Calculation of the inner energy

3.2 Calculation of the inner energy

From Eq. (2.45) we know the grand canonical potential

$Ω (T, V, μ) = \pm k T \sum_{α} ln (1 \mp exp (- \frac{ϵ_{α} - μ}{k T}))$

(3.4)

$α$ indicates independent states, the plus/minus sign depend on the particles to be Fermion or Bosons. For a continuous system the sum changes into an integral:
Let

$F_{β μ} (ϵ) : = \pm k T ln (1 \mp exp (- \frac{ϵ - μ}{k T}))$

(3.5)

and

$f_{β μ} (ϵ) : = \frac{\partial F_{β μ} (ϵ)}{\partial ϵ} = \frac{1}{exp (\frac{ϵ - μ}{k T}) \mp 1} = - \frac{\partial F_{β μ} (ϵ)}{\partial μ} .$

(3.6)

Obviously we find

$Ω (T, V, μ) = \int_{- \infty}^{+ \infty} D (ϵ) F_{β μ} (ϵ) d ϵ,$

(3.7)

$N (T, V, μ) = \int_{- \infty}^{+ \infty} D (ϵ) f_{β μ} (ϵ) d ϵ,$

(3.8)

and

$E (T, V, μ) = \int_{- \infty}^{+ \infty} ϵ D (ϵ) f_{β μ} (ϵ) d ϵ .$

(3.9)

$D (ϵ)$ is the density of states.

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