Power spectral density of trajectories of active Ornstein-Uhlenbeck particles

Abstract: The power spectral density (PSD) is a central frequency-domain descriptor of stochastic processes. While PSDs have been studied for Brownian motion and a few anomalous diffusion processes, the spectral densities of active nonequilibrium processes remain almost unexplored. Here, we present an exact theory for the PSDs of active diffusion using the model of active Ornstein-Uhlenbeck particles (AOUPs). We investigate the spectral densities of AOUPs in free space and under harmonic confinement. In free space, active motion does not alter the Brownian $f^{-2}$ spectrum, but only modifies its amplitude and introduces a crossover at the persistence frequency. Under confinement, the spectrum exhibits a rich variety of features depending on the persistence, trap relaxation, and activity strength, including two characteristic signatures that are absent in both thermal systems and free AOUPs. These are a two-plateau structure from a double-trapping mechanism due to two noise sources, and the new $f^{-4}$ spectral scaling associated with transient ballistic motion. We also investigate the finite time effects through the finite-time PSD, and find that the low-frequency plateau and high frequency oscillation exhibit distinct dependences on the observation time $T$ in free and confined systems. Finally, we discuss our results in connection with previously reported experimental studies of active systems. Our results provide an analytically tractable framework for interpreting such systems.

• The paper presents an exact analytical framework for calculating PSDs of active Ornstein-Uhlenbeck particle trajectories, covering both free and confined conditions.
• It reveals distinct spectral features, including classical f⁻² scaling and new f⁻⁴ behavior under confinement, linking activity parameters to measurable frequency-domain observables.
• Extensive simulations validate the analytical predictions, offering quantitative strategies for probing non-equilibrium dynamics in active matter experiments.

Power Spectral Density Analysis of Active Ornstein-Uhlenbeck Trajectories

Introduction

This paper introduces an exact analytical framework for the calculation and interpretation of power spectral densities (PSDs) of active Ornstein-Uhlenbeck particles (AOUPs), both in free space and under harmonic confinement (2605.09399). AOUPs represent a minimal yet powerful model for self-propelled particle dynamics, incorporating exponentially correlated active forces. The investigation targets not only the infinite-time PSDs routinely discussed in classical stochastic process theory but also the finite-time spectral densities relevant to experimental and numerical studies, mapping how activity, persistence, and confinement modify frequency-domain observables.

Theoretical Framework

The AOUP model comprises coupled Langevin equations governing particle positions and active forces, with exponentially correlated noise characterized by a persistence time $T_A$ and an activity strength parameter. Free-space dynamics yield nonstationary behavior, with distinct short-time (thermal), intermediate-time (ballistic), and long-time (active Fickian, $\text{MSD} \sim 2D_{\mathrm{eff}} t$) regimes. Harmonic confinement leads to stationary Ornstein-Uhlenbeck behavior, with position autocovariance structured as a superposition of thermal and active exponential contributions.

The paper presents closed-form expressions for trajectory autocovariances and mean-squared displacements (MSD) for both free and confined AOUPs, underpinning subsequent PSD analyses. Effective "temperature" and diffusivity emerge as operational parameters, allowing direct mapping of activity effects onto frequency domain observables.

Infinite-Time PSDs

Free AOUPs

The PSD in free space retains the classical $f^{-2}$ Brownian scaling across all frequencies, with activity manifesting as an amplitude correction and a crossover around the persistence frequency $1/T_A$. Specifically, the spectrum shifts from $4D_{\mathrm{eff}}/f^2$ (low frequencies) to $(2D + 2D_{\mathrm{eff}})/f^2$ (high frequencies). No additional scaling exponents (e.g., $f^{-4}$) arise from activity—contradicting some heuristic expectations about ballistic contributions in active motion. This reveals PSD as a sharper probe for distinguishing activity in a temporal regime where MSD remains thermal-dominated.

Confined AOUPs

Harmonic potential induces qualitatively distinct spectral features not present in thermal or free AOUPs. The infinite-time PSD becomes a sum of Lorentzian terms, each reflecting characteristic relaxation and persistence times. Depending on parameter hierarchies ($T_A \gg T_R$, $T_A \ll T_R$, $T_A \sim T_R$), several features arise:

• Double-Plateau Structure: For $\text{MSD} \sim 2D_{\mathrm{eff}} t$0 and moderate activity, two distinct plateaus appear, attributable to a double-trapping mechanism via thermal and active noise sources.
• $\text{MSD} \sim 2D_{\mathrm{eff}} t$1 Scaling: Activity generates a new $\text{MSD} \sim 2D_{\mathrm{eff}} t$2 spectral tail at high frequencies, directly linked to transient ballistic motion in the MSD. This scaling is absent in thermal systems and free AOUPs.
• Special Squared-Lorentzian Case: For $\text{MSD} \sim 2D_{\mathrm{eff}} t$3, the active component reduces to a squared Lorentzian. The crossover occurs at a single frequency, with only intermediate $\text{MSD} \sim 2D_{\mathrm{eff}} t$4 scaling.

These regimes are analytically mapped to distinct frequency behaviors and corroborated with numerical simulations.

Finite-Time Spectral Densities

Free AOUPs

Finite-time PSDs develop a window-dependent low-frequency plateau, rising as $\text{MSD} \sim 2D_{\mathrm{eff}} t$5 with increasing trajectory length. High-frequency oscillations, proportional to $\text{MSD} \sim 2D_{\mathrm{eff}} t$6, emerge, rapidly vanishing as $\text{MSD} \sim 2D_{\mathrm{eff}} t$7 increases. Both plateau and oscillatory features are window artifacts, converging to the infinite-time limit with a $\text{MSD} \sim 2D_{\mathrm{eff}} t$8 rate.

Confined AOUPs

Finite-time PSDs in confined AOUPs show a low-frequency plateau proportional to $\text{MSD} \sim 2D_{\mathrm{eff}} t$9, distinguishable from infinite-time confinement plateaus. Oscillatory components persist with $f^{-2}$0 scaling, remaining visible across broader frequency ranges compared to the free AOUP case. Analytical expressions for all finite-time corrections are provided, including frequency and window dependence.

Numerical Results and Empirical Implications

Extensive simulation results validate all analytic predictions, confirming plateau magnitudes, crossover frequencies, and oscillatory behaviors in both free and confined systems. Theoretical predictions are aligned with recent experimental studies of colloids immersed in active baths and optically trapped Janus particles, detailing how observed MSD and PSD features can be quantitatively interpreted via AOUP parameters.

The analysis highlights the practical utility of PSD as a sensitive probe for non-equilibrium activity effects, beyond conventional time-domain observables. For instance, the presence of $f^{-2}$1 tails or double-plateau PSDs can directly reveal underlying double-trapping mechanisms or transient ballistic motion in active matter experiments.

Broader Implications and Future Directions

From a theoretical perspective, the exact solvability of the AOUP spectral densities provides a reference point for more intricate models, including viscoelastic and non-Markovian environments. The ability to analytically characterize both lower moments and, prospectively, higher-order statistics of finite-time PSDs opens avenues for advancing frequency-domain probes in active matter, testing non-ergodic and aging phenomena in biological and artificial active systems.

Future extensions may include:

• Incorporation of viscoelasticity and environmental disorder (e.g., [53])
• Analysis of higher-order PSD statistics and their empirical signatures
• Application to more complex biological systems, such as cellular actin cortex dynamics or active transport modes

Conclusion

This work delivers a comprehensive analytical and numerical characterization of the PSDs of AOUP trajectories, establishing new spectral features (double plateaus, $f^{-2}$2 scalings) associated with activity and confinement. The results provide quantitative guidance for interpreting experimental and simulation data on active matter, demonstrate the sensitivity of frequency-domain observables to non-equilibrium dynamics, and lay groundwork for further theoretical developments in active stochastic processes. The framework has immediate utility for the analysis of experimental PSDs in active colloidal and biological contexts, offering new strategies for distinguishing underlying dynamical regimes.