Computational Physics

174 Classical equilibrium statistical mechanics

the pressurePis taken over by the total magnetic momentM. The other relevant
thermodynamic quantities follow from the definition ofG(N,P,T):

μ=

(

∂G

∂N

)

P,T

V=

(

∂G

∂P

)

N,T

S=−

(

∂G

∂T

)

P,N

. (7.17)

If the volume is again fixed, but the number of particles is allowed to vary, we
obtain thegrand canonical ensembleaverage:

〈A〉=

1

ZG

∑

N

eβμN

1

N!

∑

X

e−βH(X)A(X) (7.18a)

ZG(μ,V,T)=

∑

N

eβμN

1

N!

∑

X

e−βH(X). (7.18b)

Here,μis the chemical potential for the addition or removal of a particle.
ZG(μ,V,T) should not be confused with the canonical partition function
Z(N,V,T); it can be expressed in terms of the latter as

ZG(μ,V,T)=

∑

N

eβμNZ(N,V,T). (7.19)

ZGdefines thegrand canonical potentialG, analogous to similar definitions for
the other ensembles:

G(μ,V,T)=−kBTlnZG(μ,V,T). (7.20)

In equilibrium, this potential assumes its minimum value for fixedμ,TandV.
From the definition ofZGand from the expression for the average values in the
grand canonical ensemble, it follows that

G(μ,V,T)=F−μN. (7.21)

The internal energy can be written in terms of the variablesS,VandN and it
satisfies the Gibbs–Duhem equation[4]

E(S,V,N)=TS−PV+μN (7.22)

so that we have
G(μ,V,T)=−PV. (7.23)
From the grand canonical potential we can derive thermodynamic quantities:

N=−

(

∂G

∂μ

)

V,T

P=−

(

∂G

∂V

)

μ,T

S=−

(

∂G

∂T

)

V,μ

. (7.24)

Expectation values of thermodynamic quantities are calculated either as
ensemble averages or as integrals over phase space. As an example of an ensemble
average, consider the internal energy. The expectation value of this quantity in the