statistical mechanics

How the overall behavior of a system of many particles is related to the Properties of the particles themselves.

It deals with the overall particles as whole and not with the individual particle. It tells the probability that the particle has a certain amount of energy at a certain moment.

•statistical distributions – general considerations•Maxwell-Boltzmann•Bose-Einstein•Fermi-Dirac

•Maxwell-Boltzmann statistics•Maxwell-Boltzmann distribution•energies in an ideal gas•equipartition of energy

•quantum statistics•fermions and bosons•Bose-Einstein and Fermi-Dirac distribution

•comparison of the three statistical distributions•applications

•Planck radiation law•specific heats of solids•free electrons in a metal

general considerations

central question: how does the behavior of a many-particle system depend on the properties of the single particles?

therefore: look at probabilities for particle properties

but: too many single particles to describe them one by one

example: a room filled with air•number of particles >> 1023

•mainly two kinds of particles (N2 and O2)

impossible to know all coordinates and kinetic energies

but: sm allows to calculate the probability of each particle

to e.g. have a certain amount of kinetic energy at a time t

statistical distributions

most easy setting:•system of N particles in thermal equilibrium at temperature T

question: •how is the total energy E distributed over the particles?

or:•how many particles have the energy etc.

particles interact “weakly” with one another and the container wallsthermal equilibrium but no correlation

more than one particle may have a certain energy

statistical distributions

most easy case: •thermal equilibrium •constant energy (E=const.)•constant number of particles (N=const., for “classical” particles)

n()=g()f()

number of particles with energy

number of stateswith energy

(statistical weight)

probability of occupancyof each state with energy

(distribution function) or averageNumber of particles in each state of

Energy

statistical distributionsclassical system:

d , etc. g()d

MaxwellBoltzmann

FermiDirac

Bose Einstein

•identical particles•“far” apart

(no overlap of )

distinguishable

•identical particles•integral spin

(bosons)•close together(overlapping )

indistinguishable

•identical particles•odd half-integral spin

(fermions)•close together(overlapping )

indistinguishable

statistical distributions

MaxwellBoltzmann

FermiDirac

Bose Einstein

e.g. molecules in a gas e.g. photons e.g. electrons

Maxwell-Boltzmann distribution

Maxwell-Boltzmann distribution function:

fMB()=A e-/kT

k=1.381 x 10-23 J/K=8.617 x 10-5 eV/K (Boltzmann constant)

kTeAgfgn /)()()()(

N()d is the number of particles whose energy lie between And +d

energies in an ideal gas

ideal gas: •PV=RT•N is large•translational motion, quantization is irrelevant

deAgfdgdn kT/)()())(()(

(number of molecules between and + d)

dekT

Ndn kT/

2/3)(

2)(

(energy distribution)

d toenergy

having states available ofnumber

energymolecular ofon distributi continuousFor

dg

deAgdn

eAgn

KT

KT

dpBpdp

dp

mEp

2pg

dpp top

radius from cell spherical of volume pg

states ofnumber thespace, momentumIn

2

dgm

mdmBdpBpdppg

m

mddpdmpdpmp

22

,2

,22 ,2

2

2

deCdn

deBmAdn

dBmdBmdg

KT

KT

23

23

21

23

2

22

Since each momentum magnitude p corresponds to energy

2

32

3

00

2

2

232

KT

NCKT

C

deCdnN

ABmC

KT

deKT

NdnSO KT

2

3

2,

Total number of molecules is N

aadxex ax

2

1

0

THIS IS MOLECULAR ENERGY DISTRIBUTION

KT

N

E

NKT

KTKTKT

N

deKT

NdnE KT

2

3

ENERGY, MOLECULAR AVERAGE SO

2

3

4

32

2

Tat molecule gas N ofenergy internal total

2

23

0

23

230

aadxex ax

20

23

4

3

Most probable energy

0

d

dn

Molecular speed distribution

de

KT

Ndn KT

2

3

2

mvdvdmv ,2

1 2

dvevKT

mN

dvevKT

Nm

mvdvvem

KT

Ndvvn

KTmv

KTmv

KTmv

2223

22

23

23

2

23

2

2

2

4

2

2

2

Molecular speed distribution

1. Mean velocity:

m

KT

m

KT

N

dvvvn

dvvn

dvvvnv 59.1

80

0

0

1. Root mean square velocity:

m

KTvvrms

32

1. Most probable velocity:

m

KTv

dv

vdnwhenv pp

2 ,0

fermions and bosonsdistinguishable particles (non overlapping wavefunctions)

indistinguishable particles (overlapping wavefunctions)

bosons:•integral spin (0,1,2,…)•symmetric wave function

(exchange of two bosons does not change the system)•all bosons can be in the same quantum state•Photons, Phonons•Wave function of system of boson is not affected any exchange of any pair of particle

1

1)(

/ kTBE Ae

f

FERMIONSodd half integral spin (1/2,3/2,5/2,…)• antisymmetric wave function

– (exchange of two fermions changes symmetry of the system)

• only one fermion can be in a quantum state– (exclusion principle, Pauli principle)

• probability for two particles in one state: 0!

kTkTFD

kTFD

F

FeA

ef

Aef

//)(

/

,1

1)(

1

1)(

• Consider a system of two particles , 1 and 2 one of which is in state a and the other is in state b. when all particles are distinguishable there are two possibilities for occupancy of the states as

12

21

baII

baI

•When two particles are indistinguishable we can not tell which of them is in which state, and wavefunction must be a combination of the both wavefunction I AND II.

FOR BOSONS- SYMMETRIC WAVEFUNCTION

12212

1babaB

For fermions-ANTI SYMMETRIC WAVEFUNCTION

12212

1babaB

• Now let both the particles are in same state a then both wave function will become

21 aaM

Probability density

2121 ***aaaaMM

For bosons wave function

212212

2

12212

1

aaaa

aaaaB

For fermions

012212

1 aaaaB

SO two particles can not be in the same quantum state.

Bose-Einstein / Fermi-Dirac distribution

bosons:one boson of a system in a certain state increases the probabilityof finding another boson in this state!

fermions:one fermion of a system in a certain state prevents all otherfermions from being in that state!

1

1

1

1)(

//

kTkTBE

eeAef

kTkTFD

kTkTFD

F

FeA

ef

eeAef

//)(

//

,1

1)(

1

1

1

1)(

• A describes the system and may be a function of T• >>kT fBE and fFD converge into fMB

• F is the Fermi energy

comparison of the distributions

0 1 2 3 4 50

1

2

f()

kT

1

1)(

/1 kTBE ee

f

1

1)(

/1 kTFD ee

f

kTMB eef /1)(

Consider the F-D distribution at T = 0 K

AT T=0 K

01

1f )2(

11

1f )1(

FDF

FDF

e

e

0

∞

From this we conclude that all energy states above F areEmpty (fFD) and all energy states below F are occupied(fFD = 1). So F gives the energy of the highest filled state atT = 0.

fFD

T = 0

F0

1

KT = 0.1 F

F

0.5

0.75

KT = 1.0 F

0.5

F

fFD

fFD

comparison of the distributionsMaxwell

BoltzmannFermiDirac

Bose Einstein

identicaldistinguishable

classical particles

any spin don’t overlap

e.g. gas molecules

unlimited number ofparticles per state

indistinguishableno Pauli principle

bosons

spin 0,1,2 … overlap

symmetric

e.g. cavity photons (laser)liquid He at low T

unlimited number ofparticles per state,more than in MB

aproaches MB for high T

indistinguishable,Pauli principle

fermions

spin 1/2,3/2,5/2 … overlap

antisymmetric

e.g. free electrons in metalselectrons in white dwarfs

never more than 1particle per state,

less than in MBaproaches MB for high T

free electrons in a metal

3/2

3

8 2 V( )

mg d d

h

(number of electron states)

dgNF

0

free electrons in a metal

3/2

( )/3

8 2 V 1( )

1F kT

mn d d

h e

(electron energy distribution)

V

V8

3

2

3/22 NN

m

hF

: electron density

Fermi energy

free electrons in a metal

T=0

T>>0

EF

Total internal energy

FNE 5

30

AVERAGE ELECTRON ENERGY

F5

30