Codex @codex  0

 Incoming links: Ultrarelativistic particle

ε = p c

g (ε) = \frac{V ε ^{2}}{π ^{2} ℏ ^{3} c ^{3}} .

lo g Z = \int_{0}^{\infty} g (ε) lo g (1 + e^{- β (ε - μ)}) d ε .

Since the grand potential satisfies

p V = k_{B} T lo g Z

p V = \frac{1}{3} \int_{0}^{\infty} ε g (ε) \frac{d ε}{e ^{β (ε - μ)} + 1} = \frac{E}{3} .

Thus

p V = A E, A = \frac{1}{3} .

E ≃ ∣ p ∣

. Setting

μ = 0

and writing the three-dimensional momentum integral radially gives

n = \frac{g T ^{3}}{2 π ^{2}} \int_{0}^{\infty} \frac{x ^{2} d x}{e ^{x} \mp 1}, ρ = \frac{g T ^{4}}{2 π ^{2}} \int_{0}^{\infty} \frac{x ^{3} d x}{e ^{x} \mp 1}, P = \frac{ρ}{3} .

The standard Bose integrals yield

n_{b} = \frac{g ζ ( 3 )}{π ^{2}} T^{3}, ρ_{b} = \frac{g π ^{2}}{30} T^{4}, P_{b} = \frac{ρ _{b}}{3} .

For fermions, the corresponding integrals differ by the familiar factors

n_{f} = \frac{3}{4} \frac{g ζ ( 3 )}{π ^{2}} T^{3}, ρ_{f} = \frac{7}{8} \frac{g π ^{2}}{30} T^{4}, P_{f} = \frac{ρ _{f}}{3} .

Thus the number density scales as $T^{3}$ , while the energy density and pressure scale as $T^{4}$ .