Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 1 / c

The stationary solution ofisIt follows from the Itô isometry thatThus $q$ is zero-mean colored noise with correlation time ${\nu }^{-1}$. The equation $\dot{x}=\mu (-\lambda x+q)$ is an overdamped harmonic particle of mobility $\mu$ driven by that correlated random force.

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For $X=(x,q{)}^T$,The Lyapunov equation gives ${\sigma }_{qq}=\alpha /2$, ${\sigma }_{xq}=\lambda {\sigma }_{xx}$, and $(\mu \lambda +\nu ){\sigma }_{xq}=\mu {\sigma }_{qq}$. Therefore

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Take $\nu \to \infty$ withSince $(\nu /2)e^{-\nu |\tau |}\to \delta (\tau )$,Thus $\mu q$ tends to thermal white forcing of covariance $2\mu k_{B}T\delta (\tau )$, as required by the fluctuation--dissipation relation. The equal-time position variance becomesthe equilibrium harmonic variance.

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