Statistical Mechanics - Ensembles

Boltzmann and Gibbs ensemble methods for systems with interacting particles

1

J. Willard Gibbs

(1839-1903)

Ludwig Boltzmann

(1844 – 1906)

)()()()()( NN EPEPEPEPEP 1312111

How can we determine the state of systems with interacting molecules or

systems which interact with the environment?

N

ijNtot

EEE

UUUKKKE

21

131221

2

The probability for the system total energy is not the product of independent single

molecule energy probabilities

In interacting systems, the total energy does not break up into a sum of one molecule

energies. Interactions between molecules also contribute to the energy

• How do we find the distribution of velocity

and positions for systems with interactions?

• Do attractive forces increase the probability

of finding molecules closer together?

• Do repulsive forces increase the probability

of findng molecules farther apart?

3

• In quantum mechanics, the “state” is represented by a wave function;

• The energy of the states are quantized. Not all possible energy values are observed.

Quantum states with discrete energies

http://physics.stackexchange.com/

questions/102097/why-not-drop-

hbar-omega-2-from-the-quantum-

harmonic-oscillator-energy

For an N-atom quantum system, the state of the system can be represented by a single

discrete quantum index i

4

Possible distributions of microstates Macrostate

Value, X

Degeneracy of

macrostate, Ω(X)

(1,1,1)

{1,1,2}

{1,1,3}{1,2,2}

{1,1,4}{1,2,3} (2,2,2)

{1,1,5}{1,2,4}{1,3,3}{2,2,3}

{1,1,6}{1,2,5}{1,3,4}{2,2,4}{2,3,3}

{1,2,6}{1,3,5}{1,4,4}{2,2,5}{2,3,4}(3,3,3)

{1,3,6}{1,4,5}{2,2,6}{2,3,5}{2,4,4}{3,3,4}

{1,4,6}{1,5,5}{2,3,6}{2,4,5}{3,3,5}{4,4,3}

{1,5,6}{2,4,6}{2,5,5}{3,3,6}{3,4,5}(4,4,4)

{1,6,6}{2,5,6}{3,4,6}{3,5,5}{4,4,5}

{2,6,6}{3,5,6}{4,4,6}{4,5,5}

{3,6,6}{4,5,6}(5,5,5)

{4,6,6}{5,5,6}

{5,6,6}

(6,6,6)

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

1

3

6 = 3+3

10 = 3+6+1

15 = 3+6+3+3

21=3+6+6+3+3

25=6+6+3+3+6+1

27=6+6+3+6+3+3

27=6+3+6+6+3+3

25=6+6+3+3+6+1

21=3+6+6+3+3

15=3+6+3+3

10=3+6+1

6=3+3

3

1

Distributing indistinguishable dice among microstates • Constraint: Ndice = 3

• There is no constraint on the value of the macrostate

5

Distributing indistinguishable objects among bins

(1234)(1243)(1324)(1342)(1423)(1432)

(2134)(2143)(2314)(2341)(2413)(2431)

(3124)(3142)(3214)(3241)(3412)(3421)

(4123)(4132)(4213)(4231)(4312)(4321)

Ways of arranging four

distinguishable objects 4! = 4×3×2×1

Ways of binning four

distinguishable objects

into two groups of 2 and 2

{(12)(34)}{(12)(43)}{(13)(24)}{(13)(42)}{(14)(23)}{(14)(32)}

{(21)(34)}{(21)(43)}{(23)(14)}{(23)(41)}{(24)(13)}{(24)(31)}

{(31)(24)}{(31)(42)}{(32)(14)}{(32)(41)}{(34)(12)}{(34)(21)}

{(41)(23)}{(41)(32)}{(42)(13)}{(42)(31)}{(43)(12)}{(43)(21)}

{(123)(4)}{(124)(3)}{(132)(4)}{(134)(2)}{(142)(3)}{(143)(2)}

{(213)(4)}{(214)(3)}{(231)(4)}{(234)(1)}{(241)(3)}{(243)(1)}

{(312)(4)}{(314)(2)}{(321)(4)}{(324)(1)}{(341)(2)}{(342)(1)}

{(412)(3)}{(413)(2)}{(421)(3)}{(423)(1)}{(431)(2)}{(432)(1)}

Ways of binning four

distinguishable objects

into two groups of 3 and 1

How many ways are there to bin the four

distinguishable objects into two groups?

6

Distributing indistinguishable objects among bins

Distinct binning of 4

distinguishable objects

into bins in two groups of

2 and 2 regardless of order

in bins

{(12)(34)}{(12)(43)}{(13)(24)}{(13)(42)}{(14)(23)}{(14)(32)}

{(21)(34)}{(21)(43)}{(23)(14)}{(23)(41)}{(24)(13)}{(24)(31)}

{(31)(24)}{(31)(42)}{(32)(14)}{(32)(41)}{(34)(12)}{(34)(21)}

{(41)(23)}{(41)(32)}{(42)(13)}{(42)(31)}{(43)(12)}{(43)(21)}

However, if the order of the balls in the bins is not important, the distinct distributions are:

𝑊 =4!

2! 2!=

4×3×2×1

2×1×2×1 = 6

{(123)(4)}{(124)(3)}{(132)(4)}{(134)(2)}{(142)(3)}{(143)(2)}

{(213)(4)}{(214)(3)}{(231)(4)}{(234)(1)}{(241)(3)}{(243)(1)}

{(312)(4)}{(314)(2)}{(321)(4)}{(324)(1)}{(341)(2)}{(342)(1)}

{(412)(3)}{(413)(2)}{(421)(3)}{(423)(1)}{(431)(2)}{(432)(1)}

Distinct binning of 4

distinguishable objects into

two bins in two groups of 3

and 1 regardless of order in

bins

𝑊 =4!

3! 1!=

4×3×2×1

3×2×1×1 = 4

Distinct binning of N distinguishable objects into k

bins groups of a1 , a2, …, ak regardless of order in

bins

𝑊 =𝑁!

𝑎1! 𝑎2! … 𝑎𝑘!=𝑁!

𝑎𝑗!

7

Distributing systems among microstates with constraints on the macrostate

• Assume a system with equally spaced energy levels of energy 0, Δ, 2Δ, 3Δ, … …

ε6 = 5Δ

ε5 = 4Δ

ε4 = 3Δ

ε3 = 2Δ

ε2 = 1Δ

ε1 = 0

Our problem in statistical mechanics is to find the distributions of molecules

among the levels (“microstates”) satisfying two constraints:

1) The total number of molecules in the system is constant

2) The total energy of the system is constant

The discrete states are analogous to the

discrete outcomes of the role of a dice

Our goal: Given what we know about the structure of the system (the “states”

available to the system) how do we determine the most probable the distribution of

molecules / systems among the states that is consistent with our macroscopic

information on the system?

8

The macrostate: Etot = 5Δ

or equivalently E = Δ

• A system with equally spaced energy levels;

• Find the distributions of 5 independent

molecules among the levels (“microstates”)

satisfying two constraints:

…

ε6 = 5Δ

ε5 = 4Δ

ε4 = 3Δ

ε3 = 2Δ

ε2 = 1Δ

ε1 = 0

Distributing indistinguishable molecules among states

j ja

NW

!

!)(a 5 20 20 30 30 20 1

How many ways can each microstate be constructed (i.e., what is the degeneracy)?

Microstates

Distributions:

{a1, a2, a3, …, a6}

Degeneracy

(Weight)

Probability of

distribution P(di)

d1:{4,0,0,0,0,1}

d2:{3,1,0,0,1,0}

d3:{3,0,1,1,0,0}

d4:{2,2,0,1,0,0}

d5:{2,1,2,0,0,0}

d6:{1,3,1,0,0,0}

d7:{0,5,0,0,0,0}

5

20

20

30

30

20

1

5/126

20/126

20/126

30/126

30/126

20/126

1/126

j iji aNW !! ,

• Many microstates of the molecules satisfy the macrostate conditions

• All 126 microstates are equally probable but different distributions, di, show up

with different probability 9

Distributions (di) show how molecules occupy energy states

126i iW

aj : Number of molecules that occupy state j in each

distribution (occupancy of state j)

a3=2

a2=1

a1=2

Possible microstates satisfying two constraints on the macrostates

Distribution:

{a1, a2, a3, …, a8}

Degeneracy

(Weight)

Probability

{6,0,0,0,0,0,0,1}

{5,1,0,0,0,0,1,0}

{5,0,1,0,0,1,0,0}

{5,0,0,1,1,0,0,0}

{4,2,0,0,0,1,0,0}

{4,1,1,0,1,0,0,0}

{4,1,0,2,0,0,0,0}

{4,0,2,1,0,0,0,0}

{3,3,0,0,1,0,0,0}

{3,2,1,1,0,0,0,0}

{3,1,3,0,0,0,0,0}

{2,4,0,1,0,0,0,0}

{2,3,2,0,0,0,0,0}

{1,5,1,0,0,0,0,0}

{0,7,0,0,0,0,0,0}

7

42

42

42

105

210

105

105

140

420

140

105

210

42

1

7/1716

42/1716

42/1716

42/1716

105/1716

210/1716

105/1716

105/1716

140/1716

420/1716

140/1716

105/1716

210/1716

42/1716

1/1716

Sum of microstates: 1716

N = 7 and Etot = 7Δ; E = Δ determine all possible distributions Macrostate

10

Most probable

distribution

Systems with large numbers of molecules – Example 1

j iji aNW !! ,

Distributions

{a1, a2, a3, …, a10}

Degeneracy

(Weight)

{8,0,0,0,0,0,0,0,0,1}

{7,1,0,0,0,0,0,0,1,0}

{7,0,1,0,0,0,0,1,0,0}

{7,0,0,1,0,0,1,0,0,0}

{7,0,0,0,1,1,0,0,0,0}

{6,2,0,0,0,0,0,1,0,0}

{6,0,2,0,0,1,0,0,0,0}

{6,1,1,0,0,0,1,0,0,0}

{6,1,0,1,0,1,0,0,0,0}

{6,0,1,1,1,0,0,0,0,0}

{6,0,0,3,0,0,0,0,0,0}

{5,3,0,0,0,0,1,0,0,0}

{5,0,3,1,0,0,0,0,0,0}

{5,2,1,0,0,1,0,0,0,0}

{5,1,2,0,1,0,0,0,0,0}

{5,1,1,2,0,0,0,0,0,0}

9

72

72

72

72

252

252

504

504

504

84

504

504

1512

1512

1512

Distributions:

{a1, a2, a3, …, a10}

Degeneracy

(Weight)

{4,4,0,0,0,1,0,0,0,0}

{4,1,4,0,0,0,0,0,0,0}

{4,3,1,0,1,0,0,0,0,0}

{4,3,0,2,0,0,0,0,0,0}

{4,2,2,1,0,0,0,0,0,0}

{3,3,3,0,0,0,0,0,0,0}

{3,4,1,1,0,0,0,0,0,0}

{3,5,0,0,1,0,0,0,0,0}

{2,6,0,1,0,0,0,0,0,0}

{0,9,0,0,0,0,0,0,0,0}

630

630

2520

1260

3780

1680

2520

504

256

1

Sum of microstates:

21722

11

Systems with large numbers of molecules – Example 2

Macrostate:

j iji aNW !! ,

N = 9 and Etot = 9Δ; E = Δ determine all possible distributions

As N in the system increases, a single

distribution has an overwhelming role in

determining the properties of the

macrostate

Review of quantum mechanical particle in a box (ideal gas) system:

Spacing of energy levels and degeneracy

Potential energy for particle in a three-dimensional cubic box model:

zyxzyxzyx nnnnnnnnnzyxUzyxm

,,,,,,2

2

2

2

2

22

),,(2

elsewhere

LzLyLxforzyxU

0;0;00),,(

)/sin()/sin()/sin(,,,, LzjLyjLxjA zyxzjyjxjzjyjxj

222

3/2

2222

2

2

,,88

),1( zyxzyxzjyjxjjjj

mV

hjjj

mL

hV

12

Wave function of particle in a three-dimensional cubic box:

Quantized energy levels of particle in a three-dimensional

cubic box:

L

L 0

Time-independent Schrödinger equation

Degeneracy of microstates of molecules in a 3-dimensional cubic box

6

82,1,1

1,2,1

1,1,2

3/2

2

mV

h

)/2sin()/sin()/sin(2,1,12,1,1 LzLyLxA

)/sin()/2sin()/sin(1,2,11,2,1 LzLyLxA

)/sin()/sin()/2sin(1,1,21,1,2 LzLyLxA 222

3/2

2

1,1,2 1128

),1( mV

hV

Two non-interacting particles in a three-dimensional cubic box gives higher

degeneracy:

22

22

22

21

21

213/2

2

2,18

),2( zyxzyxjj jjjjjjmV

hVE

Degeneracy: Three microstates have the same energy

13

(2,1,1,1,1,1) (1,2,1,1,1,1) (1,1,2,1,1,1)

(1,1,1,2,1,1) (1,1,1,1,2,1) (1,1,1,1,1,2)

Consider the following three states: They all have an energy of:

Six degenerate microstates:

Distribution of non-interacting molecules in states (energy levels)

Take an isolated ideal gas system of N molecules

with the total energy E

Each molecule is in a particle-in-box quantum

state ji = {jxi, jyi, jzi} characterized by energy εi

In the N-molecule system:

a1 molecules are in state 1,

a2 molecules are in state 2,

…,

ai of the molecules may be in state i,

...

ε6

ε5

ε1

ε4 ε3 ε2

14

N

zyx jjjmV

hVNE

1

2,

2,

2,3/2

2

}{8

),(

j

Total energy of the N-molecule system:

L

L

0

A possible distribution of molecules among

states: a6 = 1

a5 = 2

a1= 4

a4 = 5 a3 = 1 a2 = 2

Quantum state energy ε1 ε2 ε3 ε4 ε5 …

Occupancy of states a1 a2 a3 a4 a5 …

jjj

jj

EVa

Na

)(

The collection of molecules (the “ensemble”) are binned to place each molecule into a

box corresponding to its energy state

The distribution of molecules among states

must satisfy the constraints: 15

Distribution of molecules in non-interacting states

N molecules

E total energy

ε1 ε2 ε3 ε4 ε5

ε6 ∙∙∙

εj

εj< εj+1

Make bins and label each with a molecule energy

εj+1

a1

j a

N

aaa

NW

!

!

!!!

!)(

j321 a

EVa

Na

j jj

j j

)(

• The number of ways W(a) molecules can be distributed among energy levels

increases greatly for large numbers of molecules and high energies

• The most probable distribution has the maximum number of ways it can be

achieved, W*(a) and becomes dominant as the number of molecules increases

How do we find the most probable distribution?

Maximize W(a) with respect to the occupancies, ai, subject to the constraints on the

system

16

Finding the most probable distribution for systems with large numbers

of molecules in different states

dx

xdf

xfdx

xfd )(

)(

1)(ln 0

)(

dx

xdf0

)(ln

dx

xfd

A function and its logarithm have the same maxima and minima

So we maximize ln[W(a)] instead of W(a). Why?

Some math

NNNN ln!ln

j jj aaNN

aaaaaaNNN

aaNW

lnln

lnlnln

...)!ln()!ln()!ln()(ln

222111

21

a

j a

N

aaa

NW

!

!

!!!

!)(

j321 a

17

The logarithm of W(a) can be evaluated using Stirling’s approximation for

large integer N,

Finding the distribution with the maximum degeneracy (most probable

distribution)

Giving:

0lnln

j jj

i

aaNNa

Can we just calculate the derivative of lnW(a) with respect to the occupancy of a

specific state i, ai to determine the most probably distribution?

EVa

Na

j jj

j j

)(

No! There are constraints

on the ai’s and they are not

all independent

0)(lnln

EVaNaaaNN

aj jjj jj jj

i

)()1(* Vii eea

Constraints added with Lagrange

multipliers α and β (to be later

eliminated, see below)

18

Occupancy of state i in the most probable distribution

0)(1ln Va ii

iVi

iVi

i ie

NeeeaN

)(

)1()()1(*

Applying the constraint on the occupancies allows elimination of α Lagrange

multiplier:

Add the equations of the constraints to the function and then set the derivative to equal to

0 to find the maximum of the function

Finding the distribution with the maximum degeneracy (most probable distribution):

Method of Lagrange underdetermined multipliers

),(

)(

)(

)(**

Vq

e

e

e

N

aP

Vi

jVj

Vii

i

j jjPMM

Partition function

(sum over states)

Distribution with largest probability: 19

• Probability of state i being occupied in the N molecule system in the most

probable distribution

Knowing probabilities of the different states, averages M of microscopic

quantities Mj can be determined for the most probable distribution

• The Boltzmann factor makes an appearance in the context of probability for

system occupying state j.

iVi

jVj

jj jj

e

eVVNP

)(

)()(

),(

• Energy – from statistical mechanics

• We use the thermodynamic relation for pressure to obtain a microscopic relation

VpSdTpdVd

iVi

jVjj

j jje

eV

V

VNpPp)(

)()(

),(

VNNT T

p

Tp

V,,

1

1

kT

1

Thermodynamic relation:

),(

/)(

/)(

/)(*

TVq

e

e

e

N

aP

kTVi

jkTVj

kTVii

i

• The probability is an exponential decaying function of energy (Boltzmann

factor!)

• The derivation did not depend on the ideal gas law! 20

Determining β by comparing macroscopic and microscopic relations

VNN

pp

V,,

Statistical mechanics relation:

(See Appendix 5.1 of Extended Lecture Notes for proof)

Comparing the thermodynamic relation with the statistical

mechanical relation gives

Probability distribution for occupancy of states from statistical mechanics!

Partition function q(V, T) expressed as sum over energy levels

E0 = 0

E1 = E

E2 = 2E

E0 = 0

E1 = E

E2 = 2E

kTkTkT eeeTq /2//0 423),1(

E3 =2Δ

E2 =1Δ

E1 = 0

kTkTkTkT

kTkTkTkTkT

eeee

eeeeeTq

/2/2/2/2

///0/0/0),1(

jkTjE

eTq/)1(

),1(

E

kTVNEeETq /),()(),1(

21

Gather similar terms

Sum over one-molecule states

Sum over one-molecule energy levels

Degeneracy of the energy state

Assume we have a system with nine states

Expressing the partition function as a sum over energy levels is clarifies the

interpretation / physical content of this function.

Physical interpretation of the partition function for non-interacting molecules

kTkTkT eeeq /2//0 423

E0 = 0

E1 = E

E2 = 2E

E0 = 0

E1 = E

E2 = 2EE3 =2Δ

E2 =1Δ

E1 = 0

T (in Δ/k units) 0 0.5 1.0 1.5 2.0 5.0 10.0 → ∞

q(T) 3 3.34 4.28 5.08 5.68 7.32 8.08 → 9

P1 = 3e-0/kT/q 1 0.90 0.70 0.59 0.53 0.41 0.37 → 0.33

P2 = 2e-Δ/kT/q 0 0.08 0.17 0.20 0.21 0.22 0.22 → 0.22

P3 = 4e-2Δ/kT/q 0 0.02 0.13 0.21 0.26 0.37 0.41 → 0.44 22

Assume we have a system

with nine states

The partition

function is:

Δ

Δ

j

kTVjeTVq/)(

),(

Changes in partition function and probability of energy levels with temperature

Physical interpretation of the partition function

E0 = 0

E1 = E

E2 = 2E

E0 = 0

E1 = E

E2 = 2EE3 = 2Δ

E2 = 1Δ

E1 = 0

23

Variation of partition

function with

temperature

Variation of

probabilities with

temperature

P1 = 3e-0/kT/q

P2 = 2e-Δ/kT/q

P3 = 4e-2Δ/kT/q

),(

)()(

/)(

TVq

eEEP

kTVE

Probability of

occupancy of

energy level E

The partition function gives a measure of the number of states accessible to

the system at the given temperature

States of systems with interacting molecules

rrr i

j

j

j

EUm

2

2

2

The Schrödinger equation can be written for many-

particle systems with complex interactions,

The state of the compound molecular system is

represented by the index i (e.g., the molecular

orbital) and the discrete energy levels by Ei.

http://www.meta-

synthesis.com/webbook/40_polyatomics/pol

yatomics.html

E1

E2

E3

E4

E5

E6

E7

States of systems with interacting molecules

rrr i

j

j

j

EUm

2

2

2

The Schrödinger equation can be written for even the most complex many-

molecule systems with complex interactions,

The state of the compound

system is represented by the

index i and the energy by Ei,

even though in these cases,

we cannot determine these

energy states

Probabilities for systems with interacting particles: The set-up

• The system has a constant volume V with N interacting molecules;

• The N-molecule interacting system as a whole is characterized by a quantum state

i with energy Ei;

• There may be interactions between the molecules in the system, and these are all

captured in the quantum states i;

• The system has walls which allow it to exchange heat with its surroundings;

• The temperature of the system is maintained constant.

26

Infinite bath (surroundings)

at temperature T

System:

Volume V

Molecules N

Quantum states i

Heat exchange

occurs between

system and bath

Probabilities for systems with interacting particles: The “canonical” ensemble

• A large number of replicas of the system (the “ensemble”) are put in contact with

each other and placed in the infinite heat bath. Heat exchange is possible between

systems and bath;

• The system replicas have constant volume and are maintained at constant

temperature. This ensemble (collection) of systems is called the canonical ensemble;

• A more descriptive name would be the “constant temperature – constant volume”

(isothermal-isochoric) ensemble;

• After the ensemble equilibrates with the bath (environment), it is removed from

contact with the bath and placed in an isolated container;

• The replicas of the system will have energies, Ei, from among possible quantum

states

27

• The isolated ensemble is made of A copies

of the original system;

• The total energy of the ensemble of systems is E ;

• E and A are mathematical constructs and do

not have physical meaning.

28

Introducing the ensemble as a mathematical trick

. Ej

E5 E1 E3 E20 E1

…

• The systems in the ensemble have states Ei

consistent with nature of the nature of the

interactions;

• The number of systems in each state i are

given as Ai;

• The distribution of systems in the ensemble

among states must satisfy:

E

A

j jj

j j

EA

A

29

. Ek

E5 E1 E3 E20 E1

…

Molecule

states εi

Total system

states Ei

Analogy between non-interacting system and ensemble

Nj jj

j j

Ea

Na

E

A

j jj

j j

EA

A

Energy can be exchanged

between molecules of the gas

The total gas system energy is distributed

among the states of the molecules:

The ensemble energy is distributed

among the states of the systems:

Energy can be exchanged between

systems in the ensemble

N-molecule quantum state: E1 E2 E3 E4 E5 …

Occupancy of each state: A1 A2 A3 A4 A5 …

j AAAAW

!

!

!!!

!)(

j321

AA

A

30

Ways of assigning systems in the ensemble to different possible quantum states

Constraints

Finding the most probable distribution of systems in the ensemble

...)lnln(ln)(ln 2211 AAAAW A AAFor a large ensemble (using

Stirling’s approximation) :

E

A

j jj

j j

EA

A

0)(ln

EA j

jj

jj

i

EAAWA

A

Maximize lnW(A) with respect to the occupancy of each state Ai using Lagrange

undetermined multipliers:

How do we calculate the occupancies Ai* corresponding to the maximum

W(A) and to determine the most probable distribution?

iEi eeA

)1(*

iiEi

iEi i

eeeeA

A

A )1()1(*

),,(

*

VNQ

e

e

eAP

iE

jjE

iEi

i

A

“Canonical” ensemble

(isothermal-isochoric ensemble)

partition function for system states

31

Similar to a non-interacting system, in the most probable

distribution, the number of systems in the ensemble in

state i are:

j jjPMMAverages of quantities can be calculated from knowledge

of the probabilities

Probability of observing a particular state i in most probable

distribution

Using the first constraint: Ai iA*we eliminate the undetermined multiplier α,

Most probable distribution of systems in the ensemble

iVNE

jVNE

j

j jji

j

e

eVNEVNEPVNE

),(

),(),(

),(),,(

iVNE

jVNEj

j jji

j

e

eV

E

VNpPVNp),(

),(

),(),,(

Eliminating the β undetermined multiplier by comparing

statistical mechanics prediction with thermodynamics relations

Statistical mechanics relation for energy:

Statistical mechanics relation for pressure

32

VNN

pp

V

E

,,

After some steps (see appendix of

Extended Lecture Notes)

VNNT T

p

Tp

V

E

,, 1

1

kT

1

Thermodynamic relation:

33

Comparing statistical mechanics and thermodynamics relations

VNN

pp

V

E

,,

jkTVNE jeTVNQ

/),(),,(

Energy of N-particle interacting system

Statistical mechanics relation:

Canonical ensemble partition function:

Relating thermodynamic quantities from the canonical ensemble partition function:

j

kTVNjEeTVNQ

/),(ln),,(ln

dx

xdf

xfdx

xfd )(

)(

1)(lnRecall:

Instead of Q, we deal with the logarithm

34

The partition function and thermodynamic quantities

VNT

QkTE

,

2 ln

TNV

QkTp

,

ln

Note that this relation is the statistical mechanical form for the equation of state!

We can recognize the average energy in this relation:

ikTVNiE

jkTVNjE

j

VN e

eE

kTT

TVNQ

/),(

/),(

2,

1),,(ln

The temperature derivative of lnQ gives:

Similarly, the pressure derivative of lnQ gives:

Ways to change the energy of a N-molecule

ideal gas system

N

zyx jjjmV

hVNE

1

2,

2,

2,3/2

2

}{8

),(

j

• Change the distribution of

molecules among the available

states

• Change the volume V (which

changes energy levels) without

changing the distribution

• Change the number of

molecules in the system without

changing the levels

ℎ2

8𝑚𝑉2/3 𝑗𝜈,𝑥

2 + 𝑗𝜈,𝑦2 + 𝑗𝜈,𝑧

2

𝑁

𝜈=1

ℎ2

8𝑚𝑉2/3 𝑗𝜈,𝑥

2 + 𝑗𝜈,𝑦2 + 𝑗𝜈,𝑧

2

𝑁

𝜈=1

ℎ2

8𝑚𝑉2/3 𝑗𝜈,𝑥

2 + 𝑗𝜈,𝑦2 + 𝑗𝜈,𝑧

2

𝑁

𝜈=1

The energy is a

function of the

quantum state,

the volume,

and the number

of ideal gas

molecules in

the system;

E{j}(N,V)

Statistical mechanical interpretation of work and heat

36

qwdE Thermodynamics:

j jjPEE

j jjj jj dPEdEPEd

Statistical mechanics:

Energy can change by performing

work or transfer of heat:

The ensemble average energy can varied by changing the system energy levels

or the distribution of systems among the energy levels

• The relation for energy of the particle in a box shows that changing the

energy without change of distributions is done by changing the volume of

the system:

• Therefore j jjdEP

j jjdPE

corresponds to change of energy due to work

must correspond to transfer of heat and

N

zyx jjjmV

hVNE

1

2,

2,

2,3/2

2

}{8

),(

j

Statistical mechanics and thermodynamic functions

QkT

QkTS

VN

lnln

,

Can be shown that:

(see Extended Lecture Notes)

37

From the thermodynamic relation:

),,(ln TVNQkTA

TSEA

Substituting the relation for energy

and entropy gives:

The canonical partition function is the “characteristic” function for the Helmholtz

free energy

),,(),,( TVNQTVNA

SdTpdVdA

We can use the second law of thermodynamics expression:

to relate thermodynamic quantities to the canonical ensemble partition function.

• Ideal gas with N molecules

!

2),,()(

8),1(

2/3

2

222

3/2

2

}{N

V

h

mkTTVNQjjj

mV

hVE

NN

zyx

j

Statistical mechanics of systems under constant pressure

38

• How do we treat systems at constant pressure with statistical mechanics?

• The system has constant pressure p, constant temperature T, with N interacting

molecules;

• The system has flexible walls which allow it to change volume to adjust the

pressure;

• The walls are thermally conducting, allowing exchange of heat energy with its

surroundings

• At each volume Vk, the N-molecule system as a whole is characterized by a

quantum state Ei

System:

-N molecules

-Pressure P

-Quantum states i

(for each volume)

Set-up of the isothermal – isobaric ensemble

39

• Replicas of the system are placed in contact with each other in a large thermal

bath / barostat apparatus;

• The walls of the systems are taken to be flexible allowing the volume of each

system to change within the ensemble;

• After equilibration, the replicas are placed in an isolated container with fixed

volume. This is called the isothermal-isobaric ensemble.

Systems are enclosed in a rigid container of volume V

The isothermal-isobaric ensemble

40

)(

)(

)(

22

11

VEV

VEV

VEV

i

i

i

• In the isothermal-isochoric ensemble different system volumes may be encountered.

• Each volume Vk has its corresponding energy states Ei(Vk)

N, P

E2 (V1) N, P

E3 (V2)

N, P

Ei (Vℓ)

kV jkkj

kV jkjkj

kV jkj

VVA

VEVA

VA

V

E

A

)(

)()(

)(

Constraints on

the ensemble

The continuous volume variation is

represented by the summation to simplify notation

The E, A and V are mathematical constructs

How does the system volume enter into the quantum states?

)(8

),1( 222

3/2

2

}{ zyx

k

k jjjmV

hVE j

Particle in a box quantum states depend on the volume of the box, V

41

)(12

)()( 222

3/5

2}{

}{ zyx jjjmV

h

V

VEVp

jj

V1 V2

)(8

)( 222

3/21

2

1}{ zyx jjjmV

hVE j

Pressure for a particle in a

box state {j}:

)(8

)( 222

3/22

2

2}{ zyx jjjmV

hVE j

Most probable distribution in the isothermal-isobaric ensemble

kV jkj VA

W)!(

!})({

AA

VViEi eeeVA

)(* )(

),,(

)()(

)(

)(

)(*

N

ee

ee

eeVAVP

VViE

kV j

kVkVjE

VViEi

i

A

0)()()()(})({ln)(

VEA

kVk

jkj

kVkj

jkj

kV jkj

i

VVAVEVAVAWVA

A

Isothermal – isobaric

partition function

42

Find the most probable distribution, subject to constraints of the system,

The probability includes an energy (Boltzmann) factor and a volume factor!

Number of systems in ensemble with volume Vℓ in state i in most probable distribution

The ways the members of the ensemble can be distributed

among systems of volume Vk and states j

Constraints on the

distribution

kV jkkj

kV jkjkj

kV jkj

VVA

VEVA

VA

V

E

A

)(

)()(

)(

The probability

),,(ln pTNkTG

pNTkTkS

,

lnln

VkTpV

V EkTpVkTE

eEVNQ

eeEVNpTN

/

//

),,(

),,(),,(

Isothermal-isobaric partition function

TNp

kTV

,

ln

TpNkT

,

ln

We construct the statistical mechanical equivalent. Comparing the two allows us to

identify the Lagrangian multipliers β and δ (see canonical ensemble derivation)

Characteristic thermodynamics function:

43

Sum over energy levels

pNNTT

V

TV

p

H

,,1

1

Using the thermodynamic relation as a guide,

kT

p

kT ;

1

Statistical mechanics of open systems: Grand canonical ensemble

The system set-up:

• Constant volume

• Can exchange molecules with

environment

• Can exchange heat with the

environment

44

Semi-

permeable

membrane

• A system of constant volume which can exchange heat and molecules with the

surroundings;

• For each number of molecules, the system as a whole is characterized by a

quantum state Ei(N)

E1(1), E2(1), E3(1), …, Ei(1), …

E1(2), E2(2), E3(2), …, Ei(2), …

E1(3), E2(3), E3(3), …, Ei(3), …

…

E1(N), E2(N), E3(N), …, Ei(N), …

Thermostat at T

Molecule reservoir at μ

Volume V

Molecules μ

Quantum state i

Grand canonical ensemble – Open systems

45

• The N-molecule system as a whole is characterized by a quantum state Ei

• Replicas of the system are placed in contact with each other in a large

thermal bath / molecule bath

• After equilibration, the replicas which now have the same μ, T, and V are

placed in an isolated container

• The grand canonical ensemble can be called the isopotential-isothermal-

isochoric ensemble

Isolated ensemble at T and μ

Total energy in the ensemble E,

Total number of systems in the ensemble A ,

Total number of molecules in the ensemble N

The energy states of ideal gas systems with different numbers of molecules

kN jkkj

kN jkjkj

kN jkj

NNA

NENA

NA

N

E

A

)(

)()(

)(

)(8

)1( 222

3/2

2

}{ zyx jjjmV

hE j

one-particle

quantum states

)(8

)2( 22

22

22

21

21

213/2

2

}{ zyxzyx jjjjjjmV

hE j

two-particle

quantum states

46

Find the most probable distribution for the states in the grand-canonical ensemble ,

subject to constraints of the system,

μ,V,T

E,j(N1)

μ,V,T

Ek(N7)

μ,V,T

Ei(Nℓ)

Constraints on the system:

NNiEi eeeNA

)()1(* )(

),,(

)()(

)(

)(

)(*

V

ee

ee

eeNANP

NNiE

kN j

kNkNjE

NNiEi

i

A

0)()()()(})({ln)(

NEA

kNk

jkj

kNkj

jkj

kN jkj

i

NNANENANAWNA

A

47

Most probable distribution for the Grand Canonical Ensemble

Number of systems in ensemble with N molecules in state j in most probable distribution

Probability includes energy (Boltzmann) factor and molecule number factor.

Grand canonical

partition function

Greek letter xi

The ways the ensemble can be distributed

among systems with N molecule and state i

Constraints on the

distribution

Find the most probable distribution, subject to constraints of the system,

kN jkj NA

W)!(

!})({

AA

kN jkkj

kN jkjkj

kN jkj

NNA

NENA

NA

N

E

A

)(

)()(

)(

),,(ln TVkTpV

VTkTkS

,

lnln

pV is the characteristic function of the grand canonical ensemble

kNkTkNkTVkNE

E k

kNkTkN

k

eeVNE

eTVNQTV

//),(

/

),,(

),,(),,(

Grand canonical partition function

VkT

VkTp

T

lnln

, TVkT

N

,

ln

TSpVENG

48

We construct the statistical mechanical equivalent of a thermodynamic relation

and compare the two to identify the Lagrange multipliers β and γ

Microcanonical (isoenergy, isobaric) ensemble

NVE

NVE NVE NVE NVE NVE

Each system is confined to

• a constant volume

• a constant energy

A j jA

j jAW

!

!})({

AA

0})({ln

Aj j

i

AAWA

),,(

1*

EVN

AP i

i

A ),,(ln EVNkS 49

50

Partition functions for non-interacting systems

)()2()1()(, NNE itot

jkTNkTjE

eeTVNQ

,,,

/)]()2()1([/),,(

How does the partition function behave for systems with non-interacting molecules:

We replace this expression for energy in the canonical ensemble partition function:

Quantum states for individual molecules

NkTNkTkT

kTN

qqqeee

eTVNQ

21/)(/)2(/)1(

,,,

/)]()2()1([),,(

Sum over all states of N-

molecule system

Sum over all states of

individual molecules

The N-molecule partition function is decomposed to a product of 1-molecule

partition functions

51

Partition functions for non-interacting systems

,,,

/],,,,[/),(

kji

kTelecvibkrotjtransikTeeTVq

electroniclvibrationarotationalnaltranslatioi

We can repeat the process for the internal degrees of freedom of the molecule:

The 1-molecule canonical ensemble partition function decomposes to partition

functions for different degrees of freedom:

elecvibrottrans

kTeleck

kTvibkj

kTrotji

kTtransi

qqqq

eeeeTVq

/,/,/,/,),(

If we use quantum mechanical expressions for the translational, rotational,

vibrational, and electronic energies, we can determine the partition function

for each of these degrees of freedom.

52

Phase space probability distribution of an ideal gas molecule

The phase space for an ideal gas at constant energy {r1,p1,r2,p2, …, rN, pN}:

1) Molecules are uniformly distributed in the volume of the system and their

momentum components are distributed according to the Maxwell (Gaussian)

distribution;

2) Many possible distributions of positions and momenta of

the molecules are possible which would still give the

same total energy, EN, for the system.

V

dxdydz

V

ddP

rrr)(1

Spatial distribution of molecules: Probability of finding

molecule 1 at r within the volume element, dr = dxdydz:

Momentum distribution of the molecules: Probability of molecule 1 having

momentum p within the momentum element dp = dpxdpydpz:

zyxmkTzpypxp

zyx dpdpdpekTm

dpppP2)222(

2/3

12

1),,(

p

53

Classical states and phase space trajectories

p / kg·m·s-1

(x-x0) = ξ

U =

k(x

-x0)2

/2

F =

-k(

x-x 0

)

• In classical mechanics states with a particular energy are represented by a phase

space trajectory;

• Trajectories with all possible total energy values can be observed. The energy of

trajectories can vary continuously.

• For N atom classical systems, the state is represented by a 3N-dimensional point

in phase space {r1, p1, r2, p2, …, rN, pN}

• Each energy values defines a trajectory in phase space

ξ / m

54

Switching over from the classical to the quantum description of states

For our purposes, the description of the state of an N-atom system in terms of a

discrete set of states represented by a single quantum index i is much simpler than in

terms of a continuous variation of the state {r1, p1, r2, p2, …, rN, pN} in phase space.

)()()(1

iNi

i Pff

NNNNN ddddPff prprprprprpr r

r 1111

max,1min,1

}),,,,({}),({}),({

We aim to derive the expression for the probability of observing a system in

different states and use that in calculating different thermodynamic and mechanical

properties of the system

It is easier to do calculations of averages with discrete states

than with continuous states

jkTjE

eTVNQ/

),,(

Classical limit to the canonical partition function

• Sum over N particle quantum states

NNkTNqNpqpH

Ndqdpdqdpe

hNQ 3311

),,...,1,1(

3!

1

Molecules are indistinguishable

h: volume of phase space element from quantum mechanics

• Integral over phase space elements Semi-classical equivalent

dpdq has units of energy·time (h Planck’s constant)

55

state quantumdpdq

),,,...,,,(),...,( 311131 NNNN zyxzyxUppKH

V NkTNqqU

p NmkT

Nppp

Ndqdqedpdpe

hNQ 31

)3,...,1(1

2/)23

22

21

(

3!

1

Hamiltonian of the system represents the energy in a canonical ensemble

NNkT

Nq

NpqpH

Ndqdpdqdpe

hNQ 3311

)3

,3

,...,1

,1

(

3!

1

N

N

V N

kTN

qqUN

Zh

mkT

Ndqdqe

h

mkT

NQ 3

2/3

231

)3

,...,1

(2/3

2

2

!

12

!

1

Momentum integrals are Gaussian!

Configurational integral 56

Classical partition function

We can separate the momentum and position integrals

Evaluating the momentum position integrals

NNkTqpH

kTN

pN

qpqH

NNdpdqdpdqe

epqpqP

3311),(

)3

,3

,,1

,1

(

3311),,,,(

• Classically momenta always follow the Maxwell distribution!!!

Classical expression for probability in the canonical ensemble

),,,...,,,()( 1111

2,

2,

2,2

1NNN

N

zyxmzyxzyxUpppH

NkTqU

kTN

qqU

N

mkTN

pp

NkTqU

kTN

qqU

NmkT

Npp

mkTN

pp

NNkTqpH

kTN

pN

qpqH

dqdqe

e

mkT

e

dqdqe

e

dpdpe

e

dpdqdpdqe

e

31)(

)3

,,1

(

2/3

2/)23

21

(

31)(

)3

,,1

(

312/)2

321

(

2/)23

21

(

3311),(

)3

,3

,,1

,1

(

)2(

57

• This is true for gases, liquids, and solids!

NNkTqpHkTPV

kTPVkTpqH

NNdpdqdpdqedVe

eeVpqpqP

3311),(

0

),(

3311 ),,,,,(

Classical expression for probability in isothermal - isobaric ensembles

58

NN

kTqpHkTPV

NdpdqdpdqedVe

hNVTPN 3311

),(03

0 !

1,,

NkTqUkTPV

kTPVkTNqqU

N

mkTN

pp

NNdqdqedVe

ee

mkT

eVpqpqP

31)(

0

)3,,1(

2/3

2/)23

21

(

3311)2(

),,,,,(

• Momenta always follow the Maxwell distribution!

Probability of observing system volume V and q1, p1, …, q3N, p3N and :

Classical expression for probability in the grand canonical ensemble

59

NNkTqpH

N

kTN

kTNkTpqH

NN

dpdqdpdqee

eeNpqpqP

3311),(

0

),(

3311 ),,,,,(

NN

kTqpH

NN

kTN dpdqdpdqehN

eTV 3311),(

30 !

1,,

NkTqU

N

kTN

kTNkTNqqU

N

mkTN

pp

NN

dqdqee

ee

mkT

eNpqpqP

31)(

0

)3,,1(

2/3

2/)23

21

(

3311)2(

),,,,,(

• Momenta always follow the Maxwell distribution!

Probability of observing N molecules with q1, p1, …, qN, pN :

q p

How do we link ensemble relations to MD and MC simulations?

pqddpqPpqApqA ),(),(),(

pq

pq

dde

ddeqpHE

kTpqH

kTpqH

),(

),(),(

60

Note: time does not enter these equations!

Phase space surface: Each point

(q, p) is weighted according to its

energy H(q, p)

pqpq

dde

eP

kTqpH

kTpqH

),(

),(

),(

Why does a system in a MD

simulation seek minimum

energy states?

timeN

kTqU

kTNqqU

N

mkTN

pp

pqPdqdqe

e

mkT

epqP ),(

)2(),(

31)(

)3,,1(

2/3

2/)23

21

(

Link to MD and Monte Carlo Simulations

timepqMddpqPpqMpqM ),(),(),(),( pq

NNkT

Np

NqpqH

NNkT

Np

NqpqH

dpdqdpdqe

dpdqdpdqeqpHE

3311)

3,

3,,

1,

1(

3311)

3,

3,,

1,

1(

),(

If we can guarantee that the MD or MC sampling follows the canonical

ensemble, all thermodynamic averages can be calculated by the

simulation 61

Probability of observing q1, p1, …, q3N, p3N is determined by the ergodic property of

MD:

Average value of a mechanical quantity is determined by ergodicity:

Monte Carlo simulations sample the phase space probability distribution directly

and do not depend on ergodicity