35B (Statistical Physics)

A grand canonical ensemble describes a system exchanging both energy and particles with a reservoir at fixed

T

μ

. For a noninteracting fermion level

r

, occupation is

0

1

, so

Z_{r} = 1 + e^{- β (E_{r} - μ)} .

Factorization over levels gives

Z = \prod_{r} Z_{r}

β μ

⟨ n_{r} ⟩ = \frac{1}{e ^{β (E_{r} - μ)} + 1} .

b

The number of momentum states in a shell is

\frac{g _{s} V}{( 2 π ℏ ) ^{d}} S_{d - 1} p^{d - 1} d p .

With

E = p^{2} / (2 m)

this gives

g (E) = g_{s} B V E^{(d - 2) /2}, B = \frac{S _{d - 1} m ( 2 m ) ^{(d - 2) /2}}{( 2 π ℏ ) ^{d}} .

At zero temperature all states up to

E_{F}

are occupied, hence

N = \frac{2 g _{s} B V}{d} E_{F}^{d /2}, E_{F} = (\frac{d N}{2 g _{s} B V})^{2/ d} .

The energy is

U = d N E_{F} / (d + 2)

. Since a nonrelativistic ideal gas in

d

p V = 2 U / d

p V = \frac{2}{d + 2} N E_{F} .

The number flux through the surface is

J_{z} = \frac{g _{s}}{( 2 π ℏ ) ^{3}} \int_{p_{z} > p_{0}} \frac{p _{z}}{m} \frac{d ^{3} p}{e ^{β (p^{2} /2 m - E_{F})} + 1}, p_{0} = 2 m (E_{F} + V_{0}) .

Throughout this tail the occupation is Boltzmann to leading order. The two transverse Gaussian integrals contribute

2 πm / β

, while

\int_{p_{0}}^{\infty} \frac{p _{z}}{m} e^{- β (p_{z}^{2} /2 m - E_{F})} d p_{z} = \frac{1}{β} e^{- β V_{0}} .

Therefore

J_{z} = \frac{g _{s} m}{4 π ^{2} ℏ ^{3}} (k_{B} T)^{2} e^{- V_{0} / (k_{B} T)} .

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