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Chapter 10

Grand

10.1 Grand canonical partition function

The is a generalization of the canonical ensemble where the restriction to a definite number of is removed. This is a realistic representation when then the total number of particles in a macroscopic system cannot befixed. and reservoir. Consider a sys- temA 1 in a heat and particle reservoirA 2. The two systems are in equilibrium with the . – Thermal equilibrium results form the ex- change of heat. The two are then equal:T=T 1 =T 2 – The equilibrium with respect to particle exchange leads to identical chemical poten- tials:µ=µ 1 =µ 2.

Energy and particle conservation. We assume that the systemA 2 is much larger than the systemA 1, i.e., that E E ,N N , 2 � 1 2 � 1 with N1 +N 2 =N = const.E 1 +E 2 =E = const. whereN andE are the particle number and the of the total systemA=A 1 +A 2. Hamilton function. The overall Hamilton function is defined as the sum of the Hamilton functions ofA 1 andA 2:

H(q, p) =H 1(q(1), p(1),N1) +H 2(q(2), p(2),N2).

For the above assumption to be valid, we neglect interactions among particles inA 1 and A2: H12 = 0.

117 118 CHAPTER 10. GRAND CANONICAL ENSEMBLE

This is a valid assuming for most macroscopic systems. . Since the total systemA is isolated, its distribution function is given in the microcanonical ensemble as 1 ρ(q, p) = δ(E H H ), Ω(E,N) − 1 − 2

as in (9.1), with t Ω(E,N)= d3N q d3N pδ(E H H ) − 1 − 2 � being the of states. Sub-macroscopic particle exchange. The total of the combined system is given by the microcanonical expression

Ω(E,N)Δ S=k ln ,Γ (N)=h 3N N !N !, (10.1) B Γ (N) 0 1 2 � 0 � whereΔ is the width of the energy shell. Compare ( 9.4) and Sect. 9.4.

The reason thatΓ 0(N) N 1!N2! in (10.1), and not N!, stems from the assumption that there is no particles exchange∼ on the macroscopic level.∼ This is in line with the observation made in Sect. 9.6 that energyfluctuations, i.e. the exchange of energy between a system and the heat reservoir, scale like 1/√N relatively to the . We come back to this issue in Sect. 10.3.2.

Integrating out the reservoir. We are now interested in the systemA 1. As we did for the canonical ensemble, we integrate the total probability densityρ(q, p) over the of the reservoirA 2. We obtain

ρ (q(1), p(1),N ) dq(2) dp(2)ρ(q, p) 1 1 ≡ � dq(2) dp(2)δ(E H H ,N) = − 1 − 2 Ω(E,N) � Ω (E H ,N N ) 2 − 1 − 1 (10.2) ≡ Ω(E,N)

in analogy to (9.2). Expanding inE andN . WithE E andN N, we can approximate the slowly 1 1 1 � 1 � varying logarithm ofΩ 2(E2,N2) aroundE 2 =E andN 2 =N and thus obtain:

Ω (E E ,N N )Δ S (E E ,N N ) =k ln 2 − 1 − 1 2 − 1 − 1 B Γ (N) � 0 � ∂S ∂S =S (E,N) E 2 N 2 . 2 1 ∂E 1 ∂N − 2 N2 � E2 =E − 2 E2 � E2 =E � � � � � � � N2 =N � N2 =N � � � � 10.1. GRAND CANONICAL PARTITION FUNCTION 119

Using (5.14), T dS= dU+ P dV µdN, the derivatives may be substituted by −

∂ Ω2(E,N)Δ ∂S2 1 kb ln = = (10.3) ∂E Γ0(N) ∂E2 N2 =N T � � � � E2 =E

and ∂ Ω2(E,N)Δ ∂S2 µ kb ln = = . (10.4) ∂N Γ0(N) ∂N2 N2 =N − T � � � � E2 =E

Expansion of the probability density.( 10.3) and (10.4) allow to expandΩ 2(E E ,N N ) as − 1 − 1 Ω (E E ,N N )Δ Ω (E,N)Δ E N µ k ln 2 − 1 − 1 =k ln 2 1 + 1 , B Γ (N) B Γ (N) − T T � 0 � � 0 �

which leads viaE 1 =H 1 and (10.2) to a probability distribution of the form

ρ (q(1), p(1),N ) exp H (q(1), p(1)) µN /(k T) . 1 1 ∼ − 1 − 1 B � � � � Grand canonical partition function. The constant of proportionality for the proba- bility distribution is given by the grand canonical partition function = (T, V, µ), Z Z

∞ 3N 3N d q d p β[H(q,p,N) µN] (T, V, µ) = e− − , (10.5) Z h3N N! N=0 � �

where we have dropped the index to thefirst system substitutingρ,N,q andp forρ 1, N1,q(1) andp(1). The partition function normalizes the distribution function

1 1 β[H(q,p,N) µN] ρ(q, p, N)= e− − (10.6) h3N N! (T, V, µ) Z to 1: ∞ d3N q d3N pρ(q, p, N)=1 N=0 � � Note that the normalization factorh 3N N! in (10.5) and (10.6) cancel each other. It is in any can not possible to justify this factor within classical mechanics, as discussed previously in Sect. 8.2. Internal Energy. From the definition (10.5) of the grand canonical potential function it follows that internal energyU= H is given by � � ∂ U µN= ln (T, V, µ) . (10.7) − − ∂β Z 120 CHAPTER 10. GRAND CANONICAL ENSEMBLE 10.2 Grand canonical potential

In Sect. 5.4 we defined the grand canonical potential∗ as

Ω(T, V, µ) =F(T,V,N) µN ,Ω= PV, − − and showed with Eq. (5.21) that

∂ ∂ U µN= βΩ = ln (T, V, µ), (10.8) − ∂β − ∂β Z � � where we have used in the last step the representation (10.7) ofU µN=∂ ln /∂β. − Z Grand canonical potential. From (10.8) we may determine (up-to a constant) the grand canonical potential with

βΩ Ω(T, V, µ) = k T ln (T, V, µ) , =e − (10.9) − B Z Z as the logarithm of the grand canonical potential. Note the analog to the relationF= k T lnZ valid within the canonical ensemble. − B N Calculating with the grand canonical ensemble. To summarize, in order to obtain the thermodynamic properties of our system in contact with a heat and particle reservoir, we do the following:

1 - calculate the grand canonical partition function:

∞ 3N 3N d q d p β(H µN) = e− − ; Z h3N N! N=0 � � 2 - calculate the grand canonical potential:

Ω= k T ln = P V, dΩ= SdT P dV Ndµ; − B Z − − − −

3 - calculate the remaining thermodynamic properties through the equations:

∂Ω S= − ∂T � �V,µ ∂Ω Ω P= = − ∂V − V � �T,µ ∂Ω N= − ∂µ � �T,V and the remaining thermodynamic potentials through Legendre transformations.

∗ Don’t confuse the grand canonical potentialΩ(T, V, µ) with the density of microstatesΩ(E)! 10.3. FUGACITY 121 10.3 Fugacity

By using the definition of the partition function in the canonical ensemble,

1 3N 3N βH Z = d q d p e− , N h3N N! � we can rewrite the partition function in the grand canonical ensemble as

∞ ∞ (T, V, µ) = eµNβ Z (T)= zN Z (T) , (10.10) Z N N N=0 N=0 � � wherez = exp(µ/k T ) is denoted as the fugacity. Equation (10.10) shows that (T, µ) B Z is the discrete Laplace transform ofZ N (T).

10.3.1 Particle distribution function In the canonical ensemble the particle numberN wasfixed, whereas it is a variable in the in the grand canonical ensemble. We define with Z (T) w =e βµN N (10.11) N (T, µ) Z the probability that the system at temperatureT and with chemical potentialµ contains N particles. It is normalized:

∞ ∞ eβµN Z (T) w = N = 1. n (T, µ) N=0 N=0 � � Z Average particle number. That mean particle number N is given by � � ∞ N N∞=0 N z ZN (T,V) N = N wN (T,V)= N , (10.12) � � ∞ z ZN (T,V) N=0 N=0 � � which can be rewritten as �

1 ∂ N = ln (T, V, µ) , (10.13) � � β ∂µ Z � �T,V or, alternatively, as ∂ N =z ln (T, V, µ) , (10.14) � � ∂z Z � �T,V when making use of the definition of fugacityz. . Conversely, the chemical potential for a given average particle numberN= N , � � µ=µ(T,V,N), is obtained by by inverting (10.13). 122 CHAPTER 10. GRAND CANONICAL ENSEMBLE

10.3.2 Particle numberfluctuations We start by noting that from

∂Ω Ω= PV,N= − − ∂µ � �T,V it follows that ∂P 1 ∂Ω N = − = . (10.15) ∂µ V ∂µ V � �T,V � �T,V Particle numberfluctuations. From ( 10.12),

N eβµN Z (T,V) N = N N , � � eβµN Z (T,V) � N N it follows that � ∂N =β N 2 N 2 , (10.16) ∂µ T,V � � − � � � � � � where the second term results from the derivative of the denumerator. Two systems are in equilibrium with respect to the exchange in particle if their chemical potentials are identical,µ=µ 1 =µ 2. It hence makes sense that the the dependence of the particle number on the chemical potential is proportional to the size of the particle numberfluctuations N 2 N 2, which mediate the equilibrium. � � − � � Scaling to the . Using (10.15) we can recast (10.16) into

2 ∂ P β 2 2 2 = N N . (10.17) ∂µ T,V V � � − � � � � � � The left-hand side of (10.17) is intensive, viz it scales likeV 0, since both the pressureP and the chemical potential are intensive. For the right-hand side to scale likeV 0 for large volumeV we hence need that N 2 N 2 1 N 2 N 2 √V, � � − � � , (10.18) � � − � � ∼ � N ∼ √N � where we have used thatV N forfixed . The relative particle numberfluctu- ations hence vanish in the∼ thermodynamic limit; a precondition for the grand-canonical and the canonical ensemble to yield identical results.

10.3.3 Stability conditions

The mechanical stability condition for a system implies that its compressibilityκ T cannot take negative values, i.e., 1 ∂V κ = 0. T − V ∂P ≥ � �T 10.3. FUGACITY 123

In fact, this can be proven within the grand canonical ensemble in the same way as it was proven in Sect. 9.6 that the energyfluctuations in the canonical ensemble fulfill the thermal stability criterion

2 C =k β2 H H 0. (10.19) V B −� � ≥ �� � � Intensive variables. We start be defining the free energy per particle,a= F/N, as

F(T,V,N) a(T, v) v= V/N , (10.20) ≡ N

where thev is the per particle (the specific volume).

–a= F/N is intensive and a function of the intensive variableT . It can therefore not be a independently a function of the volumeV or the particle numberN (which are extensive), only of V/N (which is intensive).

– The circumstance thata is a function of V/N only allows to transform derivative with respect toV and toN into each other, the reason of using this representation.

Chemical potential. With (10.20) wefind

∂F ∂ NF/N ∂a ∂a ∂(V/N) µ= = =a+N =a+N , ∂N � ∂N � ∂N ∂(V/N) ∂N V/N 2 − when using (5.12), namely that dF= SdT P dV+ µdN. This relation� �� yields� − −

∂a ∂µ ∂2a µ=a v , = v . (10.21) − ∂v ∂v − ∂v 2

Note, thatµ is intensive. . For the pressureP , an intensive variable, wefind likewise that

∂F ∂ NF/N ∂a ∂(V/N) P= = = N , − ∂V − � ∂V � − ∂(V/N) ∂V 1/N

which results in � �� �

∂a ∂P ∂2a ∂µ ∂P P= , = , =v , (10.22) − ∂v ∂v − ∂v 2 ∂v ∂v

where we have used in the last step the comparison with (10.21). 124 CHAPTER 10. GRAND CANONICAL ENSEMBLE

Particlefluctuations in intensive variables. Using intensive variables for ( 10.17), β ∂2P(T, v) ∂ ∂P ∂v N 2 N 2 = = , (10.23) V � � − � � ∂µ2 ∂v ∂µ ∂µ � � � � the last relation of (10.22) and (10.15), namely∂P/∂µ=1/v, we obtain ∂2P(T, v) ∂ 1 1 ∂v 1 ∂V 1 ∂v = ,κ = = . ∂µ2 ∂v v v ∂P T − V ∂P − v ∂P � � � � 1/v2 κT − − . Taking� the�� results� � �� together� we obtain

βv 2 V N 2 N 2 κ = N 2 N 2 =β � � − � � . (10.24) T V � � − � � N N � � for the compressibilityκ T .

– The compressibility is strictly positive,κ T 0, which implies that the system constricts when the pressure increases. The basic≥ stability condition. – The compressibility isfinite if the size of thefluctuations N 2 N 2 vanishes in the thermodynamic limit relative to the number of particles� N� −present � � in the � system.

Response vs.fluctuations. The specific heatC V and the compressibilityκ T are mea- sure the response of the system to a change of variables. – A temperature gradientΔT to the heat reservoir leads to a transfer of heat,

ΔQ=C V ΔT,

which is proportional toC V . Compare Sect. 3.2. – An increase in temperature byΔP leads to relative decrease of the volume by ΔV =κ ΔP, V T

which is proportional toκ T .

It is not a coincidence, that both response functions,C V andκ T , as given respectively by (10.19) and (10.24), are proportional to thefluctuations of the involved variables.

A system in which a certain variableA is notfluctuating (and hencefixed) cannot respond to perturbations trying to changeA. The response always involves thefluctuation

2 A A = A 2 A 2 . −� � � � − � � �� � � 10.4. THE IDEAL IN THE GRAND CANONICAL ENSEMBLE 125 10.4 The in the grand canonical ensemble

We consider a mono-atomic gas system in a volumeV in contact with a heat and particle reservoir at temperatureT and chemical potentialµ. Canonical partition function. One calculatesfirst the grand canonical partition func- tion, which we found in Chap. 9 to be

1 V N h ZN (T,V)= ,λ T = , N! λ3 √2πmk T � T � B whereλ T is the thermal wavelength. V Z1(T,V, 1) = 3 , λT (10.25)

Weighted sum over particle number. The grand canonical partition function is

N N ∞ ∞ µN ∞ N 1 V 1 µ/k T V = z Z (T)= e kB T = e B , Z N N! λ3 N! λ3 N=0 N=0 T N=0 T � � � � � � � wherez = exp(βµ) is the fugacity. The result is

V (T, V, µ) = exp eµ/kB T . (10.26) Z λ3 � T � Note that the factorN! comes in handy. The grand canonical potential. The grand canonical potential is then obtained as

V µ/kB T Ω= k BT ln , Ω(T, V, µ) = k BT e 3 . (10.27) − Z − λT

Particle number. Comparing the expression

∂Ω V N= , N(T, V, µ) =e µ/kB T (10.28) − ∂µ λ3 � �T,V T for the particle numberN with ( 10.27) wefind

Ω(T, V, µ) = k TN,N=N(T, V, µ). (10.29) − B This relation holds only for the ideal gas. . Comparing now ∂Ω 1 =P(T, V, µ) =k T eµ/kB T (10.30) −∂V B λ3 � �T,µ T 126 CHAPTER 10. GRAND CANONICAL ENSEMBLE with (10.28) wefind immediately the equation of state

VP=k BTN of the ideal gas Entropy. Evaluating the entropy

∂ V h Ω µ/kB T S= ,Ω(T, V, µ) = k BT e λT = , − ∂T − λ3 √2πmk T � �V,µ T B wefind Ω 3 µ Ω 5 µ S= 1 + Ω − = , − T 2 − k T 2 − T 2 − k T � � B � B � which leads with (10.29),Ω= k TN, to − B 5 µ V S(T, V, µ) =k N ,N=e µ/kB T . (10.31) B 2 − k T λ3 � B � T

Sackur-Tetrode equation. InvertingN for µ/(k BT ) we obtain for the entropy with

V 5 S=k N ln + B Nλ 3 2 � T � the Sackur-Tetrode equation, which coincides with the both the microcanonical and the canonical expressions, (9.20) and (8.25). The three ensembles yield identical results in the thermodynamic limit. 10.4. THE IDEAL GAS IN THE GRAND CANONICAL ENSEMBLE 127

10.4.1 Summary ensembles Phase space integration Thermodynamic potential

Γ Microcanonical Γ(E,V,N)= d3N q d3N p S(E,V,N)=k ln B h3N N! E

∂Γ Ω(E,V,N)= ∂E

= d3N q d3N pδ(E H) − � 3N 3N d q d p βH Canonical Z (T,V)= e− F(T,V,N)= k T lnZ (T,V) N h3N N! − B N ensemble � 1 ∞ βH βF = dEΩ(E)e − Z =e h3N N! N − �0 µNβ Grand canonical (T, µ) = N∞=0 e ZN (T) Ω(T, V, µ) = k BT ln (T, V, µ) ensemble Z − Z PV � βΩ =e − =e kB T Z 128 CHAPTER 10. GRAND CANONICAL ENSEMBLE