Introduction to Statistical Thermodynamics - 2

고려대학교 화공생명 공학과

강정원

Review of Previous Lecture

Molecular Partition Function Effective way of calculating average properties

(macroscopic) of a system with given quantum state.

Probability

Molecular partition function

Molecular partition function indicates number of possible state that are thermally available at T.

qe

ee

Nnp

i

j

i

j

ii

βε

βε

βε −

−

−

===

−=j

jeq βε

Objectives of 2nd Lecture

Ensemble Average Method

Thermodynamic properties and the Canonical Ensemble

Link between classical and quantum mechanics : Phase Space

Semi classical partition function

Very Simple example of Monte-Carlo Simulation

Introduction

Statistical Mechanics

Properties of individual molecules

PositionMolecular geometry

Intermolecular forces

Properties of bulk fluid (macroscopic properties)

PressureInternal EnergyHeat Capacity

EntropyViscosity

What we have learned from previous lecture…

Solution to Schrodinger equation (Eigen-value problem) Wave function

Allowed energy levels : En

Using the molecular partition function, we can calculate average values of property at given QUANTUM STATE.

Quantum states are changing so rapidly that the observed dynamic properties are actually time average over quantum states.

Ψ=Ψ+Ψ∇−i

ii

EUm

h 22

2

8π

Fluctuation with time…

time

Although we know most probable distribution of energies of individual molecules at given N and E (previous section – molecular partition function) it is almost impossible to get time average for interacting molecules

Thermodynamic Properties

Entire set of possible quantum states

Thermodynamic internal energy

,...,...,, 111 iΨΨΨΨ

,...,...,,, 321 iEEEE

Δ=∞→ i

ii tEUττ

1lim

Difficulties

Fluctuations are very small

Fluctuations occur too rapidly

We have to use alternative, abstract approach.

Ensemble average method

Alternative Procedure

Proposed by Gibbs

Ensemble Method

Ensemble ? : Infinite number of mental replica of the system of interest

Large Reservoir (const.T)

All the ensemble members have theSame N,V.T

Energies can be exchanged but molecules cannot.

Current N = 20 but N infinity

Two postulates

Long time average = Ensemble average at N infinity

In an ensemble , the systems of enembles are distributed uniformly (equal probability or frequency) Ergodic Hypothesis

Principle of equal a priori probability

E1 E2 E3 E4 E5

time

Averaging Method

Probability of observing particular quantum state i

Ensemble average of a dynamic property

Time average and ensemble average

=

ii

ii n

nP

ii

iPEE >=<

ii

inii PEtEU ∞→∞→=Δ= limlim

τ

Calculation of Probability in Ensemble

Several methods are available

Method of Undetermined multiplier

:

:

Maximization of Weight - Most probable distribution

Weight

∏==

iin

Nnnn

NW!

!!...!!

!

321

The Boltzmann Distribution

Task : Find the dominating configuration for given N and total energy E.

Find Max. W which satisfies ;

=

=

iiit

ii

nEE

nN

0

0

=

=

iii

ii

dnE

dn

Method of Undetermined Multipliers

Maximum weight , W Recall the method to find min, max of a

function…

Method of undetermined multiplier : Constraints should be multiplied by a constant

and added to the main variation equation.

0ln

0ln

=

∂

=

idnW

Wd

Method of undetermined multipliers

0ln

lnln

=

−+

∂=

−+

∂=

iii

i

iii

ii

ii

i

dnEdn

W

dnEdndndn

WWd

βα

βα

0ln =−+

∂i

i

Edn

W βα

∂∂

−∂

∂=

∂

∂

−=

j i

jj

ii

ii

nnn

nNN

nW

nnNNW

)ln(lnln

lnlnln

1ln1lnln +=

∂∂×+

∂∂=

∂∂ N

nN

NNN

nN

nNN

iii

+=

∂∂

×+

∂∂

=∂

∂

ji

i

j

jjj

i

j

j i

jj nnn

nnn

nn

nnn

1ln1ln)ln(

NnNn

nW i

ii

ln)1(ln)1(lnln −=+++−=∂

∂

0ln =++− ii E

Nn βα

iEi eNn βα −=

−

−

=

==

j

E

j j

Ej

j

j

ee

eNenN

βα

βα

1

−

−

==

j

E

Ei

i j

i

ee

NnP β

βBoltzmann Distribution

(Probability function for energy distribution)

Canonical Partition Function

Boltzmann Distribution

Canonical Partition Function

Qe

ee

NnP

i

j

i E

j

E

Ei

i

β

β

β −

−

−

===

−=j

E jeQ β

Thermodynamic Properties and Canonical Ensemble

Internal Energy

VNVN

qsi

Ei

VN

qsi

Eii

ii

QQQ

U

eEQ

eEQ

PEEU

i

i

,,

)(,

)(

ln1

1

∂

∂−=

∂∂−=

−=

∂∂

==>==<

−

−

ββ

ββ

β

Thermodynamic Properties and Canonical Ensemble

Pressure at i state

N

ii

iiiNi

iiNi

VEP

wdVPdxFdEdxFdVPw

∂∂−=

−=−=−===

δδ

)()(

dx

dViF

V

idE iE

Small Adiabatic expansion of system

∂=

∂∂=

∂∂

∂∂==

>==<

−

−−

VQP

eVE

QVQ

eVE

QeP

QP

PPP

i

E

N

i

N

i

E

N

i

i

Ei

ii

i

ii

lnln1

ln

11

P

,

i

β

β β

β

ββ

Thermodynamic Properties and Canonical Ensemble

Pressure

Equation of State in Statistical Mechanics

ProbabilityPressure at quantum state i

Thermodynamic Properties and Canonical Ensemble

−=

+−=

−=−=

∂∂=

+==−=

iii

ii

iii

iii

irev

N

ii

iii

iii

i iiiii

revrev

dPP

dPQdPPdPE

wPdVdVVEPdEP

dEPdPEPEddUwqdU

ln1

)lnln(1

)(

β

β

δ

δδ

Entropy

Thermodynamic Properties and Canonical Ensemble

Entropy

TdSQUdq

TdSPPdq

qdPPdEP

rev

iiirev

revi

iii

ii

ββδ

ββδ

δβ

=+=

=−=

=−=

)ln(

)ln(

ln1

The only function that links heat (path integral) and state property is TEMPERATURE.

kT/1=β

><−=−=+=

++=

PkPPkSTUQkS

STUQkS

ii lnln

/ln/ln 0

Summary of Thermodynamic Properties in Canonical Ensemble

ijNVTii

NT

NTNV

NV

NV

NQkT

VQQkTG

QkTAVQ

TQkTH

TQQkS

TQkTU

≠

∂

∂−=

∂∂−−=

−=

∂∂+

∂∂=

∂∂+=

∂∂=

,,

,

,,

,

,

ln

)lnln(ln

ln

)lnln()

lnln(

)lnln(ln

)lnln(

μ

All thermodynamic propertiesCan be obtained from “PARTITION FUNCTION”

Classical Statistical Mechanics

It is not easy to derive all the partition functions using quantum mechanics

Classical mechanics can be used with negligible error when energy difference between energy levels (Ei) are smaller thank kT.

However, vibration and electronic states cannot be treated with classical mechanics. (The energy spacings are order of kT)

Phase Space

Recall Hamiltonian of Newtonian Mechanics

Instead of taking replica of systems (ensemble members), use abstract ‘phase space’ compose of momentum space and position space (6N)

Average of infinite phase space

),...,,(2

),(

energy) alPE(potenti energy) KE(kinetic),(

21 Ni i

iNN

NN

Um

H

H

rrrppr

pr

+=

+=

ii

ii

H

H

rp

pr

=

∂∂

−=

∂∂

Phase Space

Np

Nr

1t

2t

1Γ

2Γ

{ }NN rrrrpppp ,...,,,,,...,,, 311321=Γ

Phase space

Ensemble Average

ΓΓΓ=Γ= ∞→∞→dEdEU Nn

)()(lim)(1lim0

Pτ

ττ

τ

ΓΓ dN )(P Fraction of Ensemble members in this range (Γ το Γ+dΓ)

Γ−Γ−=ΓΓdkTH

dkTHdN )/exp(...)/exp()(P

Using similar technique used forBoltzmann distribution

Canonical Partition Function

Γ−= dkTH )/exp(...T

Phase Integral

Γ−= dkTHc )/exp(...Q

Canonical Partition Function

Γ−

−=

∞→ dkTH

kTEi

i

T )/exp(...

)/exp(limc

Match between Quantum and Classical Mechanics

NFhN!1=c

For rigorous derivation see Hill, Chap.6

Canonical Partition Function in Classical Mechanics

Γ−= dkThN NF )/exp(...

!1 HQ

Example : Translational Partition Function for an Ideal Gas

=

+=

+=

N

i i

i

Ni i

iNN

NN

mpH

Um

H

H

3 2

21

2

),...,,(2

),(

energy) alPE(potenti energy) KE(kinetic),(

rrrppr

pr

No potential energy, 3 dimensionalspace.

NN

NVN

iN

iNN

i

iN

VhmkT

N

drdrdrdpmp

hN

drdrdpdpmp

hNQ

2/3

2

0321

3

3

11

2

3

2!

1

)2

exp(!1

......)2

exp(...!1

=

−=

−=

∞

∞−

π

Semi-Classical Partition Function

The energy of a molecule is distributed in different modes- Vibration, Rotation (Internal : depends only on T)- Translation (External : depends on T and V)

Hamiltonian operator can be separated into two parts (internal + center of mass motion)

Assumption 1

),(),,(

)exp()exp()exp(

int

intint

int

TNQTVNQQ

kTE

kTE

kTEEQ

HHH

CM

iCMii

CMi

opCMopop

=

−−=+−=

+=

Semi-Classical Partition Function

Internal parts are density independent and most of the components have the same value with ideal gases.

For solids and polymeric molecules, this assumption is not valid any more.

),0,(),,( intint TNQTNQ =ρ

Semi-Classical Partition Function

For T > 50K, classical approximation can be used for translational part.

Assumption 2

N

Ni

NNiziyixN

Ni

iziyixCM

drdrdrkTUZ

mkTh

ZN

drkTUdpmkT

ppphN

Q

rrrUm

pppH

321

2/12

3

33222

3

321

222

...)/(...

2

!

)/(...)2

exp(...!1

),...,,(2

−=

=Λ

Λ=

−++

−=

+++

=

−

π ConfigurationIntegral

ZQN

Q N3int!

1 −Λ=

=Ω

−

Ω=

ω

ωω

d

dddrdrdrkTUZ NN

N

......)/(...11321

For non-central forces (orientation effect)

Configuration Integral and Molecular Simulation

−

−>=<

rr

rrr

dkTU

dkTUAA

)/)((

)/)()((

Introduction to Monte-Carlo Simulation

Monte Carlo Method : Wide range of problem solving tool using RANDOM NUMBERS

Monte Carlo ? – coined after casino

In principle any method that uses random number to solve a problem is a Monte Carlo Method

A Classical Example

Calculation of pi

Trial shots are generated between 0 and 1 (x and y)

Compare x^2 + y ^2 < 1 or not

If true, add to the number of successful shot

Pi = 4H/N

N : Area of Rectangle (1*1)

4H : Area of ¼ circle (pi*1*1*1/4 )