For $q\ne0$ and $\alpha>0$, each <Fourier mode> is an <Ornstein-Uhlenbeck process> with decay rate $r_q=\alpha q^2$. Its <explicit Ornstein-Uhlenbeck solution> is
$$
h_q(t)=e^{-r_qt}h_q(0)+\sigma\int_0^t e^{-r_q(t-s)}\,dW_q(s),
$$
so the <Itô isometry> gives
$$
\langle h_q(t)^2\rangle
=e^{-2r_qt}\langle h_q(0)^2\rangle
+\frac{\sigma^2}{2r_q}(1-e^{-2r_qt}).
$$
To identify the <Overdamped Langevin dynamics>, choose a friction $\zeta_q>0$, a <harmonic oscillator> potential $V_q=\zeta_q\alpha q^2h_q^2/2$, and force noise $f_q=\zeta_q\eta_q$. The <fluctuation-dissipation relation for a Langevin particle> then defines $k_BT_{\rm eff}=\zeta_q\sigma^2/2$. Its <Boltzmann distribution> is
$$
P_q(h)=\sqrt{\frac{\alpha q^2}{\pi\sigma^2}}
\exp\!\left(-\frac{\alpha q^2h^2}{\sigma^2}\right),
\qquad \boxed{\langle h_q^2\rangle_{\rm st}=\frac{\sigma^2}{2\alpha q^2}\propto q^{-2}}.
$$
This is the <stationary spectrum of a linear fluctuating interface>, also following from the <equipartition theorem>. The numerical prefactor uses exactly the mode-noise normalization given in the paper. The nonzero modes can therefore have a stationary <Gaussian distribution> even though the unpinned zero mode cannot.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 344 / 1 / e / Solution

{title2=$\langle h_q^2\rangle=\sigma^2/(2\alpha q^2)$}

For a nonzero <Fourier mode> satisfying $dh_q=-\alpha q^2h_q\,dt+\sigma\,dW_q$ with $\alpha>0$, the <explicit Ornstein-Uhlenbeck solution> and <Itô isometry> give the stationary <second moment> $\sigma^2/(2\alpha q^2)$. The prefactor depends on the normalization of the mode noise, while the inverse-square dependence follows from diffusive relaxation.

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 Incoming links: Stationary spectrum of a linear fluctuating interface