Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 4

For $\alpha >0$, variation of constants gives the stationary Ornstein-Uhlenbeck processFor $t\ge t^{\prime }$ its covariance isSymmetry in $t,t^{\prime }$ therefore gives

The Fluctuation-dissipation theorem for mobility $\tilde{M}$ requiresComparison with the stated noise amplitude givesThis choice makes the stationary Fokker-Planck equation have the Boltzmann distribution proportional to $e^{-V/(k_{B}T)}$.

Let $\tau =|t-t^{\prime }|$. Integrating the equation for $x$ and using independence of the two noises givesWith the supplied integral,For $\alpha \tau \ll 1$, the active contribution is ballistic, $f_0^2{\tau }^2+O({\tau }^3)$, in addition to the Brownian term. For $\alpha \tau \gg 1$,so the long-time motion is diffusive with an enhanced diffusion coefficient.

Conditioned on $f$, the additive-noise trajectory has the Onsager–Machlup functionalUnder time reversal, $\dot{x}$ changes sign while $x$ and the active force $f$ are even. Subtracting the forward and backward conditional actions givesThe corresponding ratio for an Ornstein–Uhlenbeck path conditioned on its initial endpoint isConsequentlywhereIf stationary endpoint densities are included in the path measures, their ratio cancels the Ornstein–Uhlenbeck boundary term; the displayed convention is the endpoint-conditioned path probability used in the calculation.

The first part of $\Delta U$ is the change of the quadratic energy associated with the active Ornstein–Uhlenbeck force. The second is minus the change in the particle's potential energy, measured in thermal-noise units. The time integral is the work done by active propulsion, again divided by its noise scale; the full log ratio is the trajectory's time-reversal asymmetry or entropy production.

In a stationary confining state the two endpoint terms remain $O(1)$ as $T\to \infty$ and have zero mean, while the mean active work and the mean log ratio grow proportionally to $T$. Endpoint-term distributions approach time-independent distributions with positive and negative fluctuations. Under mixing assumptions, the time-integrated work has a large-deviation distribution: its central part becomes approximately Gaussian with mean and variance proportional to $T$, while its far tails scale exponentially in $T$. The log-ratio distribution obeys the corresponding fluctuation theorem symmetry.