In natural units, the equilibrium <phase-space distribution function> is
$$
f(\mathbf p)=\frac{1}{\exp[(E(\mathbf p)-\mu)/T]\mp1},
\qquad E(\mathbf p)=\sqrt{\mathbf p^2+m^2},
$$
where the minus sign gives the <Bose-Einstein distribution> for <bosons> and the plus sign gives the <Fermi-Dirac distribution> for <fermions>.

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 / i / Solution

{title2=$f(\mathbf x,\mathbf p)$}
{wiki}

A phase-space distribution function gives the occupation density of particles as a function of position and momentum. For a spatially homogeneous gas it depends only on momentum.


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