Chapter 10 Grand canonical ensemble 10.1 Grand canonical partition function The grand canonical ensemble is a generalization of the canonical ensemble where the restriction to a definite number of particles is removed. This is a realistic representation when then the total number of particles in a macroscopic system cannot befixed. Heat and particle reservoir. Consider a sys- temA 1 in a heat and particle reservoirA 2. The two systems are in equilibrium with the thermal equilibrium. – Thermal equilibrium results form the ex- change of heat. The two temperature 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 energy 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. Microcanonical ensemble. 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 density of states. Sub-macroscopic particle exchange. The total entropy 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 internal energy. 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 phase space 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. Chemical potential. 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= P V, 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 thermodynamic limit. 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.