The other diagonal entry of the same inverse gives
$$
\boxed{\langle p(q)p(-q)\rangle
=\frac{a+\kappa q^2+\gamma q^4}
{\nu\left(a+\widetilde\kappa q^2+\gamma q^4\right)}}
$$
or, more revealingly,
$$
\boxed{\langle p(q)p(-q)\rangle
=\frac1\nu+\frac{\lambda^2q^2}
{\nu^2\left(a+\widetilde\kappa q^2+\gamma q^4\right)}.}
$$
The first term is the uncoupled local orientational fluctuation. The second shows that every nonuniform composition fluctuation induces a correlated surfactant-polarization fluctuation. It vanishes at $q=0$, as the coupling contains a gradient, and is enhanced at wavevectors where the composition structure factor is large.


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 1 / g / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-344/1/g/solution