

Sample It With

r=rNVT !!

: Transition Probability per Unit Time

from to

Importance Sampling: Monte Carlo

as a Solution to the Master Equation

After a Long Time, the System Reaches Equilibrium

At Equilibrium, We Have:

This Will Occur if the Transition Probabilities p

Satisfy Detailed Balance

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).

Use:

Detailed Balance is Satisfied:

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

d

L

Periodic Boundary Conditions

If d>L/2 then d=L-d

It’s Like Doing a

Simulation on a Torus!

A

B

Interactions

eAA

eBB

eAB

0.0 -1.0kT

0.0 0.0

-1.0kT 0.0

Run it Long...

…and Longer!

Run it Big… …and Bigger!

Estimate the Error

Make a Picture!

Try Different

Initial Conditions!

Rod and Disk Systems

TMV and spheres

Nature 393, 349 (1998).

E. coli

Electronic circuits

Bottom-up assembly of spheres

Nematic

Smectic

Miscible

Nematic

Isotropic

Miscible

Isotropic

Use MC Simulation to Understand

the Phase Behavior of

Hard Rod and Disk Systems

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!

Old Configuration

Perform Move at Random

New Configuration

Ouch!

Small Moves or…

A Lot of Infeasible Trials!

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

Detailed Balance

Recall we Have p

of the Form:

The Probability a of

Choosing a Move:

The Acceptance Ratio:

Detailed Balance

Orientational Correlation

Functions

Nematic Order Parameter

1257 rods

ρ = 5.5 L-2

800 rods

ρ = 3.5 L-2

6213 rods

ρ = 6.75 L-2

8055 rods

ρ = 8.75 L-2

x

Accelerating Monte Carlo Sampling

How Can We Overcome the High

Free-Energy Barriers to Sample Everything?

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