12. Grandcanonical Ensemble Gerhard Müller University of Rhode Island, Gmuller@Uri.Edu Creative Commons License

Home , Canonical ensemble, Grand canonical ensemble, Grand potential, Particle number

University of Rhode Island DigitalCommons@URI

Equilibrium Statistical Physics Physics Course Materials

2015 12. Grandcanonical Ensemble Gerhard Müller University of Rhode Island, [email protected] Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.

Follow this and additional works at: http://digitalcommons.uri.edu/equilibrium_statistical_physics Abstract Part twelve of course materials for Statistical Physics I: PHY525, taught by Gerhard Müller at the University of Rhode Island. Documents will be updated periodically as more entries become presentable.

Recommended Citation Müller, Gerhard, "12. Grandcanonical Ensemble" (2015). Equilibrium Statistical Physics. Paper 3. http://digitalcommons.uri.edu/equilibrium_statistical_physics/3

This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in Equilibrium Statistical Physics by an authorized administrator of DigitalCommons@URI. For more information, please contact [email protected]. Contents of this Document [ttc12]

12. Grandcanonical Ensemble

• Grandcanonical ensemble. [tln60] • Classical ideal gas (grandcanonical ensemble). [tex94] • Density fluctuations and compressibility. [tln61] • Density fluctuations in the grand canonical ensemble. [tex95] • Density fluctuations and compressibility in the classical ideal gas. [tex96] • Energy fluctuations and thermal response functions. [tex103] • Microscopic states of quantum ideal gases. [tln62] • Partition function of quantum ideal gases. [tln63] • Ideal quantum gases: grand potential and thermal averages. [tln64] • Ideal quantum gases: average level occupancies. [tsl35] • Occupation number fluctuations. [tex110] • Density of energy levels for ideal quantum gas. [tex111] • Maxwell-Boltzmann gas in D dimensions. [tex112] Grandcanonical ensemble [tln60]

Consider an open classical system (volume V , temperature T , chemical potential µ). The goal is to determine the thermodynamic potential Ω(T, V, µ) pertaining to that situation, from which all other thermodynamic properties can be derived. A quantitative description of the grandcanonical ensemble requires a set of phase spaces ΓN ,N = 0, 1, 2,... with probability densities ρN (X). The interaction Hamiltonian for a system of N particles is HN (X). ∞ Z X 6N Maximize Gibbs entropy S = −kB d X ρN (X) ln[CN ρN (X)] N=0 Γ subject to the three constraints

∞ Z X 6N • d X ρN (X) = 1 (normalization), N=0 ΓN ∞ Z X 6N • d X ρN (X)HN (X) = hHi = U (average energy), N=0 ΓN ∞ Z X 6N • d X ρN (X)N = hNi = N (average number of particles). N=0 ΓN

Apply calculus of variation with three Lagrange multipliers:

" ∞ Z # X 6N δ d X {−kBρN ln[CN ρN ] + α0ρN + αU HN ρN + αN NρN } = 0 N=0 ΓN ∞ Z X 6N ⇒ d XδρN {−kB ln[CN ρN ] − kB + α0 + αU HN + αN N} = 0 N=0 ΓN 1 α0 αU αN ⇒ {· · · } = 0 ⇒ ρN (X) = exp − 1 + HN (X) + N . CN kB kB kB

Determine the Lagrange multipliers α0, αU , αN : ∞ Z α0 X 1 6N αU αN exp 1 − = d X exp HN (X) + N ≡ Z, kB CN kB kB N=0 ΓN ∞ Z X 6N d X ρN (X){· · · } = 0 ⇒ S − kB + α0 + αU U + αN N = 0 N=0 ΓN 1 α k ⇒ U + S + N N = B ln Z. αU αU αU 1 1 µ Compare with U − TS − µN = −pV = Ω ⇒ α = − , α = . U T N T

Grand potential: Ω(T, V, µ) = −kBT ln Z = −pV. ∞ X 1 Z 1 Grand partition function: Z = d6N X e−βHN (X)+βµN , β = . CN kBT N=0 ΓN 1 −βHN (X)+βµN Probability densities: ρN (X) = e . ZCN

Grandcanonical ensemble in quantum mechanics: 1 Z = Tr e−β(H−µN), ρ = e−β(H−µN), Ω = −k T ln Z. Z B

Derivation of thermodynamic properties from grand potential:

∂Ω ∂Ω ∂Ω S = − , p = − , N = hNi = − . ∂T V,µ ∂V T,µ ∂µ T,V

Relation between canonical and grandcanonical partition functions:

∞ ∞ X µN/kB T X N µ/kB T Z = e ZN = z ZN , z ≡ e (fugacity). N=0 N=0

Open system of indistinguishable noninteracting particles:

∞ 1 X 1 ˜ Z = Z˜N ,Z = zN Z˜N = ezZ N N! N! N=0 ˜ ⇒ Ω = −kBT ln Z = −kBT zZ.

Thermodynamic properties of the classical ideal gas in the grandcanonical ensemble are calculated in exercise [tex94].

2 [tex94] Classical ideal gas (grandcanonical ensemble)

PN 2 Consider a classical ideal gas [HN = l=1(pl /2m)] in a box of volume V in equilibrium with heat and particle reservoirs at temperature T and chemical potential µ, respectively. 3 (a) Show that the grand partition function is Z = exp(zV/λT ), where z = exp(µ/kBT ) is the p 2 fugacity, and λT = h /2πmkBT is the thermal wavelength. (b) Derive from Z the grand potential Ω(T, V, µ), the entropy S(T, V, µ). the pressure p(T, V, µ), and the average particle number hNi = N (T, V, µ). 3 (c) Derive from these expressions the familiar results for the internal energy U = 2 N kBT , and the ideal gas equation of state pV = N kBT .

Solution: Density ﬂuctuations and compressibility [tln61]

Average number of particles in volume V :

∞ X 1 Z 1 ∂Z 1 ∂ N = hNi = d6N X Ne−βHN (X)+βµN = = ln Z. ZCN Zβ ∂µ β ∂µ N=0 ΓN Fluctuations in particle number (in volume V ):

2 2 2 2 2 1 ∂ Z 1 ∂Z 1 ∂ ln Z 1 ∂(βhNi) ∂N hN i−hNi = 2 2 − = 2 2 = 2 = kBT . Zβ ∂µ Zβ ∂µ β ∂µ β ∂µ ∂µ TV

Here we use Z = Z(β, V, µ). V S ∂µ V ∂p Gibbs-Duhem: dµ = dp − dT ⇒ = . N N ∂(V/N ) T N ∂(V/N ) T ∂ ∂N ∂ N 2 ∂ For V = const: = = − . ∂(V/N ) ∂(V/N ) ∂N V ∂N ∂ ∂V ∂ ∂ For N = const: = = N . ∂(V/N ) ∂(V/N ) ∂V ∂V

2 N ∂µ ∂p ∂µ V −1 ⇒ − = V ⇒ = 2 κT . V ∂N TV ∂V T N ∂N TV N 1 ∂V Compressibility: κT ≡ − . V ∂p T N N 2 Fluctuations in particle number: hN 2i − hNi2 = k T κ . V B T

An alternative expression for hN 2i − hNi2 is calculated in exercise [tex95]. The density fluctuations for a classical ideal gas are calculated in exercise [tex96]. At the critical point of a liquid-gas transition, the isotherm has an inflec- tion point with zero slope (∂p/∂V = 0), implying κT → ∞. The strongly enhanced density fluctuations are responsible for critical opalescence. [tex95] Density fluctuations in the grandcanonical ensemble

Consider a system of indistinguishable particles in the grandcanonical ensemble. Derive the following two expressions for the ﬂuctuations in the number of particles N for an open system of volume V in equilibrium with heat and particle reservoirs at temperature T and chemical potential µ, respectively: ∂ ∂ ∂2p hN 2i − hNi2 = z z ln Z = k TV , ∂z ∂z B ∂µ2 where z = exp(µ/kBT ) is the fugacity, p(T, V, µ) = −(∂Ω/∂V )T µ = −Ω/V is the pressure, and Ω(T, V, µ) = −kBT ln Z is the grand potential.

Solution: [tex96] Density ﬂuctuations and compressibility of the classical ideal gas

(a) Use the results of [tex94] and [tex95] to show that the variance of the number of particles in a classical ideal gas (open system) is equal to the average number of particles:

hN 2i − hNi2 = hNi = N .

(b) Use this result to show that the isothermal compressibility of the classical ideal gas is κT = 1/p.

Solution: [tex103] Energy ﬂuctuations and thermal response functions

(a) Show that the following relation holds between the energy ﬂuctuations in the microscopic ensemble and the heat capacity of a system described by a microscopic Hamiltonian H:

2 2 h(H − hHi) i = kBT CV .

(b) Prove the following relation in a similar manner: 3 2 4 ∂CV 3 h(H − hHi) i = kB T + 2T CV . ∂T V (c) Determine the relative ﬂuctuations as measured by the quantities h(H − hHi)2i/hHi2 and h(H − hHi)3i/hHi3 for the classical ideal gas with N atoms.

Solution: Microscopic states of ideal quantum gases [tln62]

N ˆ X ˆ Hamiltonian: HN = h`. `=1 ˆ 1-particle eigenvalue equation: h`|k`i = `|k`i. ˆ N-particle eigenvalue equation: HN |k1,..., kN i = EN |k1,..., kN i.

N X 2k2 Energy: E = , = ~ ` . N ` ` 2m `=1

N-particle product eigenstates: |k1,..., kN i = |k1i ... |kN i.

(S) Symmetrized states for bosons: |k1,..., kN i .

(S) 1 • N = 2: |k1, k2i = √ (|k1i|k2i + |k2i|k1i). 2

(A) Antisymmetrized states for fermions: |k1,..., kN i .

(A) 1 • N = 2: |k1, k2i = √ (|k1i|k2i − |k2i|k1i). 2

Occupation number representation: |k1,..., kN i ≡ |n1, n2,...i.

Here k1 represents the wave vector of the ﬁrst particle, whereas n1 refers to the number of particles in the ﬁrst 1-particle state.

∞ ˆ X • energy: H|n1, n2,...i = E|n1, n2,...i,E = nkk. k=1 ∞ ˆ X • number of particles: N|n1, n2,...i = N|n1, n2,...i,N = nk. k=1

`: energy of particle `. k: energy of 1-particle state k.

Allowed occupation numbers:

• bosons: nk = 0, 1, 2,...

• fermions: nk = 0, 1. Partition function of ideal quantum gases [tln63]

0 ∞ ! X X Canonical partition function: ZN = σ(n1, n2,...) exp −β nkk .

{nk} k=1

0 ∞ X X : sum over all occupation numbers compatible with nk = N.

{nk} k=1

The statistical weight factor σ(n1, n2,...) is diﬀerent for fermions and bosons:

• Bose-Einstein statistics: σBE(n1, n2,...) = 1 for arbitrary values of nk. 1 if all n = 0, 1 • Fermi-Dirac statistics: σ (n , n ,...) = k . FD 1 2 0 otherwise

What is the statistical weight factor for the Maxwell-Boltzmann gas?

∞ !N 0 1 1 1 N! n n ˜N X −βk X −β1 1 −β2 2 ZN = Z = e = e e ··· N! N! N! n1!n2! ... k=1 {nk} 0 ∞ ! X 1 X = exp −β nkk . n1!n2! ... {nk} k=1

1 • Maxwell-Boltzmann statistics: σMB(n1, n2,...) = . n1!n2! ... Grandcanonical partition function:

∞ ∞ ! X N X X ⇒ Z = z ZN = σ(n1, n2,...) exp −β nk(k − µ) ,

N=0 {nk} k=1

∞ ! N βµ N X where we have used z = (e ) = exp βµ nk . k=1

∞ ∞ ∞ ! ∞ −1 X X X Y −βk • ZBE = ··· exp −β nk(k − µ) = 1 − ze .

n1=0 n2=0 k=1 k=1 1 1 ∞ ! ∞ X X X Y −βk • ZFD = ··· exp −β nk(k − µ) = 1 + ze .

n1=0 n2=0 k=1 k=1 ∞ ∞ ∞ ! ∞ 1 X X X Y −βk • ZMB = ··· exp −β nk(k − µ) = exp ze . n1!n2! ... n1=0 n2=0 k=1 k=1 Ideal quantum gases: grand potential and thermal averages [tln64]

Grand potential: Ω(T, V, µ) = −kBT ln Z = U − TS − µN = −pV.

∞ ∞ X −βk X −β(k−µ) • ΩMB = −kBT ze = −kBT e , k=1 k=1 ∞ ∞ X −βk X −β(k−µ) • ΩBE = kBT ln 1 − ze = kBT ln 1 − e , k=1 k=1 ∞ ∞ X −βk X −β(k−µ) • ΩFD = −kBT ln 1 + ze = −kBT ln 1 + e . k=1 k=1

Parametric representation [a = 1 (FD), a = 0 (MB), a = −1 (BE)]:

∞ pV 1 X ln Z = = ln 1 + aze−βk . kBT a k=1

Average number of particles:

∞ ∞ ∂Ω 1 ∂ ln Z X 1 X N = − = = = hnki. ∂µ β ∂µ z−1eβk + a T,V T,V k=1 k=1

Average energy (internal energy):

∞ ∞ ∂ ln Z X k X U = − = = khnki. ∂β z−1eβk + a z,V k=1 k=1

Average occupation number of energy level k:

−1 ∂ ln Z 1 hnki = −β = β( −µ) . ∂k e k + a

Fluctuations in occupation number [tex110]:

2 2 2 −2 ∂ ln Z hnki − hnki = β 2 . ∂k Average occupation numbers for MB, FD, and BE gases [tsl35]

Average occupation number of energy level ǫk: 1 hnki = eβ(ǫk−µ) + a • a = 1: Fermi-Dirac gas, • a = 0: Maxwell-Boltzmann gas, • a = −1: Bose-Einstein gas.

Range of 1-particle energies: ǫk ≥ 0. BE gas restriction: µ ≤ 0 ⇒ 0 ≤ z ≤ 1.

2.5

1.5 Maxwell-Boltzmann Bose-Einstein > k

0.5 Fermi-Dirac

0 -2 -1 0 1 2 3 β ε µ ( k - )

The BE and FD gases are well approximated by the MB gas provided the 2 thermal wavelength λT = ph /2πmkBT is small compared to the average interparticle distance:

β(ǫk − µ) ≫ 1 ⇒ − βµ ≫ 1 ⇒ z ≪ 1.

1/3 [tex94] for D = 3 : ⇒ λT ≪ (V/N ) . [tex110] Occupation number ﬂuctuations

Consider an ideal quantum gas speciﬁed by the grand partition function Z. Start from the expressions 1 ∂2Z 1 ∂Z 2 1 ∞ 2 2 −2 −1 X −βk hnki − hnki = β 2 − β , ln Z = ln(1 + aze ), Z ∂ Z ∂k a k k=1 where a = +1, 0, −1 represent the FD, MB, and BE cases, respectively, to derive the following result for the relative ﬂuctuations in the occupation numbers:

2 2 hnki − hnki 1 2 = − a. hnki hnki

Note that in the BE (FD) statistics, these ﬂuctuations are enhanced (suppressed) relative to those in the MB statistics.

Solution: [tex111] Density of energy levels for ideal quantum gas

Consider a nonrelativistic ideal quantum gas in D dimensions and conﬁned to a box of volume V = LD with rigid walls. Show that the density of energy levels is

LD m D/2 D() = D/2−1. Γ(D/2) 2π~2

Solution: [tex112] Maxwell-Boltzmann gas in D dimensions

From the expressions for the grand potential and the density of energy levels of an ideal Maxwell- Boltzmann gas in D dimensions and conﬁned to a box of volume V = LD with rigid walls,

X LD m D/2 Ω(T, V, µ) = −k T e−β(k−µ),D() = D/2−1, B Γ(D/2) 2π 2 k ~ derive the familiar results pV = N kBT for the equation of state, CV N = (D/2)N kB for the heat capacity, and pV (D+2)/D = const for the adiabate at ﬁxed N .

Solution: