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Bosons and fermions
Using the first and the second law of Thermodynamics, we will set the
expressions of the Bose-Einstein and Fermi-Dirac functions, using the
chemical potential.
1. First law:
dQ = dU + dW
U is the internal energy of the system E.
E = Σ ni Ei. Therefore:
dU = dE = Σ dni Ei + Σ ni dEi.
For a change in the volume then in the number (mass) of the gas, we have:
dW = - PdV , then:
dW = PdV = Σ dniEni
dQ = Σ ni dEi
2. Second law:
For a system that is a Grand canonical ensemble, where nothing is
fixed, we add Q (heat) and N (particles), then:
dQ + dN = dU
Therefore: dQ + μdN = dU + dW , where μ is called the Chemical potential. Then:
dQ = TdS = dU + dW - μdN
For the fermions, we have seen :
Σ dni {ln[(gi - ni)/ ni] + α - βEi]} = 0
or
Σ dni {ln[(gi - ni)/ ni]} = Σ dni {- α + βEi]}
With:
Σ dni [- α + βEi] = - αN + βΣ dni Ei]
Since : dni Ei = d(ni Ei) - ni dEi, we can write:
Σ dni [- α + βEi] = - α dN + βΣ {d(ni Ei) - ni dEi}
From S = k ln(Ω) and then dS = k dln(Ω), we have:
dS = k Σ dni {ln[(gi - ni)/ ni]}
= k Σ dni {- α + βEi]}
Then dQ = TdS = kT Σ dni {ln[(gi - ni)/ ni]}
= kT Σ dni {- α + βEi]}
= - α kT dN + kT βΣ {d(ni Ei) - ni dEi}
= dU + dW - μdN
Since kT = 1/β then:
dU = kT βΣ d(ni Ei)= Σ d(ni Ei) = dE = dU
dW = - kT βΣ nidEi = - Σ nidEi
μdN = α kT dN
Thus:
α = μ/kT
Finally,
3. Expressions of the population functions
The Fermi-Dirac population numbers is :
ni = gi/(exp [ - (μ - Ei)/kT] + 1)
The Bose-Einstein population numbers is :
ni = gi/(exp [ - (μ - Ei)/kT] - 1)
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