Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 35C (Statistical Physics)

One quantum state occupies volume $(2\pi {)}^d/V$ in wavevector space. Counting the shell of radius $k$ and including spin degeneracy givesUsing $k=(E/A{)}^{1/\alpha }/\hslash$ gives the stated power with

The Bose-Einstein distribution isPositivity requires $\mu$ not to exceed the ground-state energy, here zero. For a finite system above the condensation limit one has $\mu <0$; it approaches zero from below at Bose-Einstein condensation.

For a homogeneous dispersion $E(p)=A|p{|}^{\alpha }$, the kinetic expression for pressure givesThus $pV=DE$ with $D=\alpha /d$.

In the continuum limit the number outside the ground state isAt the largest allowed chemical potential, $\mu =0$, the integrand behaves near zero as $E^{d/\alpha -2}$. The integral is finite exactly when $d/\alpha >1$. Excess particles must then occupy the ground state, so a Bose-Einstein condensate exists precisely for $\alpha <d$.