1. Grand Canonical Ensemble
PHY 471, Statistical Physics, 2007
Lecture 09.
Grand Canonical Ensemble
Mahn-Soo Choi (Korea University)
F. Reif, Fundamentals of Statistical and Thermal Physics (1965)
Chapter 6.

2. Thermal and Chemical Equilibrium of the “System” A with the “Bath” B
"Universe"
"System"
"Environment (Bath)"
The probability PA (EA , NA ) that the “system” has the energy
EA and the particle number NA ?
The probability Pα for the “system” to be found in the
microstate α?

3. Equilibrium Condition The probability distribution PA (EA , NA ):
Γ(EA , NA |E , N )
Γ(E , N )
PA (EA , NA ) =
ΓA (EA , NA )ΓB (E − EA , N − NA )
Γ(E , N ) =
PA (EA , NA ) and hence log PA (EA , NA ) has a sharp peak at
EA = E¯A and NA = N¯ A in equilibrium.
PA(EA)
ΓA(EA)
ΓB (E − EA) EA

4. ∂ 1 1 ∂ The equilibrium condition is thus ∂ EA ⇒ TA TB log PA =0 =
EA =E¯A ∂
∂ NA
log PA =0 ⇒
µA = µB
NA =NA

5. 1 Grand Canonical Ensemble The probability Pα ΓB (E − Eα , N − Nα )
kB log Pα
∂ SB
∂ EB
∂ SB
∂ NB
≈ SB (E , N ) −
EA − NA
EB =E
NB =N Eα T
µNα T
= SB (E , N ) − +
Canonical distribution 1 ZG
Eα − µNα
kB T
Pα =
exp −
where ZG is the normalization constant:
Eα − µNα
kB T
ZG =
exp − . α

6. −kB T log ZG = E − TS − µN = G Grand Partition Function ZG
ZG is more than just a normalization constant.
From the expression for the entropy
S = −kB E T µN T
Pα log Pα = −
+ kB log ZG α
−kB T log ZG = E − TS − µN = G
the Gibbs free energy!

7. Thermodynamics with Grand Canonical Ensemble
Partition function ZG
G (T , µ, V , · · · ) = −kB T log ZG (T , µ, V , · · · )
dG = −SdT − Nd µ − PdV + · · ·
Thermodynamic quantities
∂ G
∂ T
∂ G ∂µ
∂ G
∂ V
S = − ,
N = − ,
P = −
, ···

8. S ∂ ∂ S More about the Entropy kB = − Pα log Pα = (E − µN ) + log ZG∂β
E − µN = −β
log ZG S kB ∂ ∂β =
1 − β
log ZG

9. The elementary consideration leads to
More about Pressure
The elementary consideration leads to
∂ G
∂ V
P = −
Since the Gibbs free energy is an extensive quantity, it should
be directly proportional to V . It means that
∂ G
∂ V
= constant.
The constant should be the pressure P :
G = −kB T log ZG = PV .

10. A(E , N , V , · · · ) = −kB T log Z (T , N , V , · · · )
Summary of Ensembles
Microcanonical ensemble
Γ(E , N , V , · · · ) =
δ(E − Eα ) α
S (E , N , V , · · · ) = kB log Γ(E , N , V , · · · )
Canonical ensemble
Z (T , N , V , · · · ) =
exp − Eα
kB T α
A(E , N , V , · · · ) = −kB T log Z (T , N , V , · · · )
Grand canonical ensemble
Eα − µNα
kB T
ZG (T , N , V , · · · ) =
exp − α
G (E , µ, V , · · · ) = −kB T log FG (T , µ, V , · · · )