Summing over the $g$ internal states, the <number density>, <energy density>, and isotropic <pressure> are
$$
n=g\int\frac{d^3p}{(2\pi)^3}f(\mathbf p),
\qquad
\rho=g\int\frac{d^3p}{(2\pi)^3}E(\mathbf p)f(\mathbf p),
$$
and
$$
P=g\int\frac{d^3p}{(2\pi)^3}
\frac{\mathbf p^2}{3E(\mathbf p)}f(\mathbf p).
$$
The factor $\mathbf p^2/(3E)$ 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-310/1/a/ii/solution