Post on 19-Jan-2016
An Introduction toMonte Carlo Methodsin Statistical Physics
Kristen A. FichthornThe Pennsylvania State University
University Park, PA 16803
Monte Carlo Methods: A New Way to SolveIntegrals (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
22 yxd
tot
hit
4
OABC Square of Area
CA Curve Under Area x 4
A1
CB
y
x0
1
Monte Carlo Sample Mean Integration
2
1
)( x
x
xfdxF
2
1
)((x)
)(
x
x
xxf
dxF
trials
fF
)(
)(
To Solve:
We Write:
Then: When on Each TrialWe RandomlyChoose from
Monte Carlo Sample Mean Integration:Uniform Sampling to Estimate
21,12
1)( xxx
xxx
10,1 )12(2
1
2/12
tot
tot
xx
12
01
2/12)1( 2x
x
xdxπFTo Estimate
Using a Uniform Distribution
Generate tot Uniform, Random Numbers
Monte Carlo Sample Mean Integrationin Statistical Physics: Uniform Sampling
)(exp rUrdZNVT
Quadraturee.g., with N=100 Molecules3N=300 Coordinates10 Points per Coordinate to Span (-L/2,L/2)10300 Integration Points!!!!
L
LL
tot
UV
Ztot
N
NVT
1
)(exp
Uniform Sample Mean Integration•Generate 300 uniform random coordinates in (-L/2,L/2)•Calculate U•Repeat tot times…
Problems with Uniform Sampling…
L
LL
tot
UV
Ztot
N
NVT
1
)(exp
Too Many Configurations Where
0)(exp rU
Especially for a DenseFluid!!
What is the Average Depth of the Nile?
Integration Using…
Adapted from Frenkel and Smit, “Understanding Molecular Simulation”,Academic Press (2002).
Quadrature vs. Importance Samplingor Uniform Sampling
else , 0
Nile in the if , 1)(
max
1
max
1
w
dw
wd
Importance Sampling for Ensemble Averages
NVT
NVT
NVTNVT
Z
rUr
rArrdA
)(exp)(
)()(
trialsNVT
trials
NVTNVT
AA
AA
If We Want to Estimatean Ensemble AverageEfficiently…
We Just Need toSample It With NVT !!
Importance Sampling: Monte Carloas a Solution to the Master Equation
)'(
),(
),'()'(),()'(),(
''
rr
trP
trPrrtrPrrdt
trdP
rr
: Probability to be at State at Time tr
: Transition Probability per Unit Time from to 'r
r
r
'r
The Detailed Balance Criterion
)'(),'( );(),( rrPrrP NVTNVT
xx
rPrrrPrrdt
rdP
),'()'(),()'(0
),(
After a Long Time, the System Reaches Equilibrium
)()'(exp)(
)'(
)'(
)'(
)()'()'()'(
rUrUr
r
rr
rr
rrrrrr
NVT
NVT
NVTNVT
At Equilibrium, We Have:
This Will Occur if the Transition Probabilities Satisfy Detailed Balance
Metropolis Monte Carlo
)()'( if ,
)()'( if , )(
)'(
)'(
rr')rrα(
rrr
r')rrα(
rr
NVTNVT
NVTNVTNVT
NVT
Nrrrr /1)'()'(
Use:
With:
)'()'()'( rraccrrrr
Let Take the Form:
= Probability to Choose a Particular Moveacc = Probability to Accept the Move
N. Metropolis et al. J. Chem. Phys. 21, 1087 (1953).
Metropolis Monte Carlo
Detailed Balance is Satisfied:
' if , N
1
' if , )'(exp1
)'(
UU
UUUUN
rr
Use:
)'(exp)'(
)'(UU
rr
rr
Metropolis MC Algorithm
Finished?
Yes
No
Give the Particle a RandomDisplacement, Calculate theNew Energy ')'( UrU
Accept the Move with
'exp
1min)'(
UUrr
Select a Particle at Random,Calculate the Energy UrU )(
1
)(
tottot
tottot rAAA
Calculate the Ensemble Average
tot
totAA
Initialize the Positions 0 ;0 tottotA
Periodic Boundary Conditions
L
Ld
If d>L/2 then d=L-d
It’s Like Doing aSimulation on a Torus!
Nearest-Neighbor, Square Lattice Gas
A
B
InteractionsAA
BB
AB
0.0 -1.0kT
0.0 0.0
-1.0kT 0.0
When Is Enough Enough?
0 100000 200000 300000 400000Trials
800
700
600
500
400
300
200
100
ygrenE
Run it Long...
…and Longer!
When Is Enough Enough?
0 2500 5000 7500 10000 12500 15000Trials
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
ygrenE
0 100000 200000 300000 400000Trials
0.7
0.6
0.5
0.4
0.3
0.2
0.1
ygrenE
Run it Big… …and Bigger!
2/1
1
2
1in Error
tottot
tot
xxx
Estimate the Error
When Is Enough Enough?
Make a Picture!
When Is Enough Enough?
Try DifferentInitial Conditions!
Phase Behavior in Two-DimensionalRod and Disk Systems
E. coli
TMV and spheres
Electronic circuitsBottom-up assembly of spheres
Nature 393, 349 (1998).
Use MC Simulation to Understandthe Phase Behavior of
Hard Rod and Disk Systems
Lamellar Nematic
Isotropic
MiscibleNematic
Smectic
MiscibleIsotropic
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
0)( rUold
0exp
exp
oldnewnewold UUP
)'(rUnew
New Configuration
Ouch!
A Lot of Infeasible Trials! Small Moves or…
k
jj
or bUnewW1
)(exp)(
k
jj
or )(bβUW(old)1
exp
Rosenbluth & Rosenbluth, J. Chem. Phys. 23, 356 (1955).
Move Center of Mass RandomlyGenerate k-1 New Orientations bj
New
Old
Configurational Bias Monte Carlo
Select a New Configurationwith
)(
)(exp )(
newW
bUbP n
or
n
)(
)(,1min
oldW
newWP newold
Accept the New Configurationwith
Final
Configurational Bias Monte Carlo andDetailed Balance
)(
)(exp)(
)(
)(exp)( )(
oW
oUon
nW
nUnobP n
)()(exp)(
)(oUnU
on
no
)()()( noaccnono
)(
)(
)(
)(
oW
nW
onacc
noacc
The Probability ofChoosing a Move:
Recall we Have of the Form:
The Acceptance Ratio:
Detailed Balance
Nematic Order Parameter
N
i
iuiuN
Q1
)()(21
Radial Distribution Function
Orientational CorrelationFunctions
rrg 02cos2
rrg 04cos4
N
i
N
jij
ji rrrN
Arg
1 1
22
Properties of Interest
800 rodsρ = 3.5 L-2
Snapshots
1257 rodsρ = 5.5 L-2
6213 rodsρ = 6.75 L-2
Snapshots
8055 rodsρ = 8.75 L-2
Accelerating Monte Carlo SamplingE
nerg
y
x
How Can We Overcome the HighFree-Energy Barriers to Sample Everything?
Accelerating Monte Carlo Sampling:Parallel Tempering
System N at TN
System 1 at T1
System 2 at T2
System 3 at T3
…
Metropolis Monte CarloTrials Within Each System
Swaps Between Systems i and j
TN >…>T3 >T2 >T1
))((exp
1min)(
ijji UUnoP
E. Marinari and and G. Parisi, Europhys. Lett. 19, 451 (1992).
Parallel Tempering in a Model Potential
2
275.1))2sin(1(5
75.175.0))2sin(1(4
75.025.0))2sin(1(3
25.025.1))2sin(1(2
25.12))2sin(1(1
2
)(
x
xx
xx
xx
xx
xx
x
xU
System 1 at kT1=0.05
System 2 at kT2=0.5
System 3 at kT3=5.0
90% Move Attempts within Systems10% Move Attempts are Swaps
Adapted from: F. Falcioniand 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).