Lecture 529.01.2019

Grand-canonical ensemble

1Fys4130, 2019

Statistical Equilibrium Ensembles Microcanonical ensemble: ! ", $ = &

'()) +(, ", $ − ))

• Microcanonical density of states: ' ), .,/ = ∫ 12 + , ", $ − ) , d4 = 56785679:;ℏ 67

• Describes a system at a fixed energy, volume and number of particles • Each possible state at fixed ) and N has an equal probability

• Phase space volume: = ), .,/ = ∫, ",$ >)12 + , ", $ − )

• Boltzmann’s formula (correspondence to thermodynamics) Entropy: ? @, A, B = C DE [= ), .,/ ]

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Statistical Equilibrium Ensembles • Canonical ensemble: describes a system that is in thermal equilibrium with a heat bath at a fixed temperature !

" #, % ='

( !)*+, #,%

• Canonical partition function and Helmholtz free energy ( ! = ∫ ./)*+, #,% = )*+0 !

• Energy fluctuates around the average, equilibrium value 1 = ⟨3⟩, with a probability 5(3) = 8

9:* ;(<*=>)

( !, ? = ∫ .@ )*+@A @ = )*+0 ! , 0 = B − !D B , @ = B = −EE+

FG ((!)

• Energy Fluctuations

H3I =1KLI

LMIK −

1KLLM

KI

=LI

LMIlog K

Fluctuations are much smaller than the average in the thermodynamic limit: Q@R

@∼

'

T→ V

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Statistical Equilibrium Ensembles • Microcanonical ensemble ! ", $ ∼ &'()*.

• Describes a system at a fixed energy, volume and number of particles • Each possible state at fixed , and N has an equal probability

• Canonical ensemble. ! ", $ ∼ -./ ",$01

• describes a system at a fixed volume and number of particles, and that is thermal equilibrium with a heat bath at a fixed temperature T

• The energy fluctuates according to a probability distribution function (PDF) P(E) determined by 2(4, 5)• Internal energy U of the thermodynamic system is fixed by T and determined as an average , = ⟨9⟩

• Grand canonical ensemble ! ", $, ( ∼ -./ ",$01 ;<(01

• describes a system with varying number of particles and that is in thermal and chemical equilibrium with a thermodynamic reservoir, i.e. fixed = and >

• Particle number and energy are fluctuating variables drawn from corresponding PDFs ?(9), ?(@)• The average energy and number of particles are fixed by the temperature and chemical potential

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Grand canonical ensemble• Describes a system with varying number of particles

and that is in thermal and chemical equilibrium with a thermodynamic reservoir, i.e. fixed ! and "• system+heat and particle reservoir = closed system

#$ + & = #, #$ ≫ &,#$ ∼ #

• Reservior ≡ Ideal gas (-, ., #$)• Distribution of particles between the system and the

reservoir #!&!#$!

• Ensemble density for the closed system is in the microcanonical ensemble

1 2, 3, &, -, ., #$ ∼ 4!

5!46!7(8 2, 3 + 9 - − ;$<$=>)

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9 - = ∑@ABC46 DE

F

GH, NJ

K L, MN

Grand canonical ensemble• Integrate out the d.o.f of the reservoir to find the density of state of

the system

! ", $, % ∼'!)!'*!

+,-ℏ /0*

∫ 23'*423'*56(8 ", $ + : 4 − <=>=?@)

• Integraloverreservoir’s d.o.f.=idealgasmicrocanonical density of states

+,-ℏ /0*

∫ 23'*423'*56(8 ", $ + : 4 − <=>=?@) = Σ= <=>=?@ − 8

Σ=XYZ?@ [?\ ] − 8 =

V'*

ℎ3'*`3'*,

3b=2 !

2d3'*,3b=2

] − 83'*, e+

! ", $, % ∼(b=+%)!%! b=!

V'*

ℎ3'*`3'*,

3b=2 − 1 !

2d<=>=?@3'*,

<=>=?@1 −

8 ", $<=>=?@

3'*, e+

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: 4 = ∑Xh+3'* ij

k

,l, Nn

o p, qr

Grand canonical ensemble Ξ(#, %, &)( ), *, + ∼

(-.++)!+!-.!

V23ℎ523

65237

3-.2 − 1 !

2<=.>.?@5237

=.>.?@1 −

A=.>.?@

5237 BC

• Totalenergy isdetermined bythe energy of idealgas

=.>.?@ =3-.2 W#

Hence 1 − Z[3\3]^

_`3a BC

= 1 − Z_`3a bc

_`3a BC

→ eBfgh

( ), *, + ∼(-.++)!+! -.!

V23ℎ523

65237

3-.2 − 1 !

2<=.>.?@5237

=.>.?@eBiZ(j,k)

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l m = ∑opC523 qr

a

7s, Nu

v w, xy

Grand canonical ensemble

! ", $, % ∼(()+%)!

%!()!

V./

ℎ1./

21./3

3()2− 1 !

289):);<

1./3

9):);<=>?@(A,B)

• CD ∼ C ≫ F

(()+%)!

%! ()!=

() + 1 () + 2 ⋯(() + %)

%!∼()I

%!∼(I

%!

! ", $, % ∼(I

%!

J 2289):);<

1

3

ℎ1

./

1

9):);<

1

3()2− 1 !

=>?@(A,B)

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K L = ∑NOP

1./ QRS

3T, NV

W X, Y

F

Grand canonical ensemble

! ", $, % ∼'(

%!

* 2,-./0/1234

ℎ3

67(

1

./0/12

1

3'2− 1 −

3%2

!;7<=>

• Keep only the termsdependenton n, q, and p(allthe restcan betaken care of bythe normalization condition)

1

3'2− 1 −

3%2

!≈

1

3'2− 1 !

! ", $, % ∼'(

%!

* 2,-./0/1234

ℎ3

7(

;7<(W,X)=>

! ", $, % ∼1

%!

'ℎ3

* 2,-YZ34

(

;7<(W,X)=>

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[ \ = ∑_`a36b cd

e

4f, Nh

i j, k

l

Grand canonical ensemble Ξ(#, %, &)

( ), *, + ∼1+!

/Λ1(#)%

2

345(6,7)89

• Chemical potential of the reservoir &: = &

/Λ1(#)%

≈/:Λ1(#)%:

= e>?89 = 3

>89

( ), *, + ∼1+!3>28934

589

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@ A = ∑CDE1F? GH

I

JK, NM

N O, PQ

Grand canonical ensemble Ξ(#, %, &)

• ( ), *, + ∼-

.!

012

3 45678

2

9

.

:;<

=>

• ? @, A, B =D

E

D

B!FG HB;I

• Grand-canonical partition function

• Ξ #, & = ∑.YZ[ \]^_

.!∫ ab :;cd(e,f) , g =

-

78

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h i = ∑jY-

k0l mn9

46, Np

I @, A

B

Grand canonical ensemble• Describes a system with varying number of particles and that is in thermal and chemical equilibrium with

a thermodynamic reservoir, i.e. fixed ! and "

# $, &, ' =)

*

)

+!-. /'01

• Grand-canonical partition function andLandau potential

Ξ !, " = D

EFG

HIJKE

L!∫ NOI0JP(R,S)

Ξ !, " = D

EFG

HIJKE

L!UV(!, L) = D

EFG

H

IJKEVE(!) , VE ! = I0JW

* X, / = D

+FY

H

-0.(Z0/+) = -0.[(\,/) , [ \, / = ](\, ^) − ^/

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` a = ∑cFdefg hi

j

kl, Nn

1 $, &

+

Grand-Canonical ensemble: number fluctuations

Probability ! " represents probability that the system is any microstates with " particles (macrostate)

! " = ∫ %& '((*,,)

'.! 0

123 4,5 02,. = '((*,,) 6.(7)0

2,. ,∑.:;< !(") = 1

• Average number N

I ≡ " =K.:;

<" !(")

" =1

Ξ(7, M)K.:;

<" 6.(7)0

,.N* =

O7Ξ(7, M)

PPMK.:;

<6.(7)0

,.N* =

O7Ξ(7, M)

PPM Ξ(7, M)

" = O7 QQ,l" Ξ

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Grand-Canonical ensemble: number fluctuationsProbability ! " follows from the transformation of probability density

! " =$

%(',))+,(-)./), ,∑,23

4 !(") = 1

• Number fluctuations

"E =F,23

4"E !(")

"E =1

Ξ(-, H)F

,23

4"E +,(-).

),I' =

J- E

Ξ(-, H)

KE

KHEF

,23

4+,(-).

),I' =

J- E

Ξ(-, H)

KE

KHEΞ(-, H)

Δ"E = "E − " E = J- E KE

KHEl" Ξ

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Grand-Canonical ensemble: Ideal gas

• ! ", $, % = '(

)!+,((.), Λ " = 12

3456.

• Ξ ", $, 8 = ∑:;<= >?@:!(", $, A)

• Ξ ", $, 8 = ∑:;<= B:!

'+,>?@

:= >C(.,@)', z = EFG

+,(.)is the fugacity

H I, J, K = LMJ = LNOP(I,J,K)

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Grand-Canonical ensemble: Ideal gas• Thermodynamiccorrespondence:Landau potential

Ω(T, 8) = −<= >? = −<=?@AB

ΛD(=)

• Thermodynamic identity: EF = −GE= − HE? − IE8

• I = n = JKJB = <= J

JB lL Ξ = ? NOP

QR(S) = zV → Ω = −kT⟨L⟩

• H = −JKJZ = <=> → H = [ \S

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Grand-Canonical ensemble: Ideal gas• Numberfluctuations

• Δn2 = 45 2 67

687l9 Ξ = 45 2 67

687; <=>

?@ A= ; <=>

?@(A)

• DE7

F= ; <=>

?@ A

GH7→ 0

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Grand-Canonical ensemble: Ideal gas• Distributionof number fluctuations

1 2 = 1Ξ(7, 9) ;<(7)=

>?<

1 2 ∼ 12!B<ΛD< =

>?<

1 2 ∼ E<2! =

>?<=>?< E<

2!

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