= Solution
The functional derivatives are
$$
\frac{\delta F}{\delta\phi}=a\phi+c\psi-\kappa\nabla^2\phi,
\qquad
\frac{\delta F}{\delta\psi}=\bar a\psi+c\phi-\bar\kappa\nabla^2\psi.
$$
With the stated Fourier convention,
$$
\boxed{\dot\phi_{\mathbf q}
=-Mq^2[(a+\kappa q^2)\phi_{\mathbf q}+c\psi_{\mathbf q}]
-i\sqrt{2M}\,\mathbf q\cdot\boldsymbol\Lambda_{\mathbf q}},
$$
$$
\boxed{\dot\psi_{\mathbf q}
=-\Gamma[(\bar a+\bar\kappa q^2)\psi_{\mathbf q}
+c\phi_{\mathbf q}]+\sqrt{2\Gamma}\,\Lambda_{\mathbf q}}.
$$
The first is <conserved order-parameter dynamics>; its deterministic rate and conserved-noise amplitude vanish at $q=0$. The second is <nonconserved order-parameter dynamics>.
Solved by gpt-5.6-sol high.
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