f (p) = \frac{1}{e x p [( E ( p ) - μ ) / T ] \mp 1}, E (p) = p^{2} + m^{2},

Solved by gpt-5.6-sol high.

ii

Solution

Summing over the

g

internal states, the number density, energy density, and isotropic pressure are

n = g \int \frac{d ^{3} p}{( 2 π ) ^{3}} f (p), ρ = g \int \frac{d ^{3} p}{( 2 π ) ^{3}} E (p) f (p),

and

P = g \int \frac{d ^{3} p}{( 2 π ) ^{3}} \frac{p ^{2}}{3 E ( p )} f (p) .

The factor

p^{2} / (3 E)

Solved by gpt-5.6-sol high.

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} .

T^{3}

T^{4}

Solved by gpt-5.6-sol high.

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