an introduction to monte carlo methods in statistical physics
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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).