The deterministic relaxation matrix is
$$
\boxed{R(q)=
\begin{pmatrix}
Mq^2(a+\kappa q^2)&Mcq^2\\
\Gamma c&\Gamma(\bar a+\bar\kappa q^2)
\end{pmatrix}}.
$$
Its right eigenvectors are the <hydrodynamic mode>[hydrodynamic modes]. Writing $A_q=a+\kappa q^2$ and $B_q=\bar a+\bar\kappa q^2$, their decay rates obey
$$
\boxed{(\lambda-Mq^2A_q)(\lambda-\Gamma B_q)
-M\Gamma c^2q^2=0}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 3 / ii / Solution

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A hydrodynamic mode is a long-wavelength collective mode whose decay rate tends to zero because of a conservation law. Coupling to fast nonconserved variables renormalizes its diffusivity and eigenvector.


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 Incoming links: Hydrodynamic mode