Microscopic definition of entropy

log ( )BS k W E

Microscopic definition of temperature

1log

B

dW E

k T dE

This applies to an isolated system for which all the microstates are equally probable

Some key results for QPM

Distribution of microstates ( )W E

Systems in contact with a heat bathBoltzmann distribution

1exp( / )i i Bp E k T

Z

exp( / )j Bj

Z E k T Partition FunctionThis gives the normalised probability of finding the system in a state with energy Ei when in thermal equilibrium at temperature T.

1exp /j j j j B

j j

E E p E E k TZ

Calculating thermal averages

Canonical Entropy

logB i ii

S k p p

Connection to thermodynamicslogB i i

i

S k p p 1

exp( / )i i Bp E k TZ

log( )BF k T Z

Starting from the results

leads to a connection between the Helmholtz Free Energy and the partition function:

F U TS so dF dU TdS SdT

but dU TdS PdV dF PdV SdT

T V

dF dFP and S

dV dT

It follows that

We have a route for calculating these from Z

k-states in three dimensions

( , , ) sin( / )sin( / )sin( / )x y zu x y z A n x L n y L n z L

; ;yx zx y z

nn nk k k

L L L

Standing wave solutions of the form

so k = (kx,ky,kz) with each component quantised:

This gives a cubic mesh with mesh size π/L .

2 3

3

22

4( )

8

( )2

k dk Lg k dk

Vg k dk k dk

Constant energy surface

Density of states for EM modes

( ) ( )G E dE g k dk

In the case of electromagnetic radiation we have a linear “dispersion relation”, since E = ħω = ħck.

with k= E/ħc and dk= dE/ħc :

2

32( )

2

V EG E dE dE

c

Density of single particle states (3d)Suppose we are dealing with non-relativistic particles where

2 2 2 2 1/ 212/ 2 2 / ( 2 / )E k m then k mE and dk m E dE

3/ 21/ 2

2 2

2( )

4

V mG E dE E dE

Distribution of speeds for atom in a gas

32 2

2 2( ) ( )

2 2

V V mg v dv g k dk k dk v dv

32 2 21 1

2 22

1 1( ) exp( / ) ( ) exp( / )

2B B

V mp v dv mv k T g v dv v mv k T dv

Z Z

Since ħk = mv, we can define the number of states that have speeds in the range v to v+dv as:

The probability that an atom has a speed in the range v to v+dv equals the number of states in that range times the Boltzmann probability of having the corresponding energy:

3/ 2 322 21

22

3/ 2

2 212

1 2( ) exp( / )

2

2( ) exp( / )

BB

BB

V mp v dv v mv k T dv

V mk T

mp v v mv k T

k T

Substituting the result for Z we get:

Maxwell-Boltzmann distribution 3/ 2

2 212

2( ) exp( / )B

B

mp v v mv k T

k T

Velocity distribution for a nitrogen molecule at three temperatures

The Planck Distribution

exp( / ) 1B

E nk T

The mean number of oscillators is:

The mean energy per oscillator is:

(ignoring zero point energy)

1

exp( / ) 1B

nk T

12n n Quantum Harmonic

Oscillator

This applies to any system that can be regarded as a collection of QHOs, e.g. photons in a cavity (black body radiation), lattice vibrations, etc.

Planck’s Black Body Radiation LawThe energy density of EM radiation in a cavity in the frequency range ω to ω+dω is

22 3

3

2 3

1 1( ) ( ) ( )

exp / 1

1( )

exp / 1

B

B

Vu d n G d d

V V c k T

uc k T

• Wien’s law ωmax ≈ T• The area under the curve increases asT 4 .• Power radiated by a black body is

2

4 43 260 BJ k T Tc

Stefan’s Law = 5.6710-8 Wm-2K-4

Debye approx. for lattice vibrations

1/32 22 30 0

3( ) 3 6 /

2

D D

D

VG d d N c N V

c

Debye introduced a cut-off wavevector kD which is chosen so that there are exactly 3N normal modes. The cut-off in frequency is then ωD = ckD,

The dispersion surface ωj(k) (where j=1,2,3 labels the branches) is replaced by the simple linear dispersion ω = ck, where c is an average sound velocity.

ω

ωD

kD k

ω=ck

0

3

2 3 0

( ) ( )

3

2 exp( / ) 1

D

D

B

E G n d

Vd

c k T

3NkB

Molar heat capacity of copper compared to Debye theory

• CV is proportional to T 3 as T→0

• CV →3NkB for kBT >> ħωD

Chemical Potential,E V

S

T N

Systems with variable numbers of particles

The Grand (Gibbs) Distribution

1( , ) exp ( ( )) /i i i i i B

G

p N E N E N k TZ

exp ( ( )) /G j j j Bj

Z N E N k T where

This is the equivalent of the Boltzmann distribution for the situation where the number of particles in the system is not fixed.

Bose-Einstein Distribution

1

exp[( ) / ] 1B

nk T

Fermi-Dirac Distribution

1

exp / 1B

nk T

0

0.5

1

0 1 2 3

ε/μ

<n>

kBT/μ=0kBT/μ=0.1

kBT/μ=0.5kBT/μ=0.25

0

1

2

3

4

5

0 1 2 3

<n>

ε

kBT=1.0

kBT=0.25

kBT=0.5kBT=0.1

Bose Einstein Condensation

0

3/ 2 1/ 2

2 2 0

( ) ( )

2

4 exp / 1B

N G n d

V m d

k T

To calculate the chemical potential for a Bose gas:

We find that μ→ 0 at a finite temperature. This leads to a macroscopic number of particles in the ground state. These particles cannot contribute to CV or the pressure, since they have zero momentum.

In real BEC systems the condensate forms a macroscopic quantum state that displays superfluidity etc.

Ground state energy of the fermi gas

The Fermi energy: εF = μ (T=0)

3/ 2 3/ 21/ 2

2 2 2 20 0

2/32 2

2 22 ( ) ( )

2 3

3: ( 0)

2F

V m V mN G n d d

NIt follows that T

m V

Rearranging

We need N/2 k states to accommodate N fermions. The total number of fermions is the integral of the occupancy over all the states:

3/ 2 3/ 23/ 2 5/ 2

2 2 2 20 0

2 22 ( ) ( )

2 5

F

ZP F

V m V mE G n d d

Using our result for the Fermi energy:

we find for the ground state energy:

3

5ZP FE N

Degeneracy pressure of fermi gas

2

2 2/333

5 22

5

FF

F

UP N writing N V

V V mN

It follows that PV

It contributes to the bulk modulus of metals and is the main source of stability of white dwarf and neutron stars.

This is called the degeneracy pressure.

Heat capacity of fermi gas

• Heat capacity is linear as long as T << TF.

• At high temperatures CV

approaches the classical value (3/2)NkB above T ~ 2TF.

The Classical Limit

exp / 1Bk T In the limit where , we may ignore the 1 in the denominator, compared to the exponential. This leads to:

( ) exp / exp /B Bn k T k T

Classical Distribution Function:

1 1

. . : ( ) . . : ( )exp / 1 exp / 1B B

F D n B E nk T k T

Expect <n (ε)> << 1

→Tμ

↑

We find that for both BE and FD gases the chemical potential becomes large and negative in the high temperature limit. This is the criterion that defines the classical limit.