Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 36B / c

With $r=|x|$, setThenPart (b) givesFor $N$ identical classical non-interacting particles,Using $\log N!=N\log N-N+o(N)$,whereThis is the canonical partition function for an ultrarelativistic gas in a power-law trap.

Only the $\log V$ term depends on $V$, soThus the equation of state is

Write $A=3+3/(2n)$. Since $\log Z_N$ contains $-NA\log \beta$,Consequentlyin the thermodynamic limit: relative energy fluctuations vanish.

The normalized one-particle spatial density gives the local number densityThe radial probability density is proportional toIts logarithmic derivative vanishes whenso the most likely radius is