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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

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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


An introduction to monte carlo methods in statistical physics

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?


An introduction to monte carlo methods in statistical physics

Monte Carlo Sample Mean Integration

To Solve:

We Write:

Then:

When on Each Trial

We Randomly

Choose x from r


An introduction to monte carlo methods in statistical physics

Monte Carlo Sample Mean Integration:

Uniform Sampling to Estimate p

To Estimate

Using a Uniform Distribution

Generate ttot Uniform, Random Numbers


An introduction to monte carlo methods in statistical physics

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…


An introduction to monte carlo methods in statistical physics

L

L

L

Problems with Uniform Sampling…

Too Many Configurations Where

Especially for a Dense

Fluid!!


An introduction to monte carlo methods in statistical physics

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).


An introduction to monte carlo methods in statistical physics

Importance Sampling for Ensemble Averages

If We Want to Estimate

an Ensemble Average

Efficiently…

We Just Need to

Sample It With

r=rNVT !!


An introduction to monte carlo methods in statistical physics

: 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


An introduction to monte carlo methods in statistical physics

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


An introduction to monte carlo methods in statistical physics

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).


An introduction to monte carlo methods in statistical physics

Metropolis Monte Carlo

Use:

Detailed Balance is Satisfied:


An introduction to monte carlo methods in statistical physics

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


An introduction to monte carlo methods in statistical physics

L

d

L

Periodic Boundary Conditions

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

It’s Like Doing a

Simulation on a Torus!


An introduction to monte carlo methods in statistical physics

Nearest-Neighbor, Square Lattice Gas

A

B

Interactions

eAA

eBB

eAB

0.0 -1.0kT

0.0 0.0

-1.0kT 0.0


An introduction to monte carlo methods in statistical physics

When Is Enough Enough?

Run it Long...

…and Longer!


An introduction to monte carlo methods in statistical physics

When Is Enough Enough?

Run it Big… …and Bigger!

Estimate the Error


An introduction to monte carlo methods in statistical physics

When Is Enough Enough?

Make a Picture!


An introduction to monte carlo methods in statistical physics

When Is Enough Enough?

Try Different

Initial Conditions!


An introduction to monte carlo methods in statistical physics

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


An introduction to monte carlo methods in statistical physics

Lamellar

Nematic

Smectic

Miscible

Nematic

Isotropic

Miscible

Isotropic

Use MC Simulation to Understand

the Phase Behavior of

Hard Rod and Disk Systems


An introduction to monte carlo methods in statistical physics

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!


An introduction to monte carlo methods in statistical physics

Metropolis Monte Carlo

Old Configuration

Perform Move at Random

New Configuration

Ouch!

Small Moves or…

A Lot of Infeasible Trials!


An introduction to monte carlo methods in statistical physics

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


An introduction to monte carlo methods in statistical physics

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


An introduction to monte carlo methods in statistical physics

Properties of Interest

Orientational Correlation

Functions

Nematic Order Parameter

Radial Distribution Function


An introduction to monte carlo methods in statistical physics

Snapshots

1257 rods

ρ = 5.5 L-2

800 rods

ρ = 3.5 L-2


An introduction to monte carlo methods in statistical physics

Snapshots

6213 rods

ρ = 6.75 L-2

8055 rods

ρ = 8.75 L-2


An introduction to monte carlo methods in statistical physics

Energy

x

Accelerating Monte Carlo Sampling

How Can We Overcome the High

Free-Energy Barriers to Sample Everything?


An introduction to monte carlo methods in statistical physics

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


An introduction to monte carlo methods in statistical physics

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).


An introduction to monte carlo methods in statistical physics

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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