pith. machine review for the scientific record. sign in

arxiv: 2605.09399 · v1 · submitted 2026-05-10 · ❄️ cond-mat.stat-mech · cond-mat.soft

Recognition: no theorem link

Power spectral density of trajectories of active Ornstein-Uhlenbeck particles

Yeongjin Kim , Gleb Oshanin , Jae-Hyung Jeon
Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:04 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft
keywords power spectral densityactive Ornstein-Uhlenbeck particlesactive matterharmonic confinementstochastic trajectoriesBrownian motionpersistence time
0
0 comments X

The pith

Active Ornstein-Uhlenbeck particles produce power spectra with two plateaus and an f to the minus four regime under confinement.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an exact theory for the frequency spectrum of the position trajectories of active Ornstein-Uhlenbeck particles. In free space the spectrum retains the Brownian f to the minus two scaling but changes its overall amplitude and adds a crossover frequency set by the persistence time of the active force. Under harmonic confinement the spectrum acquires two distinct plateaus because the trap filters both thermal and active noise sources separately, together with an intermediate f to the minus four scaling that arises from the short-time ballistic segment of the active trajectory. These analytic expressions allow experimentalists to read off activity parameters directly from measured spectra.

Core claim

We present closed-form expressions for the power spectral density of active Ornstein-Uhlenbeck particles both in free space and inside a harmonic trap. Free-space spectra follow the same f to the minus two decay as passive Brownian motion, only with a rescaled amplitude and a crossover at the persistence frequency. Confined spectra display a two-plateau structure caused by the double-trapping of the two independent noises and an additional f to the minus four segment linked to transient ballistic motion; the locations and heights of these features depend on the persistence time, trap frequency, and activity strength.

What carries the argument

The conversion, via the Wiener-Khinchin theorem, of the steady-state position autocorrelation function of particles driven by an exponentially correlated active force into their power spectral density.

If this is right

  • In free space the spectrum crosses over from activity-dominated to diffusive scaling at the persistence frequency.
  • Under confinement a lower plateau appears whose height grows with activity strength while the upper plateau remains thermal.
  • An intermediate frequency window shows f to the minus four scaling whose width is controlled by the ratio of trap relaxation rate to persistence rate.
  • Finite-time spectra exhibit a low-frequency plateau whose height scales linearly with observation time and high-frequency oscillations whose period depends on observation time.
  • These signatures are absent in both equilibrium Brownian motion and in free active particles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Experimentalists studying confined microswimmers could use the two-plateau height ratio to extract the relative strength of active versus thermal noise without separate calibration runs.
  • The f to the minus four regime might serve as a diagnostic for the presence of persistence in other confined stochastic systems even when the underlying equations differ from the active Ornstein-Uhlenbeck model.
  • Extending the same autocorrelation approach to many-particle active systems or to non-harmonic potentials would reveal whether the reported spectral features survive interactions and more complex geometries.

Load-bearing premise

The assumption that active forces on the particles lose correlation exponentially over time must be true for the systems being studied.

What would settle it

Measuring the power spectrum of a single harmonically trapped active particle and finding neither the predicted two-plateau structure nor the f to the minus four segment at the expected frequencies would disprove the theory.

Figures

Figures reproduced from arXiv: 2605.09399 by Gleb Oshanin, Jae-Hyung Jeon, Yeongjin Kim.

Figure 1
Figure 1. Figure 1: FIG. 1. Sample trajectory of AOUPs. (a) Two simulated [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Finite- [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Infinite- [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Finite- [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript presents an exact theory for the power spectral densities (PSDs) of active Ornstein-Uhlenbeck particles (AOUPs) in free space and under harmonic confinement. It claims that in free space, the PSD retains the f^{-2} scaling characteristic of Brownian motion, with only amplitude modification and a crossover at the persistence frequency. Under confinement, the spectrum shows a two-plateau structure arising from a double-trapping mechanism and a novel f^{-4} scaling linked to transient ballistic motion. Finite-time effects on the PSD are also analyzed, and results are connected to experimental studies.

Significance. If the derivations are accurate, this work supplies an important analytically tractable framework for the frequency-domain characterization of active diffusion. The exact expressions and the discovery of distinctive features such as the two-plateau structure and f^{-4} regime absent in thermal systems represent a notable contribution to the study of nonequilibrium stochastic processes in active matter.

major comments (1)
  1. The central claim that active motion does not alter the Brownian f^{-2} spectrum in free space is derived from the Fourier transform of the position autocorrelation function using the Wiener-Khinchin theorem. However, free AOUP position trajectories are non-stationary, with the mean-squared displacement growing linearly with time and the autocorrelation function depending on the minimum of the two times rather than their difference. The paper separately discusses finite-time PSD, but the main free-space result requires explicit demonstration that it corresponds to the appropriate limit of the non-stationary spectral density to ensure the f^{-2} conclusion is rigorously exact.
minor comments (2)
  1. All intermediate steps in the derivations of the PSD expressions from the AOUP Langevin equations should be provided in detail to facilitate verification of the 'exact theory'.
  2. The parameters (e.g., persistence time, activity strength, trap stiffness) should be explicitly defined in the text and figures for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for highlighting an important technical point regarding the definition of the power spectral density for non-stationary processes. We address the comment in detail below and have revised the manuscript to strengthen the rigor of the free-space result.

read point-by-point responses

  1. Referee: The central claim that active motion does not alter the Brownian f^{-2} spectrum in free space is derived from the Fourier transform of the position autocorrelation function using the Wiener-Khinchin theorem. However, free AOUP position trajectories are non-stationary, with the mean-squared displacement growing linearly with time and the autocorrelation function depending on the minimum of the two times rather than their difference. The paper separately discusses finite-time PSD, but the main free-space result requires explicit demonstration that it corresponds to the appropriate limit of the non-stationary spectral density to ensure the f^{-2} conclusion is rigorously exact.

    Authors: We agree that the position process of free AOUPs is non-stationary, as the two-time position correlation depends on the absolute times (via the integrated velocity autocorrelation) rather than solely on the time difference. Our original derivation applied the Wiener-Khinchin theorem to the long-time form of this correlation, which is a common approach for diffusive processes whose MSD grows linearly. However, we acknowledge that an explicit link to the non-stationary spectral density is desirable for full rigor. We have therefore added a new paragraph and accompanying derivation in the free-space section. Specifically, we define the finite-time PSD as the expectation value of the squared modulus of the Fourier transform of the trajectory (normalized by observation time T) and demonstrate analytically that, in the limit T → ∞, this quantity converges to the reported expression: a Brownian-like f^{-2} spectrum whose amplitude is set by the effective long-time diffusion coefficient, with a crossover to a flatter regime at the persistence frequency. This limiting procedure recovers our main claim exactly and is consistent with the separate finite-time analysis already present in the manuscript. The conclusions remain unchanged, but the presentation is now more precise. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation starts from standard AOUP Langevin equations and computes PSD via autocorrelation Fourier transform

full rationale

The paper begins from the linear Langevin equations of the active Ornstein-Uhlenbeck particle model, solves for the steady-state position autocorrelation function, and obtains the PSD as its Fourier transform using the Wiener-Khinchin theorem. This chain is self-contained and does not reduce any claimed result to a fitted input renamed as prediction, a self-definition, or a load-bearing self-citation. The free-space claim that active motion preserves the f^{-2} spectrum while adding a persistence-frequency crossover follows algebraically from the model equations without circularity. The skeptic concern about stationarity is a potential correctness issue for the infinite-time limit but does not constitute circularity under the defined criteria, as no equation is forced by construction or prior self-result.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard mathematical framework of linear stochastic differential equations for the AOUP model together with the Wiener-Khinchin theorem relating the PSD to the Fourier transform of the steady-state position autocorrelation; no new entities are introduced and the model parameters (persistence time, activity amplitude, trap stiffness) are treated as given inputs rather than quantities fitted to produce the reported spectra.

free parameters (2)
  • persistence time
    Characteristic decay time of the active force autocorrelation; a standard input parameter of the AOUP model, not derived or fitted within the PSD calculation.
  • activity strength
    Amplitude of the active noise term; an input parameter of the model whose value sets the scale of the low-frequency modifications but is not adjusted post hoc to match the spectral features.
axioms (2)
  • domain assumption The position autocorrelation function of the AOUP can be obtained exactly by solving the linear Langevin equations in steady state.
    Invoked to justify computing the PSD via Fourier transform without approximation.
  • standard math Time-translation invariance holds so that the Wiener-Khinchin theorem directly equates the PSD to the Fourier transform of the autocorrelation.
    Standard assumption for defining the power spectrum of a stationary process.

pith-pipeline@v0.9.0 · 5554 in / 1810 out tokens · 89337 ms · 2026-05-12T02:04:45.965953+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages

  1. [1]

    µ(f,∞) = 4D f 2

    (b) A simulated trajectory for an AOUP in a harmonic potential with stiffnessk= 1 for givenτ A = 1 andv p = 10. µ(f,∞) = 4D f 2 . Activity-induced motion yields an addi- tional Lorentzian correction that interpolates the prefac- tor beyond this free Brownian PSD. Specifically, the spec- trum crosses over fromµ(f,∞)≃4D eff/f2 forf≪τ −1 A toµ(f,∞)≃(2D+ 2D e...

    work page

  2. [2]

    In this situation, the persistence frequency τ −1 A and the trap-relaxation frequencyτ −1 R are well sepa- rated

    Long-Persistence Regime (τ A ≫τ R) and Moderate Activity ( p D/τA < v p < v ∗ p) Consider the power spectra under the condition of τA ≫τ R. In this situation, the persistence frequency τ −1 A and the trap-relaxation frequencyτ −1 R are well sepa- rated. Remarkably, as shown in Fig. 4(a)–(c), the power spectra and MSDs exhibit two plateaus when the activ- ...

    work page

  3. [3]

    In this case, albeitτ A ≫τ R, the PSD loses the interme- diate thermal plateau while keeping the zero-frequency plateau

    Long-Persistence Regime (τ A ≫τ R) and Strong Activity When the activity is substantially strong such that vp ≳v ∗ p, the crossover frequency hasf ∗ ≳τ −1 R . In this case, albeitτ A ≫τ R, the PSD loses the interme- diate thermal plateau while keeping the zero-frequency plateau. See the simulated PSDs (Fig. 4(d)) and theo- retical curves (Fig. 4(e)). Here...

    work page

  4. [4]

    4(g) and 4(h)

    Cross-over Persistence:τ A =τ R In the case ofτ A =τ R, the PSD (25) has the special form µ(f, T=∞) = 2D f2 +τ −2 R + 2DA τ 2 R f2 +τ −2 R 2 .(39) See the profile of the PSD in Fig. 4(g) and 4(h). Un- der this condition, the functional form of the active part µact, originally a sum of two Lorentzians, is changed to a single squared Lorentzian. This implie...

    work page

  5. [5]

    Short-Persistence Regime (τ A ≪τ R) Under the short-persistence conditionτ A ≪τ R, the trap-induced relaxation dynamics is much slower than the correlation time of the active force. As a result, con- finement is irrelevant over a broad range of lag times, so the MSD resembles that of a free AOUP before the final saturation set by the trap; see the MSD cur...

    work page

  6. [6]

    The oscillatory compo- nent is clearly resolved and persists up to higher frequencies

    (b) The cross-over casev p =v ∗ p. The oscillatory compo- nent is clearly resolved and persists up to higher frequencies. amplitude decreases monotonically asTdecreases. From the PSD (44), the zero-frequency value of the power spec- trum is found to be µ(0, T) = 2kBTτ R k 1 + τR T e−T /τR −1 + 2DAτ 2 R   τ 3 A e− T τA −1 −τ 3 R e− T τR −1 T(τ 2 A −τ 2 R...

    work page

  7. [7]

    For a trajectory defined in the interval of 0≤t≤T, using the two-time covariance in Eq

    Exact closed-form expression for finite-TPSD Here we provide the finite-TPSD of free AOUPs ob- served over a finite time windowT. For a trajectory defined in the interval of 0≤t≤T, using the two-time covariance in Eq. (8) together with Eq. (4) yields the exact closed-form expression: µ(f, T) = e−T /τA f3T ζ(1 +f 2τ 2 A)2 h 2f2v2 pζτ 3 A f τA +f 3τ 3 A −2f...

    work page

  8. [8]

    (23), we now find the analytic expression for the crossover fre- quencyf 0 in theµ(f, T) for free AOUPs

    Crossover frequency in the finite-TPSD Using the Lorentzian approximation as in Eq. (23), we now find the analytic expression for the crossover fre- quencyf 0 in theµ(f, T) for free AOUPs. Based on the Lorentzian interpolation (23), we define the crossover fre- quency as follows: f2 0 = limf→∞ µ(f, T)f 2 µ(0, T) ,(B2) which yields the analytic expression ...

    work page

  9. [9]

    Exact closed-form expression for finite-TPSD Here we provide the analytic expressions forµ nos(f, T) andµ os(f, T) introduced in the expression of finite-T PSD, Eq. (44), in the main text, which read µnos(f, T) = 2kBTτ 3 R(f2τ 2 R −1) ζT(1 +f 2τ 2 R)2 + 2τAτ 2 Rv2 p T(f 2τ 2 A + 1)2 (f2τ 2 R + 1)2 (τA +τ R) " τ 4 A f3τ 2 R +f 2 +τ AτR f2τ 2 R −1 +f 2τ 4 R...

    work page

  10. [10]

    M. P. Norton and D. G. Karczub,Fundamentals of Noise and Vibration Analysis for Engineers(Cambridge Uni- versity Press, Cambridge, 2003)

    work page 2003

  11. [11]

    G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Phys. Rev.36, 823 (1930)

    work page 1930

  12. [12]

    Kubo, The fluctuation-dissipation theorem, Rep

    R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys.29, 255 (1966)

    work page 1966

  13. [13]

    Berg-Sørensen and H

    K. Berg-Sørensen and H. Flyvbjerg, Power spectrum analysis for optical tweezers, Rev. Sci. Instrum.75, 594 (2004)

    work page 2004

  14. [14]

    Krapf, E

    D. Krapf, E. Marinari, R. Metzler, G. Oshanin, X. Xu, and A. Squarcini, Power spectral density of a single Brow- nian trajectory: what one can and cannot learn from it, New J. Phys.20, 023029 (2018)

    work page 2018

  15. [15]

    Metzler, J.-H

    R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: non- stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys.16, 24128 (2014)

    work page 2014

  16. [16]

    Grebenkov, R

    D. Grebenkov, R. Metzler, and G. Oshanin,Target search problems(Springer, Cham, 2024)

    work page 2024

  17. [17]

    Mandelbrot,Gaussian self-affinity and fractals: glob- ality, the earth,1/fnoise, andR/S(Springer Science & Business Media, 2002)

    B. Mandelbrot,Gaussian self-affinity and fractals: glob- ality, the earth,1/fnoise, andR/S(Springer Science & Business Media, 2002)

    work page 2002

  18. [18]

    S. B. Lowen and M. C. Teich, Fractal renewal processes generate 1/fnoise, Phys. Rev. E.47, 992 (1993)

    work page 1993

  19. [19]

    N. Wynn Watkins, Mandelbrot’s 1/ffractional renewal models of 1963–67: the non-ergodic missing link be- 14 tween change points and long range dependence, inIn- ternational Work-Conference on Time Series Analysis (Springer, 2016) pp. 197–208

    work page 1963

  20. [20]

    Marinari, G

    E. Marinari, G. Parisi, D. Ruelle, and P. Windey, Ran- dom walk in a random environment and 1/fnoise, Phys. Rev. Lett.50, 1223 (1983)

    work page 1983

  21. [21]

    Niemann, H

    M. Niemann, H. Kantz, and E. Barkai, Fluctuations of 1/fnoise and the low-frequency cutoff paradox, Phys. Rev. Lett.110, 140603 (2013)

    work page 2013

  22. [22]

    Leibovich, A

    N. Leibovich, A. Dechant, E. Lutz, and E. Barkai, Aging wiener-khinchin theorem and critical exponents of 1/f β noise, Phys. Rev. E94, 052130 (2016)

    work page 2016

  23. [23]

    Squarcini, E

    A. Squarcini, E. Marinari, G. Oshanin, L. Peliti, and L. Rondoni, Frequency–frequency correlations of single- trajectory spectral densities of Gaussian processes, New J. Phys.24, 093031 (2022)

    work page 2022

  24. [24]

    Squarcini, E

    A. Squarcini, E. Marinari, G. Oshanin, L. Peliti, and L. Rondoni, Noise-to-signal ratio of single-trajectory spectral densities in centered Gaussian processes, J. Phys. A: Math. Theor.55, 405001 (2022)

    work page 2022

  25. [25]

    Krapf, N

    D. Krapf, N. Lukat, E. Marinari, R. Metzler, G. Oshanin, C. Selhuber-Unkel, A. Squarcini, L. Stadler, M. Weiss, and X. Xu, Spectral content of a single non-Brownian trajectory, Phys. Rev. X.9, 011019 (2019)

    work page 2019

  26. [26]

    Sposini, D

    V. Sposini, D. Krapf, E. Marinari, R. Sunyer, F. Ri- tort, F. Taheri, C. Selhuber-Unkel, R. Benelli, M. Weiss, R. Metzler,et al., Towards a robust criterion of anoma- lous diffusion, Commun. Phys.5, 305 (2022)

    work page 2022

  27. [27]

    Sposini, R

    V. Sposini, R. Metzler, and G. Oshanin, Single-trajectory spectral analysis of scaled Brownian motion, New J. Phys.21, 073043 (2019)

    work page 2019

  28. [28]

    Sposini, D

    V. Sposini, D. S. Grebenkov, R. Metzler, G. Oshanin, and F. Seno, Universal spectral features of different classes of random-diffusivity processes, New J. Phys.22, 063056 (2020)

    work page 2020

  29. [29]

    K. A. Takeuchi, 1/f α power spectrum in the kar- dar–parisi–zhang universality class, J. Phys. A: Math. Theor.50, 264006 (2017)

    work page 2017

  30. [30]

    Z. R. Fox, E. Barkai, and D. Krapf, Aging power spec- trum of membrane protein transport and other subordi- nated random walks, Nat. Commun.12, 6162 (2021)

    work page 2021

  31. [31]

    Bechinger, R

    C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys.88, 045006 (2016)

    work page 2016

  32. [32]

    M. S. Song, H. C. Moon, J.-H. Jeon, and H. Y. Park, Neuronal messenger ribonucleoprotein transport follows an aging L´ evy walk, Nat. Commun.9, 344 (2018)

    work page 2018

  33. [33]

    T. Song, Y. Choi, J.-H. Jeon, and Y.-K. Cho, A machine learning approach to discover migration modes and tran- sition dynamics of heterogeneous dendritic cells, Front. Immunol.14, 1129600 (2023)

    work page 2023

  34. [34]

    Squarcini, A

    A. Squarcini, A. Solon, and G. Oshanin, Spectral density of individual trajectories of an active Brownian particle, New J. Phys.24, 013018 (2022)

    work page 2022

  35. [35]

    Szamel, Self-propelled particle in an external poten- tial: Existence of an effective temperature, Phys

    G. Szamel, Self-propelled particle in an external poten- tial: Existence of an effective temperature, Phys. Rev. E. 90, 012111 (2014)

    work page 2014

  36. [36]

    Maggi, U

    C. Maggi, U. M. B. Marconi, N. Gnan, and R. Di Leonardo, Multidimensional stationary probabil- ity distribution for interacting active particles, Sci. Rep. 5, 10742 (2015)

    work page 2015

  37. [37]

    Fodor, C

    ´E. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. Van Wijland, How far from equilibrium is active matter?, Phys. Rev. Lett.117, 038103 (2016)

    work page 2016

  38. [38]

    Wittmann, C

    R. Wittmann, C. Maggi, A. Sharma, A. Scacchi, J. M. Brader, and U. Marini Bettolo Marconi, Effective equilib- rium states in the colored-noise model for active matter I. pairwise forces in the Fox and unified colored noise approximations, J. Stat. Mech.: Theory Exp.2017(11), 113207

    work page 2017

  39. [39]

    L. L. Bonilla, Active Ornstein–Uhlenbeck particles, Phys. Rev. E100, 022601 (2019)

    work page 2019

  40. [40]

    Dabelow and R

    L. Dabelow and R. Eichhorn, Irreversibility in active mat- ter: General framework for active Ornstein–Uhlenbeck particles, Front. Phys.8, 582992 (2021)

    work page 2021

  41. [41]

    Martin, J

    D. Martin, J. O’Byrne, M. E. Cates, E. Fodor, C. Nar- dini, J. Tailleur, and F. van Wijland, Statistical mechan- ics of active Ornstein–Uhlenbeck particles, Phys. Rev. E 103, 032607 (2021)

    work page 2021

  42. [42]

    Leibovich and E

    N. Leibovich and E. Barkai, Aging wiener-khinchin the- orem, Phys. Rev. Lett.115, 080602 (2015)

    work page 2015

  43. [43]

    Squarcini, E

    A. Squarcini, E. Marinari, and G. Oshanin, Passive ad- vection of fractional Brownian motion by random layered flows, New J. Phys.22, 053052 (2020)

    work page 2020

  44. [44]

    Fodor and M

    ´E. Fodor and M. C. Marchetti, The statistical physics of active matter: From self-catalytic colloids to living cells, Phys. A: Stat. Mech. Appl.504, 106 (2018)

    work page 2018

  45. [45]

    F. J. Sevilla and M. Sandoval, Smoluchowski diffusion equation for active Brownian swimmers, Phys. Rev. E. 91, 052150 (2015)

    work page 2015

  46. [46]

    Kurzthaler, C

    C. Kurzthaler, C. Devailly, J. Arlt, T. Franosch, W. C. Poon, V. A. Martinez, and A. T. Brown, Probing the spatiotemporal dynamics of catalytic Janus particles with single-particle tracking and differential dynamic mi- croscopy, Phys. Rev. Lett.121, 078001 (2018)

    work page 2018

  47. [47]

    Martens, L

    K. Martens, L. Angelani, R. Di Leonardo, and L. Boc- quet, Probability distributions for the run-and-tumble bacterial dynamics: An analogy to the Lorentz model, Eur. Phys. J. E.35, 84 (2012)

    work page 2012

  48. [48]

    Semeraro, A

    M. Semeraro, A. Suma, I. Petrelli, F. Cagnetta, and G. Gonnella, Work fluctuations in the active Ornstein– Uhlenbeck particle model, J. Stat. Mech.: Theory Exp. 2021(12), 123202

    work page 2021

  49. [49]

    J. R. Howse, R. A. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Self-motile colloidal particles: from directed propulsion to random walk, Phys. Rev. Lett.99, 048102 (2007)

    work page 2007

  50. [50]

    Golestanian, Anomalous diffusion of symmetric and asymmetric active colloids, Phys

    R. Golestanian, Anomalous diffusion of symmetric and asymmetric active colloids, Phys. Rev. Lett.102, 188305 (2009)

    work page 2009

  51. [51]

    Maggi, M

    C. Maggi, M. Paoluzzi, N. Pellicciotta, A. Lepore, L. An- gelani, and R. Di Leonardo, Generalized energy equipar- tition in harmonic oscillators driven by active baths, Phys. Rev. Lett.113, 238303 (2014)

    work page 2014

  52. [52]

    Maggi, M

    C. Maggi, M. Paoluzzi, L. Angelani, and R. Di Leonardo, Memory-less response and violation of the fluctuation- dissipation theorem in colloids suspended in an active bath, Sci. Rep.7, 17588 (2017)

    work page 2017

  53. [53]

    Y. Kim, W. K. Kim, and J.-H. Jeon, Active diffusion of self-propelled particles in semiflexible polymer networks, Macromolecules57, 7735 (2024)

    work page 2024

  54. [54]

    Woillez, Y

    E. Woillez, Y. Kafri, and N. S. Gov, Active trap model, Phys. Rev. Lett.124, 118002 (2020)

    work page 2020

  55. [55]

    Goswami, S

    K. Goswami, S. Chaki, and R. Chakrabarti, Reconfigu- ration, swelling and tagged monomer dynamics of a sin- 15 gle polymer chain in Gaussian and non-Gaussian active baths, J. Phys. A: Math. Theor.55, 423002 (2022)

    work page 2022

  56. [56]

    H.-T. Han, S. Joo, T. Sakaue, and J.-H. Jeon, Nonequilib- rium diffusion of active particles bound to a semiflexible polymer network: Simulations and fractional Langevin equation, J. Chem. Phys.159, 024901 (2023)

    work page 2023

  57. [57]

    Semeraro, G

    M. Semeraro, G. Gonnella, A. Suma, and M. Zam- paro, Work fluctuations for a harmonically confined ac- tive Ornstein–Uhlenbeck particle, Phys. Rev. Lett.131, 158302 (2023)

    work page 2023

  58. [58]

    H.-T. Han, J. S. Lee, and J.-H. Jeon, Thermodynamic uncertainty relation for systems with active Ornstein– Uhlenbeck particles, PNAS Nexus4, pgaf160 (2025)

    work page 2025

  59. [59]

    Dechant, J

    A. Dechant, J. Garnier-Brun, and S.-i. Sasa, Thermody- namic bounds on correlation times, Phys. Rev. Lett.131, 167101 (2023)

    work page 2023

  60. [60]

    Halder and M

    S. Halder and M. Khan, Interplay between timescales governs the residual activity of a harmonically bound ac- tive Brownian particle, arXiv preprint arXiv:2505.07265 (2025)

    work page arXiv 2025

  61. [61]

    Narinder and E

    N. Narinder and E. Fischer-Friedrich, Time irreversibil- ity, entropy production, and effective temperature are in- dependently regulated in the actin cortex of living cells, Phys. Rev. X.16, 011007 (2026)

    work page 2026

  62. [62]

    Joo and J.-H

    S. Joo and J.-H. Jeon, Viscoelastic active diffusion gov- erned by nonequilibrium fractional Langevin equations: Underdamped dynamics and ergodicity breaking, Chaos Solit. Fractals.177, 114288 (2023)

    work page 2023