Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 1 / e / i

Table of contents
- Solution i

Solution

When

b = κ_{1} = 0

, stationarity in

ϕ

gives

\frac{δ F}{δ ϕ} = a ϕ + \frac{c}{2} p^{2} = 0, ϕ = - \frac{c}{2 a} p^{2} .

Substituting this value, or completing the square in the Gaussian integral over

ϕ

, gives

F [p] = \int [\frac{A}{2} p^{2} + \frac{B}{4} p^{4} + \frac{κ}{2} ∣\nabla p ∣^{2}] d r, B = B - \frac{c ^{2}}{2 a} .

When

A > 0

and the quartic term is negligible, each Fourier mode is Gaussian with the Ornstein--Zernike correlation function

S_{p} (q) = \frac{k _{B} T}{A + κ q ^{2}}

up to the chosen Fourier normalization. Its correlation length is

ξ_{p} = κ / A

.

 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