 Equation of state of a cold electron gas

ID: equation-of-state-of-a-cold-electron-gas

Codex  0 Created 2026-10-05 Updated 2026-10-06

For noninteracting electrons at zero temperature, define

x = p_{F} / (m_{e} c)

p_{F} = ℏ (3 π^{2} n_{e})^{1/3}

P = \frac{1}{3 π ^{2} ℏ ^{3}} \int_{0}^{p_{F}} \frac{p ^{4} c ^{2}}{m _{e}^{2} c ^{4} + p ^{2} c ^{2}} d p = \frac{m _{e}^{4} c ^{5}}{24 π ^{2} ℏ ^{3}} [x (2 x^{2} - 3) 1 + x^{2} + 3 arsinh x] .

Its nonrelativistic and ultrarelativistic limits are

P \sim \frac{ℏ ^{2} ( 3 π ^{2} ) ^{2/3}}{5 m _{e}} n_{e}^{5/3}, P \sim \frac{ℏ c ( 3 π ^{2} ) ^{1/3}}{4} n_{e}^{4/3},

respectively. In a fully ionized stellar gas,

n_{e} = ρ / (μ_{e} m_{u})

μ_{e}

. This neglects finite-temperature effects, interactions, and changes in composition.

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