Freiburg, Germany Albert Ludwig University IMTEK Monte-Carlo2002. 4. 21. ·...

ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG

Sta amics and o

orvink

ty

tistical ThermodynMonte-Carl

Evgenii B. Rudnyi and Jan G. KIMTEK

Albert Ludwig UniversiFreiburg, Germany


E.B. Rud

Preliminaries

Le

♦

F

♦

ST

♦

M

♦

P

♦

Re

♦

P

♦

A

P

♦

I

polis algorithm

, Comp. Sci. 000, v. 2, N 1, p. 65-69 (The

0 A ithms).

n esources

fk lecular Simulation, .ch .buffalo.edu/es 0/Text/text.html

ur of statistical ch as it pertains to le simulation

rk rocesses

n rlo simulation

p sing methods

lgor

e Re, Moeme

/ce53vey anicscularov p

te Cale bia

nyi, J.G. Korvink, Chair for Microsystem Simulation

arning Goalsrom Micro to Macrotatistical Mechanics (Statistical hermodynamics)onte-Carlo Method

olymersLattice Model and Random Walk

ferences. W. Atkins, Physical Chemistry.. R. Leach, Molecular Modelling: rinciples and Applications.

. Beichl, F. Sullivan, The

MetroEng. 2Ten 1

On-li♦ D. Ko

wwwcours♦ A s

memo

♦ Ma♦ Mo♦ Sim


E.B. Rud

Micro to Macro

M

♦

PA

♦

M

C

♦

Pm

QQ

♦

E

ies of system states and egeneracies

rg nd a number of en s depend on um

a m with a large b f particles is

e the energy levels v lose.

Ei Ωi ,

Ωi

Ei

ies astate

e. systeer o

and ery c


From

acro Propertiesressure, Diffusion coefficient, verage energy, TemperatureAtomic effects are smeared.

icro Propertieslassical Mechanics:

ositions and velocities/omenta of the atoms

uantum Mechanics / uantum Chemistry:

nergy is quantized

♦ Energtheir d♦ Ene

eigvol

♦ Fornumhugare

Ωi


E.B. Rud

Micro to Macro

Tw

MT

♦

CCN

♦

IS

SE

♦

Np

uces temperature and y.

ca quantum statistics

ss tatistics leads to ad : need quantum tis o solve them.

e to derive statistical ch from quantum tis

r : quasi-classical r

ri Monte-Carlo d

l andical soxes

tics tasieranicstics.acticeoach.se to.


From

o Approachesolecular Dynamics -

ime average:lassical Atomic Force, lassical Particle Mechanics - ewton mechanics

ntegrating transient chrödinger equation

tatistical Mechanics - nsemble average:o time, net effect of many articles

♦ Introdentrop

♦ Classi♦ Cla

parsta

♦ It ismesta

♦ In papp

♦ Givesmetho


E.B. Rud

ical Mechanics

Ov

♦

E

♦

M

♦

♦

♦

♦

C

♦

♦

♦

♦

♦

♦

O

♦

C

d-Form Solutions


Statist

erviewnsembleicrocanonical EnsembleBasic PostulateErgodic HypothesisNon-ergodic system

anonical EnsembleDefinitionTemperatureBoltzmann DistributionPartition FunctionClassical Statistics

ther Ensembleslassical vs. Quantum

♦ Close


E.B. Rud

ical Mechanics

En

♦

M

♦

M

♦ Ep

♦ De

♦ E

♦♦

D

D⟨ ⟩


Statist

sembleacro properties are the same.icro properties are different.

ach micro state has some robability.istribution function describes

verything.nsemble average for property ( is the probability)quantum

classical:

P

D⟨ ⟩ DiPii∑=

P pN rN,( )D pN rN,( ) pNd rNd∫=

ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGE.B. Rud

nical Ensemble

Ba♦ E

o

ntum statisticsumber of eigen-

o gy for system cles in volume .

Po ate: System with likely to be in any

bi

ti all such states is ic onical Ensemble.

(

V,E

V

N( )

V, 1Ω N V E, ,( )---------------------------

: Nf enerpartistul

lity

on ofrocan

E)

V E, ,

E) =


Microcano

sic Postulatenergy, volume and a number f particles are constant.

Qua♦

stateswith

♦ Basic fixed state.

♦ Proba

♦ Collecthe M

N V E, , )

Ω N,(

N

Pi N,(


nical Ensemble

Er♦ T

♦ I

averages should not d on initial conditions.b erage is equal to the

v av e: Molecular m

b erage: Monte-Carlo

D

D t(

D D⟨ ⟩=

24

68

10

20

40

60

80

100

120

140

120

140

TimeAveraging

le av

erageerag

icsle av

24

68

10

20

40

60

80

100


Microcano

godic Hypothesisime average of N, V, E:

.

nitial conditions:

♦ Time depen

♦ Ensem

time a♦ Time

Dyna♦ Ensem

1t--- D t ′( )

0t∫t ∞→

lim dt ′=

′ ) D rN t ′( ) pN t ′( ) r0N p0

N, , ,[ ]=

200 400 600 800 1000

2

4

6

8

10x

t

x


nical Ensemble

No♦ I

cee♦

♦ Kofka

r

on
st

Microcano

n-ergodic Systemf a time average does not give omplete representation of full nsemble, system is non-rgodic.

Truly nonergodic: no way there from here.Practically nonergodic: very hard to find route from here to there.

U

E = c


nical Ensemble

De♦ V

p♦ S

ct

Bolzmann: take r systems.G put attention on a ul stem, all others are n .o bable distribution oner is introduced g eatment. e e can say, that ra , volume and a er articles are n .

N(

L ∞→

V, const=

ibbs: ar sydingst pro.

aturethe trnd wture of pt: T N,


Cano

finitionolume and a number of articles are constant.ystems within the ensemble an exchange energy, but the otal energy is constant.

♦ After simila

♦ After particsurrou

♦ The mfuncti

♦ Tempdurin

♦ At thetempenumbconsta

12

1 V1, ) N2 V2,( )

E1 + E2 = const


nical Ensemble

Te♦ E

♦

♦ Nc

♦

likely distribution of y:

er :

py

E

Ω

ln

E1–( )------ -------------

N V E, ,

0=

Ω

1-----

∂ Ωln 2

∂E2----------------

N V,

==

=1

kBT---------

, kB Ωln N V E, ,( )

ature

:

E1 E,∂E1-------------

1--

N V,

β

E) =


Cano

mperaturenergy is extensive property:

over subsystems.The system interaction is neglected: energy is additive.

umber of states is multipli-ative:

Better use .

♦ Most energ

♦ Temp

♦ Entro

EE1 E2+=

E( ) Ω1 E1( ) Ω2 E E1–( )×=

Ω E( )[ ]ln

Ω E1 E E1–,( )[ ] =ln

Ω1 E1( )[ ] Ω2 E E1–( )[ ]ln+

∂ Ωln-----------

∂ ln

∂E---------

S N V,(


nical Ensemble

BoDi

♦ Sh

♦ Ad

d by

r sion about

itu and insert into bi

Pi

ΩB E Ei–( )

ΩB E Ej–( )j

∑----------------------------------=

Ei 0=

E Ei–( )] =ln

B( Ei

∂ Ωln B E( )∂E

-------------------------

Ei kBT⁄ ]e

Ej kBT⁄– ]j

∑----- -------------------------

expan

te T lity

ΩB[

E)] –

–[xp

[exp-------------


Cano

ltzmann stribution

mall A in contact with large eat bath B: .

in state i with energy (no egeneracy). Probability is

define

♦ Taylo

♦ Substproba

AB

E EA EB+=

Ei

Ω[ln

Pi =


nical Ensemble

Pa♦

♦ A

♦ H♦ T

♦

om it one can determine er equilibria properties.

ioy en up to an ar stant.

p le to determine a ric lue for the absolute y, Helmholtz energy.ett

Q

∑∑

------

F

F

Ej Eo–( ) kBT⁄– ]j

∑F kBT Q[ ]ln–

nis givy conossibal va

and er

[exp

Eo=


Cano

rtition Function

verage energy

elmholtz Energy hen

is a potential function

and frall oth

Caut♦ Energ

arbitr♦ It is im

numeenerg

♦ It is b

Ej kBT⁄–[ ]expj

∑=

E⟨ ⟩ EiPi=i∑=

Ei Ei kBT⁄–[ ]expi

Ei kBT⁄–[ ]expi

--------------------------------------------= ∂ Q[ ]ln∂ 1 kBT⁄( )------------------------–

E ∂ F T⁄( )∂ 1 T⁄( )------------------=

kBT Q[ ]ln–=

T V,( )

Q =


nical Ensemble

Cl

♦ E

♦ Pi

♦d

♦ Ph

♦ A

-Carlo)

al kinetic energy can en lytically

gu n integralg omes

Q

h

Eex D pN rN,( ) pNd rNd

βE– p pNd rNd----- -----------------------------------------------------

2---- 3N

βU– exp rNd∫

βU– D rN( ) rNd

βU– xp rNd------------------------------------------

over ana

ratioe bec

β–p

ex∫-------------

πmkT

h2--------------

exp∫e∫

-------------


Cano

assical Statistics

nergy

artition function becomes an ntegral

.

is the least action. makes imensionless.articles are indistinguishable, ence N!.verage (Starting point for

Monte

♦ Integrbe tak

♦ Confi♦ Avera

♦

E p r,( )pi

2

2mi---------

i

N

∑= U rN( )+

1

h3NN!--------------- βE– exp pNd rNd∫=

h3N Z

D⟨ ⟩ ∫--=

Q1

N!------

=

D⟨ ⟩ =


ical Mechanics

Ot Kofka


Statist

her Ensembles, I ♦ From


ical Mechanics

Ot Kofka


Statist

her Ensembles, II ♦ From


ical Mechanics

Cl♦ E

p♦ T

m♦

♦ Rm♦

tions - depends on temper-and the wave number.30 lassical approach v sm-1.

0 K c< 100


Statist

assical vs. Quantumlectron - always a quantum article.ransitional movement of olecules - always classical:Enormous number of states

(300 K - about ).otational movement of olecules - typically classical.Quantum statistics might required at low tempera-tures.

♦ Vibraature ♦ At

for

1030


ical Mechanics

ClSo♦ A

♦

♦ M♦♦

♦ Dc

♦ ID

a real crystal it is essary to know the n ectra

ie theories for liquidsne em is really ce .

on spty of of thssful


Statist

osed-Form lutionstomic ideal gasConfiguration integral is equal to volume.olecular ideal gas

Rigid rotator and harmonic oscillator

ense gases: estimates for virial oefficients.deal crystal: Einstein and ebye approximations

♦ Fornecpho

♦ A var♦ No

suc

Etot Etransl Erot Evib+ +=


-Carlo Method

Ov♦ B♦ R♦ I♦ M♦ M♦ L


Monte

erviewasic Metropolis Algorithmandom Sampling

mportance Samplingarkov Chainetropolis Method

imits of Metropolis method


-Carlo Method

BaAl♦ G

♦ F♦♦

♦

form distribution .

m new energy .

ce nsition from to ed robability

,

at

D⟨ ⟩

r ∆r+=

Ui r ′( )r r ′

a r r ′→( )=Ui r ′( ) Ui r( )– ]x )

BT

1L--- niD rN

i( )1

pute

pt tra on p

.e is

i

cceptβ–[p

i =L∑


Monte

sic Metropolis gorithmoal is to evaluate

or each Monte Carlo cycle:Select particle at random.Compute particle energy

.

Give particle random displacement based on the

uni

♦ Co

♦ Acbas

♦ Estim

β U rN( )[ ]– exp D rN( ) rNd∫β U rN( )[ ]– exp rNd∫

----------------------------------------------------------------------=

i

Ui r( )i

ri ′

min 1 e,(

β 1 k⁄=

D⟨ ⟩ ≈


-Carlo Method

Ra♦ E

♦ M

astic approach

th l

n rlo

from Kofka

δI n 2 d⁄–∼

δI n 1 2⁄–∼

odica

te Ca


Monte

ndom Samplingvaluating .

ethodical approach

♦ Stoch

♦ Error

♦ Me

♦ Mo

I f x[ ] xdab∫=


-Carlo Method

Im

-20

Modofallwhalm

.

.

, e.g.

.

.

.

xab∫b( f x[ ]⟨ ⟩

w x( ) xdab∫

)]u)]-------- ud

01∫

f x ui( )[ ]w x ui( )[ ]---------------------∑

f w⁄ )2⟩ f w⁄⟨ ⟩ 2– ]=

f x[ ] d

a– )f x[ ]w x[ ]-----------

f x u([w x([------------

i 1=L

1L--- (⟨[


Monte

portance Sampling

-15 -10 -5 5 10 15 20

0.2

0.4

0.6

0.8

1st ran-m points in region ere f(x) is ost zero.

f x( )

Random Points

♦

♦

♦

♦

♦

♦

I =

I =

I =

I =

I1L---≈

σI2

If is constant,variance vanishes

f w⁄


-Carlo Method

MS

♦ Mwir

♦ Fme

M♦ S

m

ion of next state depends n current state, and not on stass lly defined by a set s probabilities .

r ility of selecting n iven that presently e iti robability matrix ts .in bability does not d e initial distri- .

πij

Π

ij

P Π

tes. is fuition

obabext, gi.on-pall g pro on th

π

P=


Monte

arkov Chaintochastic processovement through a series of ell-defined states in a way that

nvolves some element of andomness.or our purposes,“states” are icrostates in the governing

nsemble.

arkov processtochastic process that has no emory.

♦ Selectonly oprior

♦ Proceof tran

♦ is pstate jin stat

♦ Transcollec

♦ Limitdepenbution

πij


-Carlo Method

M♦ A

do

♦ Ute

♦ Rpd

♦ M

is underlying bability to move (uniform tri n)

is the acceptance b y

then

then

os to prove that this to etailed balance.

N

π

n→ )

o( )

n) o)n) β U n( ) U o( )–[ ]– xpn) o)→ 1

butio

abilit

sible the d

n→

N(<e=

N(>n) =


Monte

etropolis Methodt limiting distribution, etailed balance or the principle f microscopic reversibility:

se any convenient underlying ransition matrix but not accept very step.ather accept a step with such a robability to satisfy the etailed balance.etropolis suggested

♦prodis

♦pro

♦ If

♦ If

♦ It is pleads

o( )π o n→( ) N n( )π n o→( )=

o n→( ) α o n→( )acc o n→( )=

α o(

acc

N(acc o →(

N(acc o(


-Carlo Method

Lime♦ C

r

p♦ M

e

♦ T

we can write

no p: it is necessary to le energy regions.l are required to

ate free energy and y

Q

D⟨ ⟩

βUe βU exp rNd∫βU– xp rNd

---- -----------------------------------------------

βU =
t helhightricks the .
–xp

e∫-------------

exp


Monte

mits of Metropolis thodan not estimate the configu-

ation integral

(or artition function).etropolis algorithm can

stimate only

rick

♦ Then

where ♦ Does

samp♦ Specia

estimentrop

conf βU– exp rNd∫=

β U rN( )[ ]– exp D rN( ) rNd∫β U rN( )[ ]– exp rNd∫

----------------------------------------------------------------------=

1 βU– β U expexp=

VQconf------------- =

D


Polymers

Ov mer, repeat unit and er

♦ E

s♦ R

♦ B

xt case for att tion of a polymer p er melt.

to c model is a ng simulate.

/ww m.Edu/macrog/index.htm

⟨

remeion:e soluolym

mistie to

w.Psrc.Us


erview ♦ Monopolym

nd-to-end distance , from

tatistics adius of gyration

ackbone chain

♦ Two esimul♦ dilu

and♦ Full a

challe

http:/

Rn2⟨ ⟩ o

nl2

S⟩ o2 Σi r ro–( )2 N⁄⟨ ⟩=


Polymers

LaRa

es (two and three dimen-), bent and crankshaft

♦ R♦♦

♦ D

fro


ttice Model and ndom Walk

♦ Latticsional

andom walksOn and off latticeSelf-cross and self-avoiding

emo from Atkins

m 99freire


Summary

♦ F♦ S

T♦♦♦♦♦♦

♦ M♦♦♦♦

tropolis Methodits of Metropolis methode

tic del and Random lk

rse Mo


rom Micro to Macrotatistical Mechanics (Statistical hermodynamics)

EnsembleMicrocanonical EnsembleCanonical EnsembleOther EnsemblesClassical vs. QuantumClosed-Form Solutionsonte-Carlo MethodBasic Metropolis AlgorithmRandom SamplingImportance SamplingMarkov Chain

♦ Me♦ Lim

♦ Polym♦ Lat

Wa

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