ELEMENTS OF STATISTICAL THERMODYNAMICS AND QUANTUM THEORY STATISTICAL MECHANICS OF INDEPENDENT PARTICLES ▪ Macrostates versus Microstates ▪ Phase Space.

ELEMENTS OF STATISTICAL

THERMODYNAMICS AND QUANTUM THEORY

STATISTICAL MECHANICS OF INDEPENDENT PARTICLES ▪ Macrostates versus Microstates

▪ Phase Space

▪ Quantum Mechanics Considerations

▪ Equilibrium Distributions for Different Statistics THERMODYNAMIC RELATIONS ▪ Heat, Work and Entropy

▪ Lagrangian Multipliers

▪ Entropy at Absolute Zero Temperature

▪ Macroscopic Properties in Terms of the Partition Function

IDEAL MOLECULAR GASES ▪ Monatomic Ideal Gases

▪ Maxwell’s Velocity Distribution

▪ Diatomic and Polyatomic Ideal gases STATISTICAL ENSEMBLES AND FLUCTUATIONS BASIC QUANTUM MECHANICS ▪ Schrödinger Equation

▪ A Particle in a Potential Well or a Box

▪ Atomic Emission and Bohr Radius

▪ Harmonic Oscillator EMISSION AND ABSORPTION OF PHOTONS BY MOLECULES OR ATOMS ENERGY, MASS AND MOMENTUM IN TERMS OF RELATIVITY

computer simulation technique where the time evolution of a set of interacting atoms (N) is followed by integrating their equations of motion

Molecular dynamics simulation

statistical mechanics method for 6N-dimensional phase space

link between the microscopic behavior and static and dynamic properties

averaging → thermodynamic properties transport properties

2

2

( )( , , )

Ni i

i j i j ij i

d r rF r r t m

dt r

3N positions

and 3N momenta

( )ir

( )iv

Temperature:

Internal Energy:

Pressure:

Thermodynamic properties

2

1

1

3

N

i iB

T m vNk

3(r )

2 B iji j i

E Nk T

1r

3 rB iji j i ij

NP k T

V V

Soft Sphere Model

Ne, Ar, Kr, Xe : Lennard-Jones (12-6) Pair two-body potential, inert gas, Van der Waals bond

Water : ST2, SPC/E, TIP4P, CC Pair two-body potential, polarization

Si, C : SW, Tersoff, Simplified Brenner three-body potential, covalent bond

Metals : EAM, FS, SC two-body potential, embeded atom, electron cloud

Intermolecular potential models

• Exponent for Repulsion• Little Theoretical Basis• Due to Mathematical Convenience

• Exponent for Cohesion (Attraction) • van der Waals Force• Keesom Force + Debye Force + London Dispersion Force• Good for Closed Shell Systems (Ar, Kr)

12 6

( ) 4ijij ij

rr r

Lennard-Jones Potential

Statistical methods

• Statistical mechanics

Equilibrium distribution of certain types of particles (molecules, electrons, photons, phonons) in the velocity space

• Kinetic theory

Nonequilibrium processes, microscopic description of transport phenomena

Classical vs. statistical thermodynamics

• classical macroscopic

• statisticalmicroscopic

matter is continuous matter is particulate

phenomenological approach

sum of molecules & atoms

equilibrium states equilibrium distribution

Statistical Mechanics of Independent Particles

independent particles

their energies: independent of each other

total energy: sum of the energies of individual particles

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0, N0

1, N1

2, N2

i, Ni

0i i

iN U

0,i

iN N

constraints

V, N, U

Macrostates versus Microstates

volume, V

number of particles, N

internal energy, U

specified by the number of particles in each of the energy levels of the system

• microstate: specified by the number of particles in each energy state• degeneracy: the number of quantum states for a given energy level.

• macrostate: corresponds with a given set of numerical values of N1, N2, …, Ni, …, and thus satisfies the two constraints

• thermodynamic probability

Number of microstates for each macrostate: number of ways in which we can choose Ni’s from N particles

→ quantum states

Degeneracy

En

erg

y

level

excited states

ground state

macrostate

microstate :

1 2 3 1 3 2: ( , , ) ( , , )jN N N N

(1), (2 1 0), (0 1 0 1 0)

Example : Consider 3 particles, labeled A, B,

and C

Equilibrium state

constraints

Phase Space

elemental volume dV in Cartesian position space

dx

dy dz

r

x

y

z

dV dxdydz

a six-dimensional space formed by three coordinates for the position and three coordinates of the momentum or velocity

r

p mv

v

d

ddAn

rsin

elemental volume dV in spherical position space

r

ndV dA dr

ndA sinr d rd

2 sinr d d

2 sindV r d d dr

dvx

dvy dvz

v

vx

vy

vz

element in Cartesian velocity (or momentum) space d

x y zd dv dv dv

for momentum

x y zd dp dp dp

3x y zm dv dv dv

3m d

element in spherical velocity (or momentum) space d

d

ddn

vsin

v

nd d dv

nd sinv d vd

2 sinv d d

2 sind v d d dv

2 2 2x y zv v v v

Phase space trajectories

concept: a trajectory to include not only positions but also particle momenta plotting the positions and momenta of N particles in a 6N-dimensional hyperspace

( )ir t

( )ip t

phase space: 3N-dimensional configuration space and 3N-dimensional momentum space

At one instant, the positions and momenta of the entire N-particle system are presented by one point in this space.

Ex) One-Dimensional Harmonic Oscillator: isolated system

r

r0

x

displacement :

0x r r

system potential energy

21( )

2u x kx

In an isolated system, ( )

( )du x

F x p mx kxdx

To obtain

( ), ( )x t p t with initial conditions

(0), (0)x p

0

-2

-4

-6

2

4

6

0 1 2

0

-2

-4

-6

2

4

6

0 1 2

time

time

posit

ion

x(

t)m

om

en

tum

p(t

)

To determine the phase-space trajectory

constkE E U

2 21 1

2 2E p kx

m : ellipse

0

-2

-4

-6

2

4

6

0 42

position x(t)

mom

en

tum

p(t

)

6-6 -4 -2

Quantum Mechanics Considerations• energy of a photon: h

h: Planck’s constant,

346.626 10 J sh

c • speed of light:

0 0, c

cn n

• rest energy of a particle: 20E mc

• photon momentum: h h

pc

• de Broglie wavelength and frequency

2

DB DB, h h mc

p mv h

for a particle moving with velocity v << c

• Heisenberg uncertainty principle

The position and momentum of a given particle cannot be measured simultaneously with arbitrary precision.

4x

hx p

• Pauli exclusion principle

In quantum theory, independent particles of the same type are indistinguishable.

For certain particles, such as electrons, each quantum state cannot be occupied by more than one particle.

Goal : Find the occupation in each energy level when the thermodynamics probability is maximum (equilibrium point).

Model Particle Properties Case

Maxwell-Boltzmann

distinguishableunlimited particles per quantum state

ideal gas molecules

Bose-Einstein

indistinguishableunlimited particles per quantum state

photon, phonon

Fermi-Dirac

indistinguishable, identicalone particle per quantum state (Pauli exclusion principle)

electron, protons

•MB can be considered as the limiting case of BE or FD

model.

Equilibrium Distributions for Different Statistics

There are many quantum states corresponding to the same energy levels and that the degeneracy of each state level is much larger than the number of particles which would be found in any one level at any time.The specification, at any one moment, that there areN0 particles in energy level 0 with degeneracy g0

N1 particles in energy level 1 with degeneracy g1

Ni particles in energy level i with degeneracy gi

.

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.

.in a container of volume V when the gas has a total number of particles N and an energy U is a description of a macrostate of the gas.

Any one particle would have the same gi choices in occupying gi different quantum states. A second particle would have the same gi choices, and so on.

Ex) Consider the Ni indistinguishable particles in any of the gi quantum states associated with the energy level i.

Thus, total number of ways in which Ni distinguishable particles could be distributed among gi different quantum states would beiN

i i ig g g

A

A

B

B

C

C

B1 2

3 4 5 6 7 8 9 1

011

12

13

14

15

16

C

A

C

A

B

C

B

A

C

A

B

6 ways (3!) in which 3 distinguishable particles can occupy 3 given quantum states → divided by 3! for indistinguishable particles

0 !

iNi

i i

g

N

: thermodynamic probability of the particular macrostate or the number of microstate

If V, N, and U are kept constant, the equilibrium state of the gas corresponds to that macrostate in which is a maximum.

The number of ways in which this macrostate may be achieved is given by

!

iNi

i

g

N

The number of ways to distribute Ni indistinguishable particles among gi quantum states

Maxwell-Boltzmann statistics

Number of ways to arrange N distinguishable particles 1 2 2 1 !N N N N

Number of ways to put N distinguishable particles on each energy level if there is no limit for the number of particles on each energy levelConsider one of N! ways of arranging N distinguishable particles

level 0

N0 N1 N2. . .

level 1

level 2

0 1

0

! !

! ! !ii

N N

N N N

Each arrangement of N0, N1, N2, … particles in each energy

level should be considered as one case.

If degeneracy is included,

Each of Ni different particles can be put on any gi. The first particle can be placed in gi ways, the second particle can also be placed in gi ways, and so on.Thus iN

i i i ig g g g

N0 particles in energy level 0 with degeneracy g0

N1 particles in energy level 1 with degeneracy g1

Ni particles in energy level i with degeneracy gi

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0 1

0 10 1

!

! !N NN

g gN N

ways of putting Ni distinguishable particles into gi distinguishable degeneracy for each arrangement 0

!

!ii

N

N

iNig

MB0

!!

iNi

i i

gN

N

Bose-Einstein statistics

Consider placing Ni copies of the same book among gi shelves.

. . . . . . . . . . . . . . .

Number of partitions: gi - 1

The problem is now to find the ways of arranging Ni + (gi - 1) with gi - 1 indistinguishable partitions.

Consider the number of ways to arrange Ni books and gi – 1 partitions together.

Number of ways to put Ni indistinguishable particles on gi distinguishable quantum states on the ith energy level, if there is no limit on the number of particles in anyquantum states

partition

1 !

! 1 !i i

i i

N g

N g

ways of putting N indistinguishable objects into g distinguishable boxes

BE

0

1 !

! 1 !i i

i i i

N g

N g

Fermi-Dirac statistics

Number of ways to put Ni indistinguishable particles on a set of gi quantum states on the ith energy level. Each quantum state can be occupied by no more than one particle. (Ni < gi)

Number of ways to select Ni objects from a set of gi distinguishable objects

g0 g1 g2. . .

Equivalent to find

!

! !i i

ig N

i i i

gC

g N N

On the ith energy level

FD0

!

! !i

i i i i

g

g N N

Ex. 3-3

4 indistinguishable particles

2 energy levels

3 degeneraciesFind:

1) thermodynamic probability of all arrangements

a) BE b) FD

2) most probable arrangements0 1 0 14, 3N N N g g

0 0 1 1BE

0 0 0 1 1

1 ! 1 ! 1 !

! 1 ! ! 1 ! ! 1 !i i

i i i

N g N g N g

N g N g N g

0 0 01 1 1

0 1

2 ! 1 22 ! 1 2

2 ! 2 ! 2 2

N N NN N N

N N

0 0 0 01 2 5 6

2 2

N N N N

2 20 0 0 0

13 2 11 30

4N N N N

4 3 20 0 0 0

18 68 60

4N N N N

3 2 2BE0 0 0 0 0 0

0

1 14 24 2 68 2 2 8 17

4 2

dN N N N N N

dN

Most probable arrangement0 1 2N N

Degeneracy

En

erg

y

level

States of maximum thermodynamic probability

2

01

i

1 1,g N

2 2,g N

,i ig N

0 0,g N

known: ,i ig

unknown: iN

0i

i

N N

0i i

i

N U

Method of Lagrange multipliers

1 2( , , , )nf x x xFor a continuous function

A procedure for determining the maximum/minimum point in a continuous function subject to one or more constraints

1

0n

ii i

fdf dx

x

At the maximum/minimum point,

If xi’s are independent,

0, 1,2, ,i

fi n

x

If xi’s are dependent and related by m (m < n) constraints,

1 2( , , , ) 0, 1,2, ,j nx x x j m

1

0, 1,2, ,n

jj i

i i

d dx j mx

n - m independent variables

1 1

0n m

jj i

i ji i

fdx

x x

j: Lagrangian multipliers

1

0, 1,2, ,m

jj

ji i

fi n

x x

n equations, n - m independent variables, m i’s ( , , ) 8f x y z xyz

2 2 2/ / / 1 0, , , 0x a y b z c a b c constrai

nt:

Ex) Positive values of x, y, z that maximize

0, 1,2,3j

i i

fi

x x

2 2 28 2 / 0, 8 2 / 0, 8 2 / 0yz x a xz y b xy z c

2 2 24 / , 4 / , 4 /a yz x b xz y c xy z 3 2 2 264a b c xyz

2 2 22 2 2

2 2 2 2 2 2, , 16 16 16

x y zb c c a a b

2 2 2/ / / 1x a y b z c

2

2 2 2

31,

16a b c

4

3

abc 2 2 2 216,

3a b c

2 2 22 2 2, ,

3 3 3

a b cx y z

, , 3 3 3

a b cx y z

2 2 2 2 2 2 2 2 2 2 2 216 , 16 , 16b c x c a y a b z

maximum probability

0 0

ln ln ! ln ln !i i ii i

N N g N

For MB statistics with degeneracy

MB0

!!

iNi

i i

gN

N

ln 0iN

Stirling’s approximation:

ln ! lnN N N N

ln ! ln2 ln3 lnx x

x

lnx

ln1

ln2

ln3

ln4ln5ln6ln7ln8

10 2 3 4 5 6 7 8

1ln ! lnxx xdx

ln 1x x x

ln ! lnx x x x

Since x >> 1

for x >> 1,

MB distribution

0 0

ln ln ln lni i i i ii i

N N N N g N N N

0

ln ln 1ii

i i

gN N N N

N

0 0

ln 1ln ln 1i

i i ii ii i i

gd dN N dN

N N N

0

ln 0ii

i i

gdN

N

0 0

0, 0i i ii i

dN dN

0 0

, i i ii i

N N N U

0

ln 0ii i

i i

gdN

N

Negative signs are chosen because and are generally nonnegative for molecular gases.

Since dNi can be chosen arbitrary,

ln 0ii

i

g

N

ln ,ii

i

g

N i ii

i

ge e e

N

i

ii

gN

e e MB distribution

or

1

i

i

i i

ii

N g e e

N g e e

Similarly,

BE distribution1i

ii

gN

e e

FD distribution1i

ii

gN

e e

Heat and Work

Thermodynamic Relationsfrom the microscopic point of view

0i i

i

N U

0 0i i i i

i i

dU dN N d

0i i

i

Q dN

redistribution of particles among

energy levels

0i i

i

W N d

shift in the energy levels

associated with volume change

heat added

iN

i Work done

iN

i

Heat added to a system moves particles from lower to higher energy level.

Work done on the system moves energy levels to higher values.

Degeneracy

En

erg

y level

0i i

i

Q dN

0i i

i

W N d

The entropy of an isolated system increases when the system undergoes a spontaneous, irreversible process. At the conclusion of such a process, when equilibrium is reached, the entropy has the maximum value consistent with its energy and volume. The thermodynamic probability also increases and approaches a maximum as equilibrium is approached.

Two subsystems of an isolated system

SA, A SB, B

Entropy

Since entropy is an extensive variable, the total entropy of the composite system is

A BS S S

The thermodynamic probability, however, is the product,

A B

If we let

( )S f

A B A B( ) ( ) ( ) ( )S f f f f

The only function that satisfies this relation is the logarithm.

lnS k k is turn out to be the Boltzmann constant kB

Lagrange Multipliers

0

ln 0ii i

i i

gdN

N

For MB statistics

0

ln ln ,ii

i i

gd dN

N

0 0 0

ln ii i i i

i i ii

gdN dN dN

N

0 0

ln i i ii i

d dN dN

hold for all thee types

of statistics

0i i

i

Q dN

lnd dN Q

In a reversible process in a closed system 0, ln / , BdN d dS k Q TdS

1

Bk T

0 0

1ln i i i

i i B

d dN dN dN dU PdVk T

BTdS k T dN dU PdV

Helmholtz function

dA d U TS dU TdS SdT

BdA SdT PdV k T dN

, , ,B

T V T V T V

A T VS P k T

N N N

Bk T

/1 1

i i B

ik T

i

N

g e e e

i

ii

gN

e e MB distribution:

BE distribution: ( ) /

1

1i B

ik T

i

N

g e

( ) /

1

1i B

ik T

i

N

g e

FD distribution:

Comparison of the distributions

i

i

Ny

g

i

B

xk T

( ) /

1i B

ik T

i

N

g e a

1

1

0

a

forFDdistribution

forBEdistribution

forMBdistribution

As x → 0, y → ∞, and for large x, y ≈ e-x . The distribution is undefined for x < 0. Particles tend to condense in regions where i is small, that

is inthe lower energy states (Bose condensation).

• BE curve: y = 1/(ex -1)

For x = 0, y = 1/2, and for large x, y ≈ e-x . As x → -∞, y → 1. At the lower levels with ei – negative, the quantum states

are nearly uniformly populated with one particles per states.

• FD curve: y = 1/(ex +1)

• MB curve: y = 1/ex

i

i

Ny

g

i

B

xk T

MB curve lies between the BE and FD curves

only valid for y << 1 (diluted gas region)lots of states are unoccupied at high temperatures (x → 0), Ni ≈ gi (y → 1)Since gi ↑with energy↑, occupation

number↑At low temperatures, the population of

thelower states is favored.

The 3rd law of thermodynamics

at very law temperature (T → 0), = 1/ kBT → ∞

Hence N = N0 : all particles will be the lowest

energy level (ground states).

If g0 = 1, as it is the case for pure substance, then

Entropy at Absolute Zero Temperature

0 1 2

For BE statistics

( ) /

1

1i B

ik T

i

N

g e

1 ln 0 0S k T and as

0 1T i as for 0

0

( )( )

( )0 0 0

10

1i

i

i i iN g gee

N g ge

Bose-Einstein condensation

from conservation of particles,

Partition function Z

Macroscopic Properties in Terms of Partition Function

For MB statistics B/i k Ti

i

Ne e

g

B/

0 0

i k Ti i

i i

N N e g e

or

B/

0

i k Ti

i

Ne

g e

B B

B

/ /

/

1

,i i

i

k T k Ti

k Tii

i

N Ne Ne

g Zg e

partition function

B/

0

i k Ti

i

Z g e

B/i k Ti i

NN g e

Z

Internal energy U

B/ ,i k Ti i

NN g e

Z B/

0

,i k Ti

i

Z g e

B/

0 0

i k Ti ii i

i i

g NU N e

Z

B/2

0, B

i k T ii

iV N

Zg e

T k T

B

B

/2

0 B

/,,

0

ln 1i

i

k T ii

i

k TV NV Ni

i

g eZ k TZ

T Z T g e

B/

0

i k Ti

i

Ne

g e

2B

Uek T

Ne

B/

0

i k Ti

i

N Ne Z Ne

Zg e

B/

0

i k Ti i

i

e g e

2B

U

Nk T

2B

,

ln

V N

ZU Nk T

T

Entropy

B

,

ln1 ln

V N

ZZS Nk T

N T

Helmholtz free energy

B 1 lnZ

A U TS Nk TN

Chemical potential

B,

lnT V

A Zk T

N N

Gibbs free energyB ln

ZG N Nk T

N

Enthalpy

B

,

ln1

V N

ZH G TS Nk T T

T

Pressure

B, ,

ln

T N T N

ZAP Nk T

V V