2.5 Calculation of the grand canonical ensemble

Maximize

S = -k ipi ln(pi) (2.25)

with the restrictions

0 = ipiUi - U, and0 = ipi - 1, and0 = ipiNi - N. (2.26)

Introducing the Lagrange parameters α, β, and γ the variation of the function

δ S- kα ipi - 1 - kβ ipiUi - U - kγ ipiNi - N = 0 (2.27)

without restrictions leads to

- ln(pi) - 1 - α - βUi - γNi = 0. (2.28)

Defining again

1 Z = exp(-1 - α) (2.29)

we find

pi = 1 Z exp(-βUi - γNi)andZ(β,V,γ) = i exp(-βUi - γNi). (2.30)

We get

U = ipiUi = i exp(-βUi - γNi)Ui i exp(-βUi - γNi) = - ln(Z) β := U(β,V,γ) (2.31)

and

N = ipiNi = i exp(-βUi - γNi)Ni i exp(-βUi - γNi) = - ln(Z) γ := N(β,V,γ) (2.32)

i.e.

S = k ln(Z) + βkU + γkN. (2.33)

The total derivative is:

dS k = ln(Z) β dβ + ln(Z) γ dγ + ln(Z) V dV + Udβ + βdU + Ndγ + γdN = ln(Z) V dV + βdU + γdN (2.34)

So

S = S(V,N,U) (2.35)

and S is the Legendre transformed of k ln(Z).
Let

S U := 1 Tand S N := -μ T. (2.36)

So

β = 1 kT, andγ = - μ kT. (2.37)

Following again the procedure for the calculation of the free energy we find

Ω = U - μN - TS (2.38)

and

Ω(T,V,μ) = -kT ln(Z(T,V,μ)). (2.39)


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© J. Carstensen (Stat. Meth.)