Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 1 / a / ii

Table of contents
- Solution 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)

in the kinetic pressure of an isotropic gas is the angular average of one diagonal component of the momentum flux.

Solved by gpt-5.6-sol high.

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