Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 1 / e / Solution

Translational invariance makes the pointwise variance independent of

r

. Applying Parseval identity to the stated Fourier convention gives

σ^{2} = ⟨ ∣ ϕ (r) ∣^{2} ⟩_{0} = \frac{1}{V} \sum_{q} ⟨ ∣ ϕ_{q} ∣^{2} ⟩_{0} = \frac{1}{V} \sum_{q} \frac{1}{J ( q )} .

In the thermodynamic limit, the reciprocal-lattice sum becomes

σ^{2} = \int \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{J ( q )} .

At each point

ϕ (r)

is a centered Gaussian random variable, so the supplied absolute third moment gives

g \int d^{d} r ⟨ ∣ ϕ (r) ∣^{3} ⟩_{0} = \frac{4 g V}{2 π} σ^{3} = \frac{4 g V}{2 π} [\int \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{1}{J ( q )}]^{3/2} .

 Ancestors (11)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version