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

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- Solution iv

Solution

The normalized one-particle spatial density gives the local number density

n (x) = \frac{N}{4 πV ( k _{B} T ) ^{3/ (2 n)} I _{n}} exp [- \frac{∣ x ∣ ^{2 n}}{V ^{2 n /3} k _{B} T}] .

The radial probability density is proportional to

r^{2} exp [- \frac{r ^{2 n}}{V ^{2 n /3} k _{B} T}] .

Its logarithmic derivative vanishes when

\frac{2}{r} - \frac{2 n r ^{2 n - 1}}{V ^{2 n /3} k _{B} T} = 0,

so the most likely radius is

r_{mode} = V^{1/3} (\frac{k _{B} T}{n})^{1/ (2 n)} .

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