Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 2 / b

Table of contents
- Solution b

Solution

The Fourier transform sends each spatial derivative to

i q_{i}

, so at Gaussian level

G (q) = a + κ q^{2} + γ q^{4} .

For

κ < 0 < γ

, minimizing over

s = q^{2} \geq 0

gives

s_{0} = - \frac{κ}{2 γ}, q_{0} = - \frac{κ}{2 γ} .

The minimum kernel is

G (q_{0}) = a - κ^{2} / (4 γ)

. Gaussian fluctuations first diverge when this vanishes, so the nonzero-wavevector soft-mode sphere becomes unstable at

a_{c} = \frac{κ ^{2}}{4 γ} .

 Ancestors (10)

 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