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Statistical Mechanics

Fermi-Dirac Distribution

The grand canonical potential for non-interacting fermions:

Ω[β,μ]=−∑i1βlog⁡(∑N=01e−β(Nϵi−Nμ))=\Omega[\beta, \mu] = -\sum_i {1\over\beta} \log\left(\sum_{N=0}^1 e^{-\beta\left(N\epsilon_i - N\mu\right)}\right) =
(1)
=−∑i1βlog⁡(1+e−β(ϵi−μ))== -\sum_i {1\over\beta} \log\left(1 + e^{-\beta\left(\epsilon_i - \mu\right)}\right) =
(2)
=∑iΩi ,= \sum_i \Omega_i\,,
(3)

Where Ωi\Omega_i is the grand canonical potential for a single particle state:

Ωi=−1βlog⁡(1+e−β(ϵi−μ)) .\Omega_i = - {1\over\beta} \log\left(1 + e^{-\beta\left(\epsilon_i - \mu\right)}\right)\,.
(4)

The total (average) number of particles in the state ii is:

Ni=−(∂Ωi∂μ)T,V=N_i = -\left(\partial\Omega_i\over\partial\mu\right)_{T,V} =
(5)

=−∂(−1βlog⁡(1+e−β(ϵi−μ)))∂μ== -{\partial \left(- {1\over\beta} \log\left(1 + e^{-\beta\left(\epsilon_i - \mu\right)}\right) \right)\over\partial\mu} =
(6)

=1β11+e−β(ϵi−μ)e−β(ϵi−μ)β== {1\over\beta} {1\over 1 + e^{-\beta(\epsilon_i - \mu)}} e^{-\beta(\epsilon_i - \mu)} \beta =
(7)

=1eβ(ϵi−μ)+1 .= {1\over e^{\beta(\epsilon_i - \mu)} + 1} \,.
(8)

The total (average) number of particles in the full system is:

N=−(∂Ω∂μ)T,V=−(∂∑iΩi∂μ)T,V=∑i(−(∂Ωi∂μ)T,V)=N = -\left(\partial\Omega\over\partial\mu\right)_{T,V} = -\left(\partial\sum_i\Omega_i\over\partial\mu\right)_{T,V} = \sum_i \left(-\left(\partial\Omega_i\over\partial\mu\right)_{T,V}\right) =
(9)

=∑iNi=∑i1eβ(ϵi−μ)+1 .= \sum_i N_i = \sum_i {1\over e^{\beta(\epsilon_i - \mu)} + 1} \,.
(10)

Bose-Einstein Distribution

The grand canonical potential for non-interacting bosons:

Ω[β,μ]=∑i1βlog⁡(1−e−β(ϵi−μ))=∑iΩi ,\Omega[\beta, \mu] = \sum_i {1\over\beta} \log\left(1 - e^{-\beta\left(\epsilon_i - \mu\right)}\right) = \sum_i \Omega_i\,,
(11)

Where Ωi\Omega_i is the grand canonical potential for a single particle state:

Ωi=1βlog⁡(1−e−β(ϵi−μ)) .\Omega_i = {1\over\beta} \log\left(1 - e^{-\beta\left(\epsilon_i - \mu\right)}\right)\,.
(12)

The total (average) number of particles in the state ii is:

Ni=−(∂Ωi∂μ)T,V=N_i = -\left(\partial\Omega_i\over\partial\mu\right)_{T,V} =
(13)

=−∂(1βlog⁡(1−e−β(ϵi−μ)))∂μ== -{\partial \left({1\over\beta} \log\left(1 - e^{-\beta\left(\epsilon_i - \mu\right)}\right) \right)\over\partial\mu} =
(14)

=−1β11−e−β(ϵi−μ)(−e−β(ϵi−μ))β== -{1\over\beta} {1\over 1 - e^{-\beta(\epsilon_i - \mu)}} (-e^{-\beta(\epsilon_i - \mu)}) \beta =
(15)

=1eβ(ϵi−μ)−1 .= {1\over e^{\beta(\epsilon_i - \mu)} - 1} \,.
(16)

The total (average) number of particles in the full system is:

N=−(∂Ω∂μ)T,V=−(∂∑iΩi∂μ)T,V=∑i(−(∂Ωi∂μ)T,V)=N = -\left(\partial\Omega\over\partial\mu\right)_{T,V} = -\left(\partial\sum_i\Omega_i\over\partial\mu\right)_{T,V} = \sum_i \left(-\left(\partial\Omega_i\over\partial\mu\right)_{T,V}\right) =
(17)

=∑iNi=∑i1eβ(ϵi−μ)−1 .= \sum_i N_i = \sum_i {1\over e^{\beta(\epsilon_i - \mu)} - 1} \,.
(18)
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