Statistical physics

Projekt współfinansowany ze środków Unii Europejskiej w ramach

Europejskiego Funduszu Społecznego

ROZWÓJ POTENCJAŁU I OFERTY DYDAKTYCZNEJ POLITECHNIKI WROCŁAWSKIEJ

Wrocław University of Technology

Nanoengineering

Grzegorz Harań

STATISTICAL PHYSICS

Wrocław University of Technology

Nanoengineering

Grzegorz Harań

STATISTICAL PHYSICS

Introduction to Micromechanics

Wrocław 2011

Copyright © by Wrocław University of Technology

Wrocław 2011

Reviewer: Antoni Mituś

ISBN 978-83-62098-93-4

1 Density matrix 7 1.1 Statisti alaverage . . . 7 1.2 Densitymatrix . . . 7 2 Entropy 11 3 Thermodynami s 13 3.1 Fundamentallaws . . . 13 3.2 Thermodynami fun tions . . . 14 4 Mi ro anoni alensemble 19 4.1 Densitymatrix . . . 19 4.2 Entropy . . . 20 4.2.1 Usefulformulas . . . 21 4.2.2 Properties . . . 23

4.3 Ideal lassi algas . . . 24

5 Canoni alensemble 31 5.1 Subsystemof ami ro anoni alsystem . . . 31

5.2 Densitymatrix . . . 34

5.3 Entropyandotherthermodynami fun tions . . . 35

5.4 Ideal lassi algas . . . 37

6 Grand anoni al ensemble 41 6.1 Densitymatrix . . . 41

7 Idealquantum gas 47

7.1 Fermi-Dira andBose-Einsteindistributions . . . 47

7.2 Equationof state . . . 51

7.3 Densityofstates . . . 52

7.3.1 Three-dimensionaldensityofstates. . . 53

7.3.2 Two-dimensionaldensityofstates. . . 53

7.3.3 One-dimensionaldensityofstates. . . 54

7.4 Thermodynami s . . . 54

7.4.1 Equationofstate . . . 54

7.4.2 Thermodynami potentialandinternalenergy . . . 56

7.4.3 Entropy . . . 58

7.4.4 Freeenergy . . . 60

7.4.5 Heat apa ity

C

V

. . . 60

8 Degeneratequantum gas 63 8.1 Weakdegenera y . . . 65

8.2 Strongdegenera y . . . 67

8.2.1 Chemi alpotentialandparti le on entration . . . 71

8.2.2 FermitemperatureandFermienergy . . . 72

8.2.3 Temperaturedependen e ofthe hemi alpotential . . . . 74

8.3 Roleofsystemdimensionality . . . 75

8.3.1 Two-dimensionalidealgas . . . 75

8.3.2 One-dimensionalidealgas . . . 78

9 Thermodynami sof a Fermi gas 81 9.1 Weakdegenera y . . . 82

9.2 Strongdegenera y . . . 83

10 Bose-Einstein ondensation 87 10.1 Ma ros opi o upan yofthelowestenergylevel . . . 87

10.2 Uniquenessofthelowestenergylevel . . . 90

10.3 Chemi alpotential . . . 92 10.4 Condensate . . . 93 10.5 Equationof state . . . 95 10.6 Thermodynami properties . . . 97 10.6.1 Internalenergy . . . 97 10.6.2 Entropy . . . 98 10.6.3 Spe i heat

c

V

. . . 98

10.7 Roleofasystemdimensionality[7℄ . . . 100

11 Kineti equation approa hto nonequilibrium pro esses [5℄ 105 11.1 Boltzmannequationand ollisionintegral . . . 105

11.2 Ele tri al ondu tivity . . . 107

11.3 Thermal ondu tivity . . . 108

Density matrix

1.1 Statisti al average

Quantumaveragevalueofoperator

G

ˆ

in astate

ψ

k

(x, t)

D

ˆ

G

E

=

Z

ψ

∗

k

Gψ

ˆ

k

dτ =

D

ψ

k







G

ˆ





 ψ

k

E

(1.1)

Statisti al ensemble onsists of N hypotheti al physi al systems of the same Hamiltonianwhosestatesaredeterminedbythewavefun tions

ψ

k

(x, t)

,where

x = r, p

symbolizesageneralizedvariable. Weusesu hastatisti alensemble to al ulateastatisti al averagevalue

D ¯ˆ

_G

E

=

1

N

N

X

k=1

D

ψ

k







G

ˆ





 ψ

k

E

=

1

N

N

X

k=1

Z

ψ

k

∗

Gψ

ˆ

k

dτ

(1.2) We have here two types of averaging: quantum and statisti al over a given ensemble.

1.2 Density matrix

Chosinga ompletesystemoforthonormalfun tions

ϕ

n

(x)

we anrepresent

ψ

k

(x, t) =

X

n

andwritethestatisti alaverageof

G

ˆ

D ¯ˆGE = 1

N

N

X

k=1

X

n,m

a

k∗

n

a

k

m

Z

ψ

∗

n

Gψ

ˆ

m

dτ

|

{z

}

G

n,m

=

=

X

n,m

1

N

N

X

k=1

a

k

m

a

k∗

n

G

nm

=

X

n,m

ρ

mn

G

nm

= Tr(ˆ

ρ ˆ

G)

(1.4) wherewehavedenedadensitymatrix

ρ

ˆ

ina ertainrepresentation,thatisfor agivensetofbasisfun tions

{ψ

n

}

ρ

mn

=

1

N

N

X

k=1

a

k

m

a

k∗

n

(1.5)

Although

ρ

ˆ

is dened fora ertain hoi eof

{ψ

n

}

, the averagevalue -an ob-servablequantity - doesnot depend on this hoi e, what one aneasily show transforming

{ψ

n

}

intoanotherorthonormaland ompletebasis

{ψ

′

n

}













ψ

′

1

. . .

ψ

′

n

. . .













= ˆ

U













ψ

1

. . .

ψ

n

. . .













(1.6)

where

U

ˆ

is aunitary operator. In this newrepresentation the density matrix reads

ˆ

ρ

′

= ˆ

U

−1

ρ ˆ

U

(1.7) and

ˆ

G

′

= ˆ

U

−1

G ˆ

ˆ

U

(1.8) Theaveraged

G

ˆ

′

value

Tr



ρ

ˆ

′

G

ˆ

′



= Tr



U

ˆ

−1

U ˆ

ˆ

U

−1

G ˆ

ˆ

U



= Tr



U

ˆ

−1

ρ ˆ

ˆ

G ˆ

U



=

= Tr



ρ ˆ

ˆ

G ˆ

U ˆ

U

−1



= Tr



ρ ˆ

ˆ

G



(1.9) isthesameasthetheaveraged

G

ˆ

. Therefore,wehaveshownthattheaveraging

wehavedenedaquantum statisti alaverageofanoperator

G

ˆ

D ¯ˆ

_G

E

= Tr



ρ ˆ

ˆ

G



(1.10)

where

ρ

ˆ

isadensitymatrix. Wenote thatthenormalization ondition

hψ

k

| ψ

k

i = 1,

ϕ

k

n





ϕ

k

m



= δ

nm

(1.11) leadsto

1 = hψ

k

| ψ

k

i =

X

n,m

a

k

n

a

k

m

ϕ

k

n





ϕ

k

m



=

X

n,m

a

k∗

n

a

k

m

δ

nm

=

X

n





a

k

n





2

(1.12) andin onsequen eto

Trˆ

ρ =

X

n

1

N

N

X

k=1





a

k

n





2

=

1

N

N

X

k=1

X

n





a

k

n





2

=

1

N

N

X

k=1

1 = 1

(1.13)

whi h means that

ρ

ˆ

an be regarded as a probability density matrix. Let's onsideradensitymatrixofapurestate

ψ

whenallwavefun tionsofastatisti al ensembleareidenti al,i.e.,

ψ

k

= ψ

and

ρ

mn

=

1

N

N

X

k=1

a

m

a

∗

n

= a

m

a

∗

n

(1.14) Cal ulating

ρ

ˆ

2

matrixelement

(ˆ

ρ

2

)

mn

=

X

l

ρ

ml

ρ

ln

=

X

l

a

m

a

∗

l

a

l

a

∗

n

=

= a

m

X

l

a

∗

l

a

l

!

a

∗

n

= a

m

a

∗

n

= ˆ

ρ

mn

(1.15) weobtain

ρ

ˆ

2

_{= ˆ}

_ρ

forapurestatesystem. Ifwetaketheenergyrepresentation, that isasetof eigenfun tions

ϕ

n

ˆ

Hϕ

n

= Eϕ

n

(1.16)

H

ml

=

D

ϕ

m







H

ˆ





 ϕ

l

E

= Eδ

ml

(1.17)

thetimeevolutionof

ρ

ˆ

i~

∂ ˆ

ρ

∂t

=

h

ˆ

H, ˆ

ρ

i

=

h

H, ˆ

ˆ

ρ



H

ˆ

i

= 0

(1.18) that isforanymatrixelement

ρ

mn

0 = i~

∂ρ

mn

∂t

=

h

ˆ

H, ˆ

ρ

i

mn

=

D

ϕ

m







H ˆ

ˆ

ρ





 ϕ

n

E

−

D

ϕ

m





 ˆ

ρ ˆ

H





 ϕ

n

E

=

=

X

l

D

ϕ

m







H

ˆ





 ϕ

l

E

hϕ

l

| ˆ

ρ | ϕ

n

i − hϕ

m

| ˆ

ρ | ϕ

l

i

D

ϕ

l







H

ˆ





 ϕ

n

E

=

=

X

l

(H

ml

ρ

ln

− ρ

ml

H

ln

) =

X

l

(E

m

δ

lm

ρ

ln

− ρ

ml

E

n

δ

ln

) =

= (E

m

− E

n

)ρ

mn

(1.19) andfornondegeneratestateswegetadiagonaldensitymatrix

ρ

mn

= δ

mn

ρ

n

(1.20) where

ρ

n

= ρ

nn

=

1

N

N

X

k=1





a

k

n





2

(1.21)

ρ

n

isaprobabilityofndingamongthestatisti alensembleasysteminastate

|ϕ

n

i

,inotherwords,itisaprobabilitythatasystemresidesinaquantumstate

|ϕ

n

i

. Therefore,inabasisofenergyeigenfun tionswe anrepresents

ρ

ˆ

as

ˆ

ρ =

X

n

|ϕ

n

i ρ

n

hϕ

n

|

(1.22)

We will use the energy representation of the density matrix throughout this

book. At theend let us note that

0 6





a

k

n





2

6

1

for ea h

k

and a ordingto (1.21)wehave

0 6 ρ

n

6

1

. Thedensitymatrixelementtakesitsmaximalvalue,

ρ

n

= 1

,only if





a

k

n





2

= 1

forea h

k

, that iswhen allsystemsofthestatisti al ensembleareinthestate

|ϕ

n

i

andthismeansapurestatesystemforwhi h

ˆ

ρ =











1 0

· · ·

0

0 0

· · ·

0

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

0 0

· · ·

0











(1.23)

Note,that for

ρ

n

= 1

also

ρ

2

n

= 1

and

ρ

ˆ

2

_{= ˆ}

_ρ

Entropy

Theentropy anbethoughtoasameasureofthedisorderinasystemandis obtainedbystate ounting. TheentropyofanNparti lesystemisproportional to thelogarithmofthenumberofstatesavailableto thesystemandisdened bytheGibbsformula

S = −k

B

Tr(ˆ

ρ ln ˆ

ρ)

(2.1)

where

k

B

= 1.38 · 10

−23

_[

J/K

]

istheBoltzmann onstant. Intheenergy repre-sentationwehave

S = −k

B

X

n,m

ρ

nm

ln ρ

mn

= −k

B

X

n,m

ρ

n

δ

mn

ln(ρ

n

δ

mn

)

(2.2) andtheentropyisgivenbyastraightsummationovertheenergystates

S = −k

B

X

n

ρ

n

ln ρ

n

(2.3)

Veryoftenitismore onvenienttoperformanintegrationinsteadofa summa-tionusingasubstitutionforasumoverthestatesofanN-parti lesystem

X

n

−→

(2s + 1)V

_h

3N

Z

d

3N

p

(2.4)

where

V

is avolume ofasystemand

s

is aparti lespin, we anrewrite(2.3) in aform

S = −k

B

(2s + 1)V

h

3N

Z

where

p = (p

1

, . . . , p

N

)

. The density matrix, whi h be omes a ontinuous momentum fun tion

ρ(p)

,obeysanormalization ondition

(2s + 1)V

h

3N

Z

d

3N

pρ(p) = 1

(2.6)

and anbe alledaprobabilitydensityinthe

3N

-dimensionalmomentumspa e. Weneedtoelu idatethatapplyingtherepla ement(2.4)onemustpaya parti -ularattentiontopossiblesingularitiesofanintegratedfun tionandwhileause ofanintegralisallowedforfermionsor lassi alparti lesina aseofbosonsone must in lude thelowest energy state element in addition to anintegral(2.4). Thisissuewillbedis ussedthoroughlyinChapters7and10. Theentropy def-inition (2.5) is very usefull, asit anbe applied after aslight modi ationto quantumaswellas lassi alsystems. Althoughallparti lesobeyquantumlaws, a lassi alapproa hisa onvenientapproximationforsystemswhosequantum features an be negle ted. Therefore, one must use an appropriate lassi al states ountingpro edure tohavethesamenumberof statesasin aquantum system and be ause asingle quantum state of

N

parti les orrespondsto

N !

statesofdistinguishable lassi alparti lesthenumberof lassi alstatesshould bediminishedbyafa tor

N !

whi ha ountsforanumberof

N

parti les per-mutations. Forthat purpose weintrodu e a oe ient

c

N

of an a tion units whi h dis riminates quantum and lassi al systems:

c

N

= h

3N

for quantum,

c

N

= N !h

3N

for lassi al. We annowintrodu eaprobabilitydensity

ρ(p, r)

, where

r = (r

1

, . . . , r

N

)

,inthe

6N

-dimensionalmomentumand position spa e, that is the

Γ

-spa e, whi h gives a probabilityof nding a lassi alsystem in astatewith parti lesmomenta andpositionsgivenby

p

and

r

ve tors respe -tively,andforaquantumsystemisdenedas

ρ(p, r) = ρ(p)

. Thenormalization onditiononthe

Γ

-spa ereads

Z dpdr

c

N

ρ(p, r) = 1,

(2.7)

wherewehaveusedasimpliednotation

p = p

3N

,

r = r

3N

. Wenote,thatina quantumsystemtheprobabilitydensityissolelyamomentumfun tionandthe position integralgivessimply volume to the N-th power, whi h together with the

c

N

oe ientform thedensityof statesin themomentum spa e

V

N

_/h

3N

. A ordingto(3.9)theentropyisdened

S = −k

B

Z dpdr

c

N

Thermodynami s

3.1 Fundamental laws

We will use the rst lawof thermodynami s, that is the energy onservation law,

dU = δQ − P dV + µdN

(3.1)

andthese ondlawofthermodynami s

dS >

δQ

T

(3.2)

where theequalityholds if hangesin thethermodynami statearereversible, and the inequality appliesto spontaneus orirreversiblepro ess. In this book wedealwithreversiblepro esses,ex eptforthelastparagraph,andwehave

dS =

δQ

T

(3.3)

unless it is nototherwise stated. Therefore, we an ombine both thermody-nami lawsinto asingleequation

T dS = dU + P dV − µdN

(3.4)

3.2 Thermodynami fun tions

Athermodynami denition oftheentropyintherelation(3.4)yields

dS =

1

T

dU +

P

T

dV −

µ

T

dN

(3.5)

We note, that the entropy

S = S(U, V, N )

is a fun tion of three variables: internalenergy

U

,volume

V

andnumberofparti les

N

,hen eaninnitesimal hangeof theentropy

dS =

 ∂S

∂U



V,N

dU +

 ∂S

∂V



U,N

dV +

 ∂S

∂N



U,V

dN

(3.6)

yieldsthefollowingthermodynami identities

 ∂S

∂U



V,N

=

1

T

(3.7)

 ∂S

∂V



U,N

=

P

T

(3.8)

 ∂S

∂N



U,V

= −

_T

µ

(3.9)

A hangeoftheinternalenergy

U

,whi ha ordingto Eq. (3.4)reads

dU = T dS − P dV + µdN

(3.10)

denes

U

asathreevariablefun tion

U = U (S, V, N )

. Therefore,itsdierential

dU =

 ∂U

∂S



V,N

dS +

 ∂U

∂V



S,N

dV +

 ∂U

∂N



S,V

dN

(3.11)

isdetermidedbythefollowingrelations

 ∂U

∂S



V,N

= T

(3.12)

 ∂U

∂V



S,N

= −P

(3.13)

 ∂U

∂N



S,V

= µ

(3.14)

Subsequently,wedenetheHelmholtzfreeenergy

F = U − T S

(3.15)

whosedierential hange

dF = d(U − T S) = dU − T dS − SdT

(3.16) dependsontheinternalenergydierential(3.10) andreads

dF = −SdT − P dV + µdN

(3.17)

Therefore, we have the free energy as a fun tion of

T

,

V

, and

N

variables

F = F (T, V, N )

anditsdierential

dF =

 ∂F

∂T



V,N

dT +

 ∂F

∂V



T,N

dV +

 ∂F

∂N



T,V

dN

(3.18)

leadstothefollowingthermodynami identities

 ∂F

∂T



V,N

= −S

(3.19)

 ∂F

∂V



T,N

= −P

(3.20)

 ∂F

∂N



T,V

= µ

(3.21)

Athermodynami denitionofthegrandpotential(thermodynami potential)

Ω(T, V, µ) = F − µN

(3.22)

ombinedwithEq. (3.17)givesadierential hangeofthegrandpotential

dΩ = −SdT − P dV − Ndµ

(3.23)

whi h isafun tion of

T

,

V

, and

µ

. A dierentialof athree variablefun tion

Ω = Ω(T, V, µ)

dΩ =

 ∂Ω

∂T



V,µ

dT +

 ∂Ω

∂V



T,µ

dV +

 ∂Ω

∂µ



T,V

dµ

(3.24)

leadstothermodynami identities

 ∂Ω

∂T



V,µ

= −S

(3.25)

 ∂Ω

∂V



T,µ

= −P

(3.26)

 ∂Ω

∂µ



T,V

= −N

(3.27)

Up to now, we have dened the dierentials of the internal energy, entropy, freeenergyandthermodynami potential. Usingas alingpropertyofextensive quantities we will derive anexpli it internal energy formulaand subsequently obtain the rest of dened thermodynami fun tions. First, we note that the internal energy

U = U (S, V, N )

is afun tion of extensivequantities:

S

,

V

,

N

whi hareproportionalto themassand asize ofasystem,therefore

U

itself is alsoanextensivequantity. Letusdoas alingtransformationofthesystemby extendingthesize ofasystem

λ

times

S

−→ λS

V

−→ λV

N

−→ λN

whi halsogives

U

−→ λU

Wemaywritethis transformationasfollows

U (λS, λV, λN ) = λU (S, V, N )

(3.28) Takingaderivativewithrespe tto

λ

at

λ = 1

oftheright-handsideof (3.28)

d

dλ

U (λS, λV, λN ) =

d

dλ

λU (S, V, N ) = U (S, V, N )

(3.29) andofitsleft-handside

d

dλ

U (λS, λV, λN ) =



∂

∂λS

U (λS, λV, λN )



V,N

 dλS

dλ



+



∂

∂λV

U (λS, λV, λN )



S,N

 dλV

dλ



+



_∂

∂λN

U (λS, λV, λN )



V,S

 dλN

dλ



(3.30) that is

d

dλ

U (λS, λV, λN ) =



∂

∂λS

U (λS, λV, λN )



V,N

S+



_∂

∂λV

U (λS, λV, λN )



S,N

V +



_∂

∂λN

U (λS, λV, λN )



V,S

N

(3.31) Hen e,weobtain

U (S, V, N ) =

 ∂U

∂S



V,N

S +

 ∂U

∂V



S,N

V +

 ∂U

∂N



V,N

N

(3.32)

and using the thermodynami identities (3.12)-(3.14) we nally get aformula whi hdenestheinternalenergy

U (S, V, N ) = T S − P V + µN

(3.33) whi hyieldstheexpli itformulasfortheHelmholtzfreeenergy

F = U − T S = −P V + µN

(3.34)

andthethermodynami potential

Ω = F − µN = −P V

(3.35)

or

Ω = F − µN = U − T S − µN

(3.36)

Theentropyfun tionisto bedeterminedmi ros opi allywithin thestatisti al physi sapproa h.

Mi ro anoni al ensemble

4.1 Density matrix

Ami ro anoni alensemble onsistsofallavailablestatesofanisolatedsystem, that is statesof a onstant energy

E

and axed number of parti les

N

. We dene the densitymatrix

ρ

ˆ

in the energyrepresentation

{ϕ

n

}

,

Hϕ

ˆ

n

= E

n

ϕ

n

, forwhi h

ρ

mn

= ρ

n

δ

mn

and thediagonalelementsread

ρ

n

=

δ

E

n

,E

Γ(E)

(4.1) where

Γ(E) =

P

n

δ

E

n

,E

isanumberoftheenergy

E

states. Takingintoa ount asmall, omparedtotheenergy,dis ernibilityoftheenergymeasurement,

∆ ≪

E

, we write a physi ally justied denition of the mi ro anoni al ensemble densitymatrix

ρ

n

=







1

Γ(E)

for

E < E

n

< E + ∆

0

otherwise (4.2)

wherenow

Γ(E)

isanumberofstatesintheenergyinterval

E < E

n

< E + ∆

. Wenote,that

Trˆ

ρ =

X

n

ρ

n

=

1

Γ(E)

X

n

δ

E

n

,E

=

Γ(E)

Γ(E)

= 1

(4.3)

whi hmeansthat

ρ

ˆ

isaproperlydenedprobabilitydensitymatrix. Weshallsee that the mi ro anoni alensembleextremizes theGibbs entropy,that isobeys

these ondlawofthermodynami s. WeusetheLangragemultipliersmethodto look foranextremumoftheentropy(2.8)fora onstantenergystates

S = −k

B

Z

E<H(p,r)<E+∆

dpdr

c

N

ρ(p, r) ln ρ(p, r)

(4.4)

subje ttothenormalization ondition(2.7)

Z

E<H(p,r)<E+∆

dpdr

c

N

ρ(p, r) = 1,

(4.5)

Forthesakeofsimpli ity,fromnowonwewilldropothelimitsintheintegral notationmindingthatallintegralsaretakeninthesamelimitsasin(4.5)unless otherlimitsarespe ied. Wetakeavariation

δ



S(ρ) + α

Z dpdr

c

N

ρ(p, r) − 1



= 0,

(4.6) whi hreads

Z dpdr

c

N

[−k

B

ln ρ(p, r) − k

B

+ α] δρ(p, r) = 0.

(4.7) Sin e

δρ

isanarbitraryquantity,wehave

ρ(p, r) = e

α−kB

kB

_{= const}

(4.8)

A Lagrange multiplier

α

isdetermined from the normalization ondition(4.5) whi hfora onstantdensitymatrixgives

ρ(p, r) =

Z dpdr

c

N



−1

= Γ

−1

_(E)

(4.9)

where

Γ(E)

isanumberof statesoftheenergy

E

.Thereforewehaveobtained themi ro anoni alensembleprobabilitydensity(4.2).

4.2 Entropy

S = −k

B

Z dpdr

c

N

ρ(p, r) ln ρ(p, r) =

− k

B

Z dpdr

c

N

Γ

−1

(E) ln Γ

−1

(E) = k

B

ln Γ(E)

(4.10) and on luding, we may say that we have obtained a useful formula for the entropyinthemi ro anoni alensemble

S = k

B

ln Γ(E)

(4.11)

We analsorederivetheentropyformulabytakingatra ein theGibbs deni-tion(2.3)

S = −k

B

X

n

ρ

n

ln ρ

n

=

− k

B

X

n

ρ

n

ln

1

Γ(E)

= k

B

ln Γ(E)

X

n

ρ

n

= k

B

ln Γ(E)

(4.12) 4.2.1 Useful formulas

Sometimesitismore onvenienttouseavolume

Ω(E)

o upiedbytheavailable statesoftheenergy

E

in the

Γ

-spa einsteadof anumberofstates

Γ(E)

,that is

Ω(E)

isathin shellvolume

Ω =

Z

E<H(p,r)<E+∆

dpdr

(4.13)

Insu h anotationwehave

S = k

B

ln

Ω(E)

c

N

(4.14)

We shall nowpresent asomewhat more onvenient method of omputing the entropy by dening a volume

Φ(E)

in the

Γ

-spa e o upied by the states of theenergynotex eedingtheenergy

E

,whi hisusuallyeasiertoevaluatethan

Ω(E)

Φ(E) = Φ(E, V, N ) =

Z

H(p,r)6E

dpdr,

(4.15)

whi h an be representedby a sum of the

E

i

energy thin shell volumes(Fig. 4.1)

Φ(E, V, N ) =

E

∆

X

i=1

Ω(E

i

, V, N )

(4.16)

where

Ω(E

i

, V, N )

istheenergy

E

i

thin shellvolume. Wenotethat thelargest

Ω(E

i

, V, N)

Φ(E, V, N )

E

i

E

E

i

+ ∆

Figure 4.1:

Ω(E

i

, V, N )

shell volume and

Φ(E, V, N )

volume of states of the energynotex eeding

E

shellvolume

Ω(E

i

, V, N )

isthatwithintheenergyinterval

(E, E + ∆)

,thuswe anwrite

Ω(E, V, N ) 6 Φ(E, V, N ) 6

E

∆

Ω(E, V, N ),

(4.17) orbytakingalogarithmwhi h isamonotoni fun tion

ln Ω(E) 6 ln Φ(E) 6 ln Ω(E) + ln

E

∆

.

(4.18)

Be ause

Φ(E)

is avolumein the

6N

-dimensional spa e

ln Φ(E) ∼ N

and the energy

E

ofasystemisalsoproportionaltothenumberofparti les

ln

E

∆

∼ ln N

wegetinthethermodynami limit

ln

E

∆

ln Φ(E)

∼

ln N

N

−−−−→

N →∞

V →∞

0,

(4.19)

and

ln Φ(E) = ln Ω(E).

(4.20)

Thereforeinthethermodynami limittheentropy(4.14) reads

S = k

B

ln

Φ(E, V, N )

c

N

(4.21)

where

Φ(E, V, N )

isvolumeofstatesoftheenergynotex eeding

E

.

4.2.2 Properties

1. Entropy

S(E) = S(U, V, N )

is a ontinuous and dierentiable fun tion of

U

,

V

,

N

.

2. Entropyisanadditivefun tion,thatis,forasystem onsistingofindependent subsystems

A

,

B

theentropy

S(U, V, N ) = S(U

A

, V

A

, N

A

) + S(U

B

, V

B

, N

B

)

.

U

B

,

V

B

,

N

B

U

A

,

V

A

,

N

A

Γ

A

– number of states in

A,

Γ

B

– number of states in

B

A

B

Proof: Letusshowitforthe lassi alentropy

S(U, V, N ) = k

B

ln

Ω

A

Ω

B

N

A

!N

B

!h

3(N

A

+N

B

)

= k

B

ln

Ω

A

N

A

!h

3N

A

+ k

B

ln

Ω

B

N

B

!h

3N

B

=

= S(U

A

, V

A

, N

A

) + S(U

B

, V

B

, N

B

).

(4.22) 3. Entropy isan extensivequantity:

S(λU, λV, λN ) = λS(U, V, N )

,for

λ > 0

. Proof: Forthe lassi alentropy

S(λU, λV, λN ) = k

B

ln

Φ(λU, λV, λN )

(λN )!h

3λN

,

(4.23)

where

Φ(λU, λV, λN )

isavolumeina

6λN

-dimensional

Γ

-spa e.

Φ(λU, λV, λN ) ∼ (λV )

λN

p

λN

λ

(4.24) sin etheinternalenergy

U =

N

X

n=1

wehave

p =

 U

N



1

2

(4.26)

andfortheextendedsystem

p

λ

=

 λU

λN



1

2

=

 U

N



1

2

,

(4.27) therefore

Φ(λU, λV, λN ) ∼ (λV )

λN

 U

_N



λN

2

.

(4.28)

Theentropyoftheextendedsystemreads

S(λU, λV, λN ) = k

B

ln

(λV )

λN

 U

N



λN

2

(λN )!h

3λN

(4.29)

andusingtheStirling'sapproximation,

ln N ! ≈ N ln N −N

,wehavetheentropy

S(λU, λV, λN ) = k

B

λ ln

V

N

 U

N



N

2

h

3N

+ k

B

λN ln λ − k

B

λN ln λN + k

B

λN,

(4.30) whi hisequivalentto

S(λU, λV, λN ) = λk

B

ln

V

N

 U

N



N

2

N !h

3N

= λS(λU, λV, λN )

λ=1

= λS(U, V, N )

(4.31)

4.3 Ideal lassi al gas

Weusean exampleofanideal lassi algas todemonstratethe methodof the mi ro anoni alensemble. Inorder to determinethe entropy[3℄(pp. 299-300, 348)

S = k

B

ln

Φ(E)

weneedto omputethevolume

Φ(E)

in the

Γ

-spa eo upiedbystatesofthe energylessthan

E

,that is

Φ(E) =

Z

V

dr

1

· · ·

Z

V

dr

N

Z

dp

1

· · ·

Z

dp

N

(4.33)

wherethemomentaarelimitedby

H =

N

X

i=1

p

2

i

2m

6

E

(4.34) Deningaradius

R =

√

2mE

we anwrite

Φ(E) = V

N

_Φ

p

,

(4.35) where

Φ(E) =

∞

Z

−∞

dp

1

· · ·

∞

Z

−∞

dp

N

θ

R

2

−

N

X

i=1

p

2

i

!

(4.36)

is avolumeen losed by

R

in a

3N

-dimensional momentum spa e, thus it an berepresentedas

Φ

p

= A

3N

R

3N

.

(4.37)

Weobtain

A

3N

oe ientbyevaluatingtheintegral

∞

Z

0

dR

dΦ

p

dR

e

−R

2

_,

(4.38) where

dΦ

p

dR

= 3N A

3N

R

3N −1

_.

(4.39)

We andothat withoutusinganexpli itformof

dΦ

p

dR

butitsdenition(4.36)

∞

Z

0

dR

dΦ

p

dR

e

−R

2

₌

∞

Z

0

dR

dΦ

p

dE

dE

dR

e

−R

2

₌

∞

Z

0

dE

dΦ

p

dE

e

−R

2

₌

=

∞

Z

0

dE

d

dE

Z

dp

1

· · ·

Z

dp

N

Θ

R

2

−

N

X

i=1

p

2

i

!

e

−R

2

=

=

∞

Z

0

dE2m

Z

dp

1

· · ·

Z

dp

N

δ

R

2

−

N

X

i=1

p

2

i

!

e

−R

2

=

=

∞

Z

0

dR

2

Z

dp

1

· · ·

Z

dp

N

δ

R

2

−

N

X

i=1

p

2

i

!

e

−R

2

=

=

Z

dp

1

· · ·

Z

dp

N

e

−

N

P

i=1

p

2

i

=

=

∞

Z

−∞

dp

1x

∞

Z

−∞

dp

1y

∞

Z

−∞

dp

1z

· · ·

∞

Z

−∞

dp

N x

∞

Z

−∞

dp

N y

∞

Z

−∞

dp

N z

e

„

−

N

P

i=1

p

2

ix

+p

2

iy

+p

2

iz

«

=

=

∞

Z

−∞

dp

1x

e

−p

2

1x

∞

Z

−∞

dp

1y

e

−p

2

1y

∞

Z

−∞

dp

1z

e

−p

2

1z

_{· · ·}

· · ·

∞

Z

−∞

dp

N x

e

−p

2

N x

∞

Z

−∞

dp

N y

e

−p

2

N y

∞

Z

−∞

dp

N z

e

−p

2

N z

₌

=





∞

Z

−∞

dpe

−p

2





3N

= π

3N

2

(4.40)

Ontheotherhandusing(4.39)

∞

Z

0

dR

dΦ

p

dR

e

−R

2

= 3N A

3N

∞

Z

0

dRR

3N −1

e

−R

2

=

3

2

N A

3N

Γ

 3

2

N



(4.41) where

Γ(x) =

∞

Z

0

t

x−1

e

−t

dt

(4.42)

isthegammafun tion. Therefore, omparing (4.40)and(4.41)weget

3

2

N A

3N

Γ

 3

2

N



= π

3N

2

,

(4.43) that is

A

3N

=

π

3N

2

Γ

3

₂

N + 1

(4.44)

Therefore,thevolumein themomentum spa ereads

Φ

p

=

π

3N

2

Γ

3

2

N + 1

R

3N

=

π

3N

2

Γ

3

2

N + 1

(2mE)

3N

2

=

(2πmE)

3N

2

Γ

3

2

N + 1

(4.45)

andthevolumeinthe

Γ

-spa e

Φ(E) =

V

N

_(2πmE)

3N

2

Γ

3

₂

N + 1

.

(4.46)

Wearenowin apositionto al ulatetheentropy

S = k

B

ln

"

V

N

_(2πmE)

3N

2

N !h

3N

_Γ

3

2

N + 1

#

.

(4.47) Be ause

Γ

3

2

N + 1

=

3

2

N

!

andforlarge

N

we anusetheStirling's approx-imation

ln N ! ≈ N ln N − N

or

N ! ≈

 N

e



N

,

(4.48) wehave

Γ

 3

2

N + 1



≈

 3N

_2e



3N

2

(4.49)

andtheentropy

S = k

B

ln





V

N

_(2πmE)

3N

2

N !h

3N

3N

2e

3N

2





(4.50)

Theentropyformula anberearanged,usingagaintheStirling'sapproximation asfollows

S =

3

2

N k

B

+ N k

B

ln

"

V

 4πmE

3h

2

_N



3

2

#

− k

B

ln N ! =

=

3

2

N k

B

+ N k

B

ln

"

V

 4πmE

3h

2

_N



3

2

#

− Nk

B

ln N + N k

B

=

=

5

2

N k

B

+ N k

B

ln

"

V

N

 4πmE

3h

2

_N



3

2

#

(4.51)

anddening theinternal energy

U = E

theentropyreads

S = N k

B

"

5

2

+ ln

V

N

 4πmU

3h

2

_N



3

2

!#

.

(4.52)

Weuse theaboveequation to obtainthetemperatureand pressure. Taking a partialderivative

∂

∂S

of (4.52)at onstant

V

and

N

wegettheequation

1 = N k

B

"

V

N

 4πmU

3h

2

_N



3

2

#

−1

V

N

 4πmU

3h

2

_N



3

2

₃

2

U

1

2

 ∂U

∂S



V,N

,

(4.53) whi hsimpliesto

U =

3

2

N k

B

 ∂U

∂S



V,N

=

3

2

N k

B

T

(4.54)

where wehave used a thermodynami relation

T =

 ∂U

∂S



V,N

. Wehave

ob-tainedtheinternalenergyofamonatomi idealgas

U =

3

2

N k

B

T

(4.55)

in agreementwith the energy equipartition theorem. A similar pro edure of takingapartialderivative

∂

∂V

at onstant

S

,

N

yieldsarelation

0 =

N

V

 4πmU

3h

2

_N



−

3

2

"

₁

N

 4πmU

3h

2

_N



3

2

+

V

N

 4πmU

3h

2

_N



3

2

₃

2

U

1

2

 ∂U

∂V



S,N

#

(4.56) whi hisequivalentto

−U =

3

₂

V

 ∂U

∂V



S,N

(4.57)

Usingathermodynami identity

P = −

 ∂U

∂V



S,N

,weget

U =

3

2

P V

(4.58)

Finally, ombining(4.55) and(4.58) weobtaintheideal lassi algas equation ofstate

We analsondtheheat apa ityat onstantvolume

c

V

=

 ∂U

∂T



V

=

∂

∂T

 3

2

N k

B

T



=

3

2

N k

B

(4.60)

andtheHelmholtzfreeenergy

F = U − T S = −Nk

B

T − Nk

B

T ln

"

V

N

 2πm

h

2

k

B

T



3

2

#

=

= −Nk

B

T − Nk

B

T ln

V

N

−

3

2

N k

B

T ln(2πmk

B

T ) + N k

B

T ln h

3

(4.61)

whi hyieldsthe hemi alpotential

µ =

 ∂F

∂N



T,V

=

− k

B

T − k

B

T ln

V

N

+ k

B

T −

3

2

k

B

T ln(2πmk

B

T ) + k

B

T ln h

3

₌

k

B

T ln

nh

3

(2πmk

B

T )

3

2

,

(4.62) where

n =

N

V

isaparti le on entration.

Canoni al ensemble

5.1 Subsystem of a mi ro anoni al system

We onsider a system onsisting of

N

parti les in a thermal onta t with a mu h larger system ontaining

N

0

parti les,

N

0

≫ N

, whi h we all a heat bath. Althoughbothsystemsareseparateandtheirparti lesdonotmix,that is parti lenumbers

N

and

N

0

andvolumesofthesystems

V

,

V

0

are onstant, they anex hangetheenergy.

E, N , V

E

0

− E, N

0

,

V

0

heat bath

Theenergy

E

ofasmallersystemdeterminedbytheHamiltonian

H(p, r) = E

is mu h smallerthantheenergyoftheheatbath

E

0

− E

,so in slowlyvarying energy fun tions we anassume

E

0

− E ≈ E

0

= const

, although

E

is allowed to hange. Bothsystems anbe onsideredsubsystemsofa onstantenergy

E

0

systemdes ribedbyami ro anoni alensemblewhosenumberofstates

Γ

m

(E

0

)

be auseof aspatial separationofthe subsystemsis aprodu t ofthe numbers

ofstates

Γ

and

Γ

0

inthe

6N

-and

6N

0

-dimensionalspa esrespe tively

Γ

m

(E

0

) = Γ(E)Γ

0

(E

0

− E).

(5.1) The variation of the mi ro anoni al ensemble entropy

S(E

0

) = k

B

ln Γ

m

(E

0

)

withrespe tto theenergy

E

mustfulll theextremum ondition

δS(E

0

) = 0

(5.2)

δk

B

ln Γ

m

(E

0

) = 0

(5.3)

δ ln [Γ(E)Γ

0

(E

0

− E)] = 0

(5.4)

δ ln Γ(E) + δ ln Γ

0

(E

0

− E) = 0

(5.5)



1

Γ(E)

∂Γ(E)

∂E

−

1

Γ

0

(E

0

− E)

∂Γ

0

(E

0

− E)

∂(E

0

− E)



δE = 0

(5.6)

foranarbitrary

δE

value,that is,

1

Γ(E)

∂Γ(E)

∂E

−

1

Γ

0

(E

0

− E)

∂Γ

0

(E

0

− E)

∂(E

0

− E)

= 0

(5.7)

1

Γ(E)

∂Γ(E)

∂E

−

∂

∂(E

0

− E)

ln Γ

0

(E

0

− E) = 0

(5.8) Be ause

E

0

≫ E

we anassumethattheenergyoftheheatbath

E

0

− E ≈ E

0

therefore

1

Γ(E)

∂Γ(E)

∂E

=

1

k

B

∂

∂(E

0

)

k

B

ln Γ

0

(E

0

).

(5.9)

On the right-hand side of Eq.(5.9) we have the energy derivative of the heat bathentropy

S

0

(E

0

) = k

B

ln Γ

0

(E

0

)

whi hdenestheabsolute temperature

T

oftheheatbath

1

T

=

 ∂S

0

(E

0

)

∂E

0



N,V

.

(5.10) Therefore

1

Γ(E)

∂Γ(E)

∂E

=

1

k

B

T

.

(5.11)

Let us notethat

T

is a ommon temperatureof bothsystemssin e Eq. (5.9) yields

∂

∂E

ln Γ(E) =

∂

∂E

0

ln Γ

0

(E

0

).

(5.12)

It means that both systemsremain at the sametemperature, in other words, the systems are in a thermal equilibrium. Eq. (5.11) is a simple dierential equation

dΓ

Γ

=

dE

k

B

T

,

(5.13)

whi hintegratedgives

ln Γ =

E

k

B

T

+ ln C,

(5.14)

where

C

is a onstant. Finally, a number of states of the energy

E

in the

6N

-dimensional spa ereads

Γ(E) = Ce

kB T

E

_.

(5.15)

Ifwelimitour onsiderationstothestatesofaxedenergy

E

thenwestilldeal with ami ro anoni al ensemble and the probability density is determined by thenormalization ondition

X

n

δ

E

n

,E

ρ(E

n

) =

Z

H(p,r)=E

dpdr

c

N

ρ(p, r) = 1

(5.16)

where

ρ

is onstantfora onstantenergysheet,thatis,itredu estothe mi ro- anoni aldensityfora onstantenergy onstraint. We anexpressthis ondition bytherelation

ρ(p, r) = ρ(H(p, r))

. Therefore,foragivenenergyweget

ρ(p, r) = Γ

−1

(E)

(5.17)

We annowin ludeallstatesoftheenergy

E

notex eedingtheheatbathenergy

E

0

andbasedonEqs. (5.15),(5.17) wewritethenormalization ondition

C

−1

Z

H(p,r)<E

0

dpdr

c

N

e

−

H(p,r)

kB T

_{= 1}

(5.18)

C =

Z

H(p,r)<E

0

dpdr

c

N

e

−

H(p,r)

kB T

_,

(5.19) where wehaveextendedtheintegraltoin lude allstatesoftheenergysmaller than the heat bathenergy. Although the aboveintegralshould be limitedby theenergyoftheheatbathwe anextendittoinnityasinfa tonlyanarrow

maysaythat theenergy

E

0

hasbeen hosenmu hlargerthantheenergy

E

of a onsidered system sowe an assumeit to beinnite. Insummary,wehave obtainedtheprobabilitydensity

ρ(p, r) =

1

Q

N

(V, T )

e

−

H(p,r)

kB T

_,

(5.20) where

Q

N

(V, T ) = C =

Z dpdr

c

N

e

−

H(p,r)

kB T

(5.21)

is alled the partition fun tion or the statisti al sum for a system onsisting of

N

parti les at thetemperature

T

en losed in avolume

V

. Theprobability density (Eq. (5.20)) denes a anoni al statisti al ensemble. Con luding, we write

Q

N

(V, T )

expli itely fora lassi alsystem

Q

N

(V, T ) =

Z

_dpdr

N !h

3N

e

−

H(p,r)

_{kB T}

(5.22)

andforaquantum one

Q

N

(V, T ) = V

Z

_dp

h

3N

e

−

H(p)

_{kB T}

(5.23) 5.2 Density matrix

The density matrixelementsin the energy representationare given by (5.20) andfortheenergy

E

n

state

ρ

mn

= ρ

n

δ

mn

=

1

Q

N

δ

mn

e

−βE

n

(5.24) where

β =

1

k

B

T

,andthepartitionfun tion reads

Q

N

=

X

n

e

−βE

n

|

{z

}

sumoverstates

=

X

n

′

g

n

′

e

−βE

n

′

|

{z

}

sumoverenergylevels

where wehavedis riminatedbetweentwopossiblesummations: overall avail-ablestatesoroverallenergylevelsinwhi hwehavein ludedapossible

degener-and ompleteset ofeigenfun tions

ϕ

n

,

P

n

|ϕ

n

i hϕ

n

| = 1

,

ˆ

ρ =

1

Q

N

X

n

|ϕ

n

i e

−βE

n

hϕ

n

| =

1

Q

N

e

−β ˆ

H

X

n

|ϕ

n

i hϕ

n

| =

1

Q

N

e

−β ˆ

H

(5.25) Therefore,weobtain

ˆ

ρ =

1

Q

N

e

−β ˆ

H

(5.26)

wherethepartitionfun tion

Q

N

=

X

n

e

−βE

n

₌

X

n

D

ϕ

n





 e

−β ˆ

H





 ϕ

n

E

=

=

X

n



e

−β ˆ

H



nn

= Tre

−β ˆ

H

(5.27)

is determined by a tra eof the density matrix. Quantum statisti al average

value of an observable

G :

ˆ

D ¯ˆ

G

E

= Tr



ρ ˆ

ˆ

G



is given by a tra eof a

G

ˆ

and

ρ

ˆ

produ t

D ¯ˆ

_G

E

= Tr

 1

Q

N

e

−β ˆ

H

G

ˆ



=

1

Q

N

X

n

D

ϕ

n





 e

−β ˆ

H

G

ˆ





 ϕ

n

E

=

=

1

Q

N

X

n

e

−βE

n

D

_ϕ

n







G

ˆ





 ϕ

n

E

=

1

Q

N

X

n

G

nn

e

−βE

n

(5.28) where

G

nn

=

Z

ϕ

∗

n

Gϕ

ˆ

n

dτ =

D

ϕ

n







G

ˆ





 ϕ

n

E

(5.29) and

ˆ

Hϕ

n

= E

n

ϕ

n

(5.30)

5.3 Entropy and other thermodynami fun tions

Afundamental thermodynami fun tion-entropy

S = −k

B

Tr {ˆ

ρ ln ˆ

ρ}

(5.31) in the anoni alensembleisgivenbyarelation

S = −k

B

Tr

n

ρ



− ln Q

N

− β ˆ

H

o

=

1

T

D ¯ˆ

H

E

+ k

B

ln Q

N

(5.32)

where astatisti alaverage ofaHamiltonianrepresentsanaverage energyofa

system,thatisaninternalenergy

U =

D ¯ˆ

H

E

. Therefore,weobtainarelation

U = T S − k

B

T ln Q

N

(5.33)

whi h omparedwithaphenomenologi althermodynami denitionoftheHelmholtz freeenergy(3.15)givesastatisti aldenition ofthefreeenergy

F = −k

B

T ln Q

N

(V, T )

(5.34) where wehaveexpli itely expressed aparti lenumber

N

, volume

V

and tem-perature

T

dependen eofthefreeenergy. We anuse

F

toexpressthepartition fun tion

Q

N

= e

−βF

andthedensitymatrix

ˆ

ρ = e

β(F − ˆ

H)

(5.35)

Astatisti al formulafortheinternalenergyfollowsfromitsdenition

U = Tr

n

ρ ˆ

ˆ

H

o

=

X

n

E

n

ρ

n

|

{z

}

sumoverstates

=

1

Q

N

X

n

E

n

e

−βE

n

(5.36) Usinganidentity

∂

∂T

ln Q

N

=

∂

∂T

ln

X

n

e

−βE

n

!

=

=

1

Q

N

X

n



−

E

_k

n

B

 

−

_T

1

2



e

−βE

n

₌

1

k

B

T

2

1

Q

N

X

n

E

n

e

−βE

n

(5.37) weobtain

U = k

B

T

2

∂

∂T

ln Q

N

(V, T ) = −

∂

∂β

ln Q

N

(V, T )

(5.38) Finally,withauseof (5.33)and(5.38)we anderivetheentropyformulainthe anoni alensemble

S = k

B

T

∂

∂T

ln Q

N

+ k

B

ln Q

N

(5.39) whi hleads

S = k

B

∂

∂T

(T ln Q

N

(V, T ))

(5.40)

We have shown that the statisti al and phenomenologi al denitions of the thermodynami fun tionsare onsistent. Moreover,usingastatisti alapproa h we an onrmsomebasi thermodynami identities. Comparingthestatisti al denitionsoftheentropy(5.40)andthefreeenergy(5.34),giveninthe anoni al ensemblethat isfora onstantvolumeanda onstantnumberof parti les,we get

 ∂F

∂T



V,N

= −S

(5.41)

In summary, the thermodynami fun tions an be expressed in the anoni al ensemblebythestatisti al sum

Q

N

F (V, T ) = −k

B

T ln Q

N

(V, T )

(5.42)

S(U, V ) = k

B

∂

∂T

T ln Q

N

(V, T )

(5.43)

U (S, V ) = k

B

T

2

∂

∂T

ln Q

N

(V, T )

(5.44)

5.4 Ideal lassi al gas

Wewillusethe anoni alensemblemethod inadis ussionofanideal lassi al gasdened bytheHamiltonian

H =

1

2m

N

X

i=1

p

2

i

.

(5.45)

Asthebasi quantitythatdeterminesthethermodynami fun tionsisthe par-titionfun tionwestartour onsiderationswithanevaluationof

Q

N

(V, T )

. The partition fun tionofasystemofavolume

V

Q

N

=

V

N

N !h

3N

∞

Z

−∞

exp

"

−

_2m

1

N

X

i=1

p

2

ix

+ p

2

iy

+ p

2

iz

k

B

T

#

dp

1x

· · · dp

N x

dp

1y

· · · dp

N y

dp

1z

· · · dp

N z

=

V

N

N !h

3N





∞

Z

−∞

e

−

2mkB T

p2

_dp





3N

(5.46)

isdeterminedbytheGaussintegral

∞

Z

−∞

dxe

−λx

2

=

r

π

λ

(5.47) andreads

Q

N

=

V

N

N !h

3N

(2πmk

B

T )

3N

2

(5.48)

Forfurtherevaluationsweneedalogarithmofthepartitionfun tion

ln Q

N

= N ln V +

3N

2

ln (2πmk

B

T ) − ln N! − N ln h

3

(5.49)

whi hfor

N ≫ 1

intheStirling's approximationreads

ln Q

N

= N ln V +

3N

2

ln (2πmk

B

T ) − N ln N + N − N ln h

3

(5.50) or

ln Q

N

= N ln

V

N

+

3

2

N ln (2πmk

B

T ) − N ln h

3

_{+ N}

(5.51)

Now,we an writedownthefreeenergy

F = −k

B

T ln Q

N

= −Nk

B

T ln

V

N

−

3

2

N k

B

T ln (2πmk

B

T ) + N k

B

T ln h

3

_{− Nk}

B

T

(5.52) theinternalenergy

U = k

B

T

2

∂

∂T

ln Q

N

= k

B

T

2

 3N

2



1

2πmk

B

T

2πmk

B

=

3

2

N k

B

T

(5.53) andtheentropy

S = k

B

∂

∂T

(T ln Q

N

) = N k

B

ln

V

N

+

3

2

N k

B

ln (2πmk

B

T ) +

3

2

N k

B

− Nk

B

ln h

3

_{+ N =}

= N k

B

ln V +

3

2

N k

B

ln T + N k

B

"

ln

(2πmk

B

)

3

2

N h

3

+

5

2

#

.

(5.54)

It isinstru tiveto he kthattheobtainedfun tionsfulll thethermodynami relations

F = U − T S

and

1

T

=

 ∂S

∂U



V,N

.

(5.55)

We note also, that theresultsagree with theones obtained within the mi ro- anoni alapproa h.

Grand anoni al ensemble

6.1 Density matrix

Wedenethegrandpartitionfun tionasaweightedsumofapartitionfun tion

Q

N

(V, T )

arriedoveravarying numberofparti les

N

θ(z, V, T ) =

∞

X

N =0

z

N

Q

N

(V, T )

, (6.1) where

z = e

βµ

_{= e}

_{kB T}

µ

isafuga ityand

µ

isa hemi alpotential. Wemayalso write

θ(z, V, T ) =

∞

X

N =0

e

βµN

Q

N

(V, T )

(6.2)

Aquantumstatisti alaverageofanoperator

G

ˆ

inthegrand anoni alensemble isdenedalsoasaweightedaverageover anoni alensembles orrespondingto varying numbersofparti les

D ¯ˆ

_G

E

=

1

θ

∞

X

N =0

z

N

Tr

n

Ge

ˆ

−β ˆ

H

N

o

(6.3)

where

H

ˆ

N

isaHamiltonianofan

N

-parti lesystem. Intheenergy representa-tion,

H

ˆ

N

|ϕ

N,i

i = E

N,i

|ϕ

N,i

i

,wehave

D ¯ˆGE = 1

θ

∞

X

N =0

z

N

X

i

D

ϕ

N,i







G

ˆ





 ϕ

N,i

E

e

−βE

N,i

=

1

θ

∞

X

N =0

z

N

X

i

G

N,ii

e

−βE

N,i

=

1

θ

∞

X

N =0

G

N,ii

e

µN −EN,i

kB T

(6.4)

and dening aprobability

w

N,i

of an

N

-parti le systemto reside in the

E

N,i

energystate

w

N,i

=

1

θ

e

µN − E

N,i

k

B

T

(6.5) where

θ =

∞

X

N =0

X

i

e

µN −EN,i

kB T

(6.6) we anwrite

D ¯ˆ

_G

E

=

∞

X

N =0

X

i

G

N,ii

w

N,i

(6.7)

Ifweassume,thatforany

N

n

ˆ

G, ˆ

H

N

o

= ˆ

G ˆ

H

N

− ˆ

H

N

G = 0

ˆ

,thatisboth opera-torssharethesamesetofeigenfu tions,we anassignto(6.7)aninterpretation of an expe ted value of

G

ˆ

. Probability

w

N,i

an be regarded as a diagonal elementofthedensitymatrixofthegrand anoni alensemble

ˆ

ρ =

1

θ

e

−β

(

H−µ ˆ

ˆ

N

)

(6.8)

whereweuseasymbol

H

ˆ

foraHamiltonianwithavaryingnumberofparti les. Infa t,

ρ

ˆ

fullls thenormalization ondition

Trˆ

ρ =

1

θ

Tre

−β

₍

H−µ ˆ

ˆ

N

_{) = 1}

θ

∞

X

N =0

X

j

D

ϕ

N,j





 e

−β

(

ˆ

H−µ ˆ

N

₎





 ϕ

N,j

E

=

1

θ

∞

X

N =0

X

j

e

−β(E

N,j

−µN )

₌

1

θ

∞

X

N =0

e

βµN

X

j

e

−βE

N,j

₌

1

θ

∞

X

N =0

z

N

Q

N

= 1

(6.9)

andusing

ρ

ˆ

we anrepresentanaveragevalueof

G

ˆ

givenby(6.7)asatra eof aprodu t

G

ˆ

and

ρ

ˆ

overtheenergystatesofallpossible

N

-parti lesystems

h

Gi = Tr

¯ˆ

n

ρ ˆ

ˆ

G

o

(6.10) Eq. (6.8)denes adensitymatrixfor agrand anoni al ensemble. Obviously, foraxednumberofparti les

N = N

′

θ(z, V, T ) = z

N

′

Q

N

′

(V, T )

(6.11)

andthegrand anoni aldensitymatrix

ˆ

ρ =

1

θ

z

N

′

e

−β ˆ

H

N

′

=

1

z

N

′

Q

N

′

z

N

′

e

−β ˆ

H

N

′

=

1

Q

N

′

e

−β ˆ

H

N

′

(6.12)

redu es tothe anoni aldensitymatrix,whi hmeansthatthegrand anoni al ensembleredu estothe anoni alone.

6.2 Entropy and other thermodynami fun tions

Theentropyin thegrand anoni alensemble

S = −k

B

Tr

n

ˆ

ρ



− ln θ − ln e

β( ˆ

H−µ ˆ

N )

o

=

k

B

ln θ + k

B

Tr

n

ˆ

ρ



β( ˆ

H − µ ˆ

N )

o

=

k

B

ln θ +

1

T

U − µ ¯

N

(6.13) where

U = Tr

n

ˆ

ρ ˆ

H

o

representstheinternal energyand

N = Tr

¯

n

ˆ

ρ ˆ

N

o

stands for the average parti le number,allowsus to formulate astatisti al denition ofthegrandpotential(thermodynami potential)in agreementwith(3.36)as

Ω = −k

B

T ln θ(z, V, T )

(6.14) Fromtheabovedenition wehavethegrandpartition fun tion

θ = e

−βΩ

and thedensitymatrix

ˆ

ρ = e

β(Ω− ˆ

H+µ ˆ

N )

(6.15) Therefore,aquantumstatisti alaverageofanoperator

G

ˆ

inthegrand anoni al ensemblereads

D ¯ˆ

_G

E