Ornstein–Uhlenbeck relaxation

Write the Itô equation $dx=-\kappa(x-x_*)dt+\sqrt{2D}\,dW_t$, with positive relaxation rate kappa and constant D. The fixed point is x star. Kappa has units of inverse seconds, unlike a spring stiffness, and D has units of length squared per second. Multiplying by the exponential integrating factor converts the deterministic part into a total differential. The stochastic integral then collects each past kick multiplied by the fraction of its effect that survives relaxation up to the observation time.

A fixed initial position gives a mean that approaches the fixed point exponentially. Itô isometry gives the variance by integrating the squared exponential weight. Early times recover free-diffusion variance because the restoring force has had little time to act. At late times the variance approaches D divided by kappa. The process therefore forgets its initial mean without losing all fluctuations. Because the noise is additive Gaussian and the equation is linear, the transition distribution is Gaussian; this conclusion does not extend automatically to nonlinear restoring forces or non-Gaussian kicks.

If the initial distribution already has mean x star and variance D divided by kappa, its one-time distribution remains unchanged. Its two-time fluctuations are nevertheless correlated. A fluctuation at an earlier time influences the conditional mean later by the same exponential factor that governs deterministic relaxation. In the stationary ensemble the covariance is therefore the stationary variance multiplied by an exponential of the time separation. The absolute value of the separation makes this equilibrium correlation symmetric under exchanging the two observation times.

For a nonstationary initial distribution, the variance includes a decaying contribution from the initial spread as well as newly accumulated noise. One must not insert a stationary covariance formula into an experiment prepared at an exact position without checking the waiting time. Likewise, recording many points at intervals short compared with the relaxation time does not create many independent samples. Correlations reduce the information gained from repeated measurements, even though a histogram collected over a sufficiently long stationary trajectory can reproduce the correct Gaussian distribution.

For the overdamped harmonic potential $U=k(x-x_*)^2/2$, the mobility gives $\kappa=\mu k$. If the bath is equilibrium thermal noise, $D=\mu k_BT$, so the steady variance becomes $k_BT/k$. Mobility controls how fast the particle relaxes, while stiffness and temperature control its equilibrium width. Increasing friction slows relaxation but leaves the width unchanged when the noise amplitude changes consistently with the same bath temperature. Changing friction while holding an incompatible noise strength fixed describes a different effective temperature or a nonthermal model.

The Ornstein–Uhlenbeck model can also describe velocity relaxation, but then x is replaced by velocity and the diffusion coefficient has velocity-squared per time units. Always declare the modeled variable before interpreting D. A stable linear drift guarantees a stationary Gaussian for this scalar additive-noise model. It does not prove that every stationary noisy system is thermal, nor that a measured Gaussian histogram alone establishes equilibrium. Later lessons will show stationary densities with nonzero probability circulation, which cannot be diagnosed from their shape alone.