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Identifying the signatures of residual activity in harmonically bound active Brownian dynamics

T0 review · 2 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A trapped active particle's position distribution lies about its activity

desk verdict A useful corrective to the 'distribution shape = activity' reading of trapped ABPs, but the categorical 'devoid of residual activity' claim for regime II overstates what the paper's own equations and simulations show. read the letter →

arxiv 2607.22222 v1 pith:6NL5WWJB submitted 2026-07-24 cond-mat.soft cond-mat.stat-mech

Sanatan Halder , Manas Khan This is my paper · ORCID
classification cond-mat.softcond-mat.stat-mech
keywords activeBrownianparticleharmonictrapresidualactivitypositiondistributionpowerspectraldensitymeansquaredisplacementeffectiveconfinementpersistencetime
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether the shape of the steady-state position distribution of a self-propelled particle in a harmonic trap can tell whether the particle is still actively swimming or has been effectively tamed. The answer it reaches is no: in the regime where the distribution is Boltzmann-like (persistence time much shorter than the trap equilibration time), the trapped particle is still dominated by residual activity—the distribution broadens with propulsion speed, the residual velocity components add up to the full propulsion speed, and the power spectrum departs from the equilibrium prediction. In the opposite regime, where the distribution is bimodal or annular (persistence time much longer than trap equilibration), the particle's motion becomes statistically identical to a passive Brownian particle in a displaced harmonic well, with the propulsion exactly balanced by the restoring force at a distance Vτk. The paper therefore replaces the common 'passive vs active' reading of the distribution crossover with a timescale-controlled crossover from activity-dominated to activity-depleted dynamics.

What carries the argument

The central object is the residual velocity vres of the confined particle, i.e., the actual velocity of the trap-bound particle after the restoring force acts. Decomposed into radial and azimuthal components, vres directly reveals whether propulsion survives confinement: in regime I it retains the full propulsion speed, in regime II it reduces to the Brownian velocity of a passive particle. The argument is carried by the competition between the two timescales—the persistence time τR=1/DR and the trap equilibration time τk=γT/k—and by the pair of closed-form position distributions derived in the appendix under τR→0 or τR→∞ limits, together with the effective harmonic confinement parameters (k

What would settle it

Measure the radial power spectrum of an HBABP with τR≫τk at several propulsion speeds, from V=0 up to V=10 μm/s; the paper predicts all curves collapse onto the passive Lorentzian. If the PSD changes with V, or if the measured effective stiffness from Boltzmann inversion changes with V, the claim that residual activity vanishes in this regime is refuted. The complementary check in regime I is that keff must follow k/(1+Pe²/2).

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Extended reading notes

Core claim

The central claim is that the resultant dynamics of a harmonically bound active Brownian particle (HBABP) are controlled by the ratio of the orientation persistence time τR to the trap equilibration time τk, and that the shape of the steady-state position distribution is not a reliable activity diagnostic. For τR≪τk, the distribution is Gaussian/Boltzmann-like but activity-dominated: its variance grows as (kBT/k)(1+Pe²/2), residual radial and azimuthal velocities show bimodal distributions whose mean-square values sum to V², the MSD shows a ballistic rise, and the PSD deviates from fluctuation-dissipation. For τR≫τk, the 2D distribution is an annulus that looks bimodal in projection, yet the

Load-bearing premise

The conclusion that regime II is devoid of residual activity rests on the Appendix B, Case-2 idealization that τR→∞ (sinϕ(t)=0), so the propulsion is exactly balanced by the restoring force; the paper's own steady-state picture includes slow orientational diffusion (τR=100 s) producing a slow azimuthal drift and a marginal widening of P(vθ_res) with propulsion speed, so the clean dichotomy softens if that drift counts as residual activity.

Editorial extensions

If this is right

  • In weak traps or fast reorienting swimmers (τR≪τk), a Gaussian/Boltzmann-like position distribution is consistent with strong residual activity, so experiments cannot infer passivity from distribution shape alone.
  • In strong traps or slow reorienting swimmers (τR≫τk), an annular/bimodal distribution does not imply far-from-equilibrium dynamics; the particle is effectively a passive Brownian particle in a shifted well at r=Vτk.
  • Peclet number Pe alone is not a valid measure of the activity that survives confinement; the controlling dimensionless ratio is τR/τk.
  • The effective trap stiffness keff and center displacement rc give two clean experimental handles: keff=k/(1+Pe²/2) with rc=0 in regime I, and keff=k with rc=Vτk in regime II.
  • In regime II the power spectral density collapses onto the Lorentzian of the passive trapped particle at all propulsion speeds, providing a frequency-domain test that the active contribution has been neutralized.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's own Fig. 6(c) shows that the azimuthal residual velocity distribution in regime II widens slightly with propulsion speed; this suggests 'devoid of residual activity' is exact only in the τR→∞ limit, and on timescales approaching τR a weak orientational-activity channel remains—a point worth testing by measuring the azimuthal PSD.
  • The timescale-ratio criterion likely carries over to other self-propelled models in harmonic traps, such as run-and-tumble or Ornstein-Uhlenbeck active particles, where a similar residual-velocity decomposition could expose hidden activity beneath Boltzmann-like position distributions.
  • A practical extension for single-particle experiments: rather than fitting only the position distribution, collect simultaneous orientation and position trajectories and compute vres(t); the sum of mean-square radial and azimuthal residual velocities should equal V² in regime I and only the Brownian value in regime II, giving a direct per-trajectory activity meter.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The manuscript analyzes a two-dimensional overdamped active Brownian particle in an isotropic harmonic trap (HBABP), combining analytic limiting calculations with Langevin dynamics simulations in three regimes: τR ≪ τk, τR ≫ τk, and τR = τk. It derives closed-form position distributions, the MSD, effective harmonic confinement parameters, and compares simulated residual velocity components, MSDs, and power spectral densities with analytic predictions. The central claim is that the shape of the steady-state position distribution is not a reliable signature of activity: the regime with a Boltzmann-like position distribution (τR ≪ τk) is actually dominated by residual activity, whereas the regime with a bimodal position distribution (τR ≫ τk) is claimed to be effectively a harmonically bound Brownian particle at a displaced position, 'devoid of residual activity.'

Significance. If the central claim holds, the paper provides a useful correction to the common interpretation of the Boltzmann-to-bimodal crossover in trapped active particles and offers several experimentally accessible diagnostics: the widening of the position distribution with propulsion speed, the residual-velocity distributions, the ballistic MSD rise, the activity-dependent effective stiffness, the displaced harmonic center, and the collapse of the PSD onto the HBBP Lorentzian. The analytical limiting derivations in Appendices A–C are standard and the simulations agree with them in the three representative regimes. The paper also makes falsifiable predictions, e.g., keff decreases with Pe in regime I and rc = Vτk with unchanged keff in regime II.

major comments (2)
  1. [§III B 2, Appendix B Case-2, Eq. (11)] The claim that regime II is 'devoid of residual activity' is stronger than what the model equations imply. At the simulated parameter values (τR = 100 s, τk = 1 s), the orientation is not frozen: Eq. (11) contains the active azimuthal term Vθ, and Appendix B Case-2 obtains the displaced HBBP solution only by taking τR → ∞ and setting sinϕ(t) = 0 for t ≲ τR. The paper itself acknowledges a 'slow azimuthal drift' (Supp. Video 2) and a marginal widening of P(vθ_res) with V (Fig. 6(c)). This azimuthal drift is activity-driven and absent for V = 0, and it is removed from the PSD analysis by computing spectra from radial strips (Fig. 7(b)). To support the absolute 'devoid' statement, the authors need a quantitative bound on the azimuthal residual-velocity contribution, e.g., ⟨vθ_res²⟩_V − ⟨vθ_res²⟩_{V=0} compared with the HBBP Brownian contribution at the largest V used. Without such a bound,
  2. [Eqs. (5), (10); Figs. 2(d), 5(d)] The observable P(r) is not unambiguously defined. Eq. (5) is the 2D joint density P(x)P(y); the true marginal radial density for the Boltzmann-like regime is (r/σ²) exp(−r²/(2σ²)), not the zero-centered Gaussian in Eq. (5). Likewise, Eq. (10) omits the polar Jacobian for a marginal radial distribution, although it is a local Gaussian if P is measured along a narrow radial strip. The text in §II B mentions trajectory segments in radial strips, but the captions of Figs. 2(d) and 5(d) say 'radial position distributions P(r)' and fit them to Eqs. (5)/(10). This ambiguity matters because the Boltzmann-inversion fits for keff and rc use this quantity. Please define P(r) precisely — marginal radial density versus conditional density along a radial strip — and display the correct normalization or curve in each figure.
minor comments (3)
  1. [§III A 2, Fig. 3] The rms-sum identity ⟨(vr_res)²⟩ + ⟨(vθ_res)²⟩ ≈ V² is introduced without an analytical estimate or error bars. A short derivation or an uncertainty statement would make the claim quantitatively robust.
  2. [§III C 1, Fig. 10] The super-Gaussian shape parameter n is a fittable exponent; its reported values (n = 1.44 and 9.3) are descriptive. Please state explicitly that n is not derived and report fit uncertainties.
  3. [§II B] The velocity averaging window of 0.1 s is a free parameter in the residual-velocity analysis. It is at least ten times shorter than τR and τk, so the choice is reasonable, but the sensitivity of P(vr_res) and P(vθ_res) to this window should be reported in an appendix or supplement.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the analytical/simulation chain is self-contained. The only self-citation of note is companion Letter [35], which provides experimental corroboration, and the 'devoid of residual activity' phrasing is partially softened by the paper's own azimuthal-drift caveat.

full rationale

The paper's derivation chain does not reduce to its inputs. Closed-form position distributions (Eqs. 3-10) are derived from the Langevin equations (Eq. 1) via Gaussian-noise lemmas in Appendix B, with the limiting assumptions stated explicitly (tau_R << tau_k and tau_R >> tau_k). The MSD (Eq. 12) is derived in Appendix C and cross-checked against an external literature result [37]. The effective confinement parameters keff and rc in Eqs. 17-18 are read off those closed-form distributions; comparing them with Boltzmann-inverted simulated P(r) is a consistency check between analytics and numerics, not a fitted quantity used as a prediction. Simulation parameters (V, tau_R, tau_k, k, DT) are inputs, and no target observable is adjusted to force agreement. The central self-citation is the companion Letter [35], which is claimed to provide experimental validation of the crossover, but the analytical and simulation results stand independently, so this is a non-load-bearing self-citation rather than a circular step. The phrase 'devoid of residual activity' for regime II is stronger than the paper's own results: Sec. III B 2 reports P(vtheta_res) 'marginally widens with increasing propulsion speed' and Supp. Video 2 describes 'free and slow dynamics along the azimuthal direction.' This is a calibration/overstatement concern, not a circularity, because the residual-velocity statistics are computed from simulated trajectories rather than assumed by the derivation. Overall, no prediction is equivalent by construction to its input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No model parameters are fitted to data in this manuscript; V, τR, τk, DT, and k are simulation inputs. The only empirical fits are descriptive: super-Gaussian exponents n in Eq. 20 for the intermediate regime and keff/rc extracted from Boltzmann-inverted U(r), both corroborating rather than generating the theory. No new particles, forces, or fields are postulated; 'residual activity' and 'effective confinement' are summary observables, not entities.

free parameters (3)
  • super-Gaussian shape exponent n (radial) = 1.44
    Eq. 20 fit to P(vr_res) in the intermediate regime (Fig. 10); descriptive, not used to derive the central claim.
  • super-Gaussian shape exponent n (azimuthal) = 9.3
    Eq. 20 fit to P(vθ_res) in the intermediate regime (Fig. 10); descriptive, not used to derive the central claim.
  • velocity averaging window = 0.1 s
    Hand-chosen smoothing time in Sec. II B used to compute vres from trajectories; affects the reported distributions.
assumptions (5)
  • standard math The 2D HBABP obeys the Langevin equations (Eq. 1) with constant-speed propulsion and Gaussian white orientational noise.
    This is the standard ABP model, cited to refs. [5-8].
  • domain assumption Stokes-Einstein relation DT = kBT/γT and the Stokes drag coefficient set simulation parameters.
    Sec. II B sets DR, DT, and k using Stokes radius and water viscosity; standard external physics.
  • ad hoc to paper In regime I, the active noise is approximated as delta-correlated with coefficient 2DT + V²τR (Eq. B2).
    This white-noise reduction enables the Gaussian position distributions; valid only for τR/τk → 0.
  • ad hoc to paper In regime II, the orientation is frozen during equilibration (τR→∞, sinϕ(t)=0).
    Appendix B, Case-2; produces the shifted HBBP distributions but ignores slow azimuthal drift.
  • standard math Chandrasekhar's lemma (ref. [41]) for Gaussian integrals of delta-correlated noise holds.
    Used in Appendix B to derive the closed-form position distributions.

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Cite this review

Pith. "Pith review of Identifying the signatures of residual activity in harmonically bound active Brownian dynamics." pith.science (2026). https://pith.science/paper/6NL5WWJB

@misc{pith2026260722222,
  author       = {Pith},
  title        = {Pith review of: Identifying the signatures of residual activity in harmonically bound active Brownian dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6NL5WWJB}},
  note         = {Machine review of arXiv:2607.22222}
}
read the original abstract

A confined self-propelled particle exhibits a range of intriguing dynamical phenomena dictated by the interplay between the intrinsic activity of the particle and the imposed confinement. This competition manifests as a crossover in the steady-state position distribution of a harmonically bound active Brownian particle (HBABP) from Boltzmann-like to bimodal, commonly recognized as the passive and active regimes, respectively, upon variations in activity and confinement strength. We present a comprehensive analysis of the resultant dynamics of an HBABP employing analytical calculations and numerical simulations, examining the variations in the position distribution, residual or resultant velocity, mean square displacement, power spectral density, and effective harmonic confinement at varying activities in the characteristic regimes across the crossover. These analyses provide a reliable identification of the signature of residual or remnant activity in ABP dynamics after being impeded by the harmonic confinement. Our results show that the resultant HBABP dynamics in the regime with a Boltzmann-like position distribution is dominated by residual activity, and the motion in the other regime, with a bimodal position distribution, is similar to that of a harmonically bound Brownian particle--devoid of residual activity--at a displaced position, where the activity is balanced by the restoring force field.

Figures

Figures reproduced from arXiv: 2607.22222 by the authors.

Figure 1
Figure 1. An active Brownian particle (ABP) in a harmonic well [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Position distributions in regime - I, where [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Residual, i.e., resultant velocity distributions for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Mean square displacement (MSD) and power spectral [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Position distributions in regime - II, where [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Residual velocity distributions of the HBABP for [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 8
Figure 8. Figure 8: Dependence of the effective harmonic confinement on [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 7
Figure 7. Figure 7: MSD and PSD of the HBABP in regime - II ( [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 9
Figure 9. Figure 9: Simulated trajectories and position distributions of an HBABP in the intermediate regime with [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: Residual velocity distributions for the intermediate [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: MSD and PSD of HBABP dynamics, where [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Active Brownian motion in a single-relaxation viscoelastic fluid

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    An active Brownian particle in a Maxwell-Voigt fluid has the same mean-square displacement as an active particle in a diffusing harmonic trap, demonstrated experimentally with a Janus colloid.

Reference graph

Works this paper leans on

67 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [37]

    A. R. Sprenger, C. Bair, and H. L¨ owen, Active brown- ian motion with memory delay induced by a viscoelastic medium, Physical Review E105, 044610 (2022)

    2022

  2. [35]

    Ribeiro and F

    H. Ribeiro and F. Potiguar, Lane formation and crystal- lization of active matter in a narrow channel, Physica A: Statistical Mechanics and its Applications503, 849 (2018)

    2018

  3. [1]

    − x−x 0e−t/τk −V τk 1−e −t/τk 2 2σ2 2,t # (6) and P(y, y0;t) = 1q 2πσ 2 2,t ×exp

    with orientation ϕ0 at t = 0, the position and orientation of the Janus particle evolve to ( x, y) and ϕ, respectively, at time t. The particle position in plane-polar coordinates (gray) is given by (r, θ). along with thermally exited Brownian motion. Therefore, the persistence time of the active dynamics, over which the correlation of the propulsion dire...

  4. [2]

    3 6 and shown in Fig

    Position distribution When τR ≪τ k, P (x; t) and P (y; t) have the same form and widen symmetrically with time, as given by Eq. 3 6 and shown in Fig. 2(a), until they reach a steady-state. At steady-state, the 1D and 2D position distributions become Gaussian (Eq. 4) and Boltzmann-like (Eq. 5), respectively, with variance σ2 1 = kBT k 1 + Pe2 2 , which inc...

  5. [3]

    Residual velocity We obtained the components of the resultant or resid- ual velocity vr res(t) and vθ res(t) of the HBABP from the simulated trajectories at various propulsion speeds V . In this regime, both vr res and vθ res exhibit wide fluctuations with bimodal distributions consisting of two symmetri- cally placed Gaussians at the corresponding rms va...

  6. [4]

    Mean Square Displacement The MSD for this regime is expressed by Eq. 13. The MSD varies linearly with τ at very short times ( τ≪ τR), where Brownian diffusion dominates, and again at τR < τ < τk, before eventually saturating to a plateau value, as given by Eq. 14, at τ > τk. At intermediate time-lags, the last term dominates, providing a ballistic 7 Fig. ...

  7. [5]

    Effective harmonic confinement The effective harmonic confinement keff of the HBABP in this regime is given by Eq. 17. Here, a monotonic decrease in keff with Pe is a clear signature of the activity- governed resultant dynamics. We obtained the effective harmonic potential U (r) through the Boltzmann inversion of the simulated steady-state position distri...

  8. [6]

    The PSDs match the FDT prediction only at higher frequencies and deviate from it where the activity dominates over spontaneous Brownian dynamics, as shown in Fig

    Power Spectral Density We further computed the PSD from the simulated HBABP dynamics at varied V and compared them with that of the FDT prediction for passive Brownian diffusion, which is given by DT/π2f 2. The PSDs match the FDT prediction only at higher frequencies and deviate from it where the activity dominates over spontaneous Brownian dynamics, as s...

Show all 67 references
  1. [7]

    8 and shown in Fig

    Position distribution In this regime, the peak of the Gaussian position dis- tribution along ˆx, the initial direction of propulsion, i.e., P (x; t), shifts progressively outward until it reaches x = V τk at t≫τ k, as given by Eq. 8 and shown in Fig. 5(a). The position distrib...

  2. [8]

    For τR ≫τ k, the distributions of both vr res and vθ res are Gaussian with significantly smaller rms values ( ∼ 1 µm s−1) than the propulsion speed (10 µm s−1), as exhibited in Fig

    Residual velocity The residual velocity components vr res(t) and vθ res(t) of the HBABP were obtained from the simulated trajec- tories at various propulsion speeds, V . For τR ≫τ k, the distributions of both vr res and vθ res are Gaussian with significantly smaller rms values...

  3. [9]

    Mean Square Displacement At τR ≫τ k, the analytical closed-form MSD is given by Eq. 15. The MSD is the sum of two similar terms, where it grows linearly and eventually saturates at timescales τk and τR, respectively. At a shorter time, τ≫τ k, the ABP equilibrates in the harmon...

  4. [10]

    Effective harmonic confinement In this regime, the stiffness of the effective harmonic con- finement keff and its position rc, representing the steady- state bound dynamics of the HBABP, are expressed by Eq. 18. Here, the activity of the ABP does not alter the effective stiffn...

  5. [11]

    19), in Fig

    Power Spectral Density The PSDs computed from the simulated radial HBABP dynamics (considering segments of a trajectory from within a radial trip) in this regime at various V val- ues are compared with those of the FDT prediction for passive Brownian diffusion ( DT/π2f 2) and ...

  6. [12]

    We simulated the HBABP dynamics with τR = τk = 10 s

    Position distribution In this regime, the positional distribution does not have an analytical closed-form solution. We simulated the HBABP dynamics with τR = τk = 10 s. The simulated trajectories are bound, space-filling, and center-avoiding (Supp. Video 3), as shown in Fig. 9...

  7. [13]

    Residual velocity We further computed the residual velocity components vr res(t) and vθ res(t) of the HBABP from the simulated tra- jectories with τR = τk. Here, the probability distributions of both vr res and vθ res are neither bimodal nor purely Gaus- sian, rather described...

  8. [14]

    12, diverges at τR = τk

    Mean Square Displacement The generic MSD expression, which is given by Eq. 12, diverges at τR = τk. We derived the closed-form analytical solution for MSD considering τR = τk = τeq (Appendix C), and it is expressed as, ⟨∆r2(τ)⟩= 4kBT k 1−e −τ /τeq +V 2τ 2 eq 1−e −τ /τeq 1− τ τ...

  9. [15]

    − x−x 0e−t/τk 2 2σ2 1,t # .(B6) The same equivalence relations hold for y axis. Following the same steps, we obtain P(y, y0;t) = 1q 2πσ 2 1,t exp

    Power Spectral Density We further computed PSDs from the simulated HBABP dynamics at τR = τk = 10 s and compared them with the 12 Fig. 10. Residual velocity distributions for the intermediate regime ( τk = τR = 10 s). A short segment (50 s) of the time series of the radial and...

    2020

  10. [16]

    Starting from the center, the particle exhibits bound dynamics, where the instantaneous direction of the resultant or residual velocity vres (Eq

    is represented by an orange color gradient. Starting from the center, the particle exhibits bound dynamics, where the instantaneous direction of the resultant or residual velocity vres (Eq. 11) is shown by a red arrow. Simulation time-step = 1 ms. (playback speed: 1×) Supp. Vi...

  11. [17]

    Ramaswamy, The mechanics and statistics of active matter, Annual Review of Condensed Matter Physics1, 323 (2010)

    S. Ramaswamy, The mechanics and statistics of active matter, Annual Review of Condensed Matter Physics1, 323 (2010)

    2010

  12. [18]

    Romanczuk, M

    P. Romanczuk, M. B¨ ar, W. Ebeling, B. Lindner, and L. Schimansky-Geier, Active brownian particles: From in- dividual to collective stochastic dynamics, The European Physical Journal Special Topics202, 1 (2012)

    2012

  13. [19]

    M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrody- namics of soft active matter, Reviews of Modern Physics 85, 1143 (2013)

    2013

  14. [20]

    M. E. Cates and J. Tailleur, Motility-induced phase sepa- ration, Annual Review of Condensed Matter Physics6, 219 (2015)

    2015

  15. [21]

    Bechinger, R

    C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, Active Particles in Complex and Crowded Environments, Reviews of Modern Physics88, 045006 (2016)

    2016

  16. [22]

    Fodor and M

    E. Fodor and M. Cristina Marchetti, The statistical physics of active matter: From self-catalytic colloids to living cells, Physica A: Statistical Mechanics and its Ap- plications504, 106 (2018)

    2018

  17. [23]

    U. Basu, S. N. Majumdar, A. Rosso, and G. Schehr, Active Brownian motion in two dimensions, Physical Review E 98, 062121 (2018)

    2018

  18. [24]

    Ten Hagen, S

    B. Ten Hagen, S. Van Teeffelen, and H. L¨ owen, Brownian motion of a self-propelled particle, Journal of Physics: Condensed Matter23, 194119 (2011)

    2011

  19. [25]

    A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, Hy- drodynamic attraction of swimming microorganisms by surfaces, Physical Review Letters101, 038102 (2008)

    2008

  20. [26]

    Elgeti and G

    J. Elgeti and G. Gompper, Wall accumulation of self- propelled spheres, EPL (Europhysics Letters)101, 48003 (2013)

    2013

  21. [27]

    C. F. Lee, Active particles under confinement: Aggrega- tion at the wall and gradient formation inside a channel, New Journal of Physics15, 055007 (2013)

    2013

  22. [28]

    B¨ auerle, R

    T. B¨ auerle, R. C. L¨ offler, and C. Bechinger, Formation of stable and responsive collective states in suspensions of active colloids, Nature Communications11, 2547 (2020)

    2020

  23. [29]

    F. A. Lavergne, H. Wendehenne, T. B¨ auerle, and C. Bechinger, Group formation and cohesion of active particles with visual perception–dependent motility, Sci- ence364, 70 (2019)

    2019

  24. [30]

    Hennes, K

    M. Hennes, K. Wolff, and H. Stark, Self-Induced Polar Order of Active Brownian Particles in a Harmonic Trap, Physical Review Letters112, 238104 (2014)

    2014

  25. [31]

    A. P. Solon, M. E. Cates, and J. Tailleur, Active brownian particles and run-and-tumble particles: A comparative study, The European Physical Journal Special Topics224, 1231 (2015)

    2015

  26. [32]

    Ginot, I

    F. Ginot, I. Theurkauff, D. Levis, C. Ybert, L. Bocquet, L. Berthier, and C. Cottin-Bizonne, Nonequilibrium equa- tion of state in suspensions of active colloids, Physical Review X5, 011004 (2015)

    2015

  27. [33]

    Y. Fily, A. Baskaran, and M. F. Hagan, Dynamics of self- propelled particles under strong confinement, Soft Matter 10, 5609 (2014)

    2014

  28. [34]

    Ben-Isaac, E

    E. Ben-Isaac, E. Fodor, P. Visco, F. van Wijland, and N. S. Gov, Modeling the dynamics of a tracer particle in an elastic active gel, Physical Review E92, 012716 (2015)

    2015

  29. [36]

    Narinder, J

    N. Narinder, J. R. Gomez-Solano, and C. Bechinger, Ac- tive particles in geometrically confined viscoelastic fluids, New Journal of Physics21, 093058 (2019)

    2019

  30. [38]

    Moore, J

    F. Moore, J. Russo, T. B. Liverpool, and C. P. Royall, Active brownian particles in random and porous envi- ronments, The Journal of Chemical Physics158, 104907 (2023)

    2023

  31. [39]

    Caraglio and T

    M. Caraglio and T. Franosch, Analytic Solution of an Active Brownian Particle in a Harmonic Well, Physical Review Letters129, 158001 (2022)

    2022

  32. [40]

    U. Basu, S. N. Majumdar, A. Rosso, and G. Schehr, Long- time position distribution of an active Brownian particle in two dimensions, Physical Review E100, 062116 (2019)

    2019

  33. [41]

    Pototsky and H

    A. Pototsky and H. Stark, Active Brownian particles in two-dimensional traps, EPL (Europhysics Letters)98, 50004 (2012)

    2012

  34. [42]

    Dauchot and V

    O. Dauchot and V. D´ emery, Dynamics of a Self-Propelled Particle in a Harmonic Trap, Physical Review Letters 122, 068002 (2019)

    2019

  35. [43]

    Chaudhuri and A

    D. Chaudhuri and A. Dhar, Active Brownian particle in harmonic trap: Exact computation of moments, and re-entrant transition, Journal of Statistical Mechanics: Theory and Experiment2021, 013207 (2021)

    2021

  36. [44]

    Malakar, A

    K. Malakar, A. Das, A. Kundu, K. V. Kumar, and A. Dhar, Steady state of an active Brownian particle in a two-dimensional harmonic trap, Physical Review E 101, 022610 (2020)

    2020

  37. [45]

    Santra, U

    I. Santra, U. Basu, and S. Sabhapandit, Direction revers- ing active brownian particle in a harmonic potential, Soft Matter17, 10108 (2021)

    2021

  38. [46]

    Baldovin, D

    M. Baldovin, D. Gu´ ery-Odelin, and E. Trizac, Control of active brownian particles: An exact solution, Physical Review Letters131, 118302 (2023)

    2023

  39. [47]

    S. Das, G. Gompper, and R. G. Winkler, Confined active Brownian particles: Theoretical description of propulsion- induced accumulation, New Journal of Physics20, 015001 (2018)

    2018

  40. [48]

    Buttinoni, L

    I. Buttinoni, L. Caprini, L. Alvarez, F. J. Schwarzen- dahl, and H. L¨ owen, Active colloids in harmonic optical potentials (a), Europhysics Letters140, 27001 (2022)

    2022

  41. [49]

    S. C. Takatori, R. De Dier, J. Vermant, and J. F. Brady, Acoustic trapping of active matter, Nature Communica- tions7, 10694 (2016)

    2016

  42. [50]

    Schmidt, H

    F. Schmidt, H. ˇS ´ ıpov´ a-Jungov´ a, M. K¨ a ll, A. W¨ urger, and G. Volpe, Non-equilibrium properties of an active nanoparticle in a harmonic potential, Nature Communi- cations12, 1902 (2021)

    1902

  43. [51]

    Halder and M

    S. Halder and M. Khan, Interplay between timescales governs the residual activity of a harmonically bound active brownian particle (2026)

    2026

  44. [52]

    J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Self-Motile Colloidal Particles: From Directed Propulsion to Random Walk, 19 Physical Review Letters99, 048102 (2007)

    2007

  45. [53]

    Jiang, N

    H.-R. Jiang, N. Yoshinaga, and M. Sano, Active Motion of a Janus Particle by Self-Thermophoresis in a Defocused Laser Beam, Physical Review Letters105, 268302 (2010)

    2010

  46. [54]

    Halder and M

    S. Halder and M. Khan, Dynamically stable optical trap- ping of thermophoretically active janus colloids, ACS Photonics13, 757 (2026)

    2026

  47. [55]

    Doi,Soft Matter Physics(Oxford university press, Oxford, 2013)

    M. Doi,Soft Matter Physics(Oxford university press, Oxford, 2013)

    2013

  48. [56]

    Halder, D

    S. Halder, D. Chanda, D. Mondal, S. Kundu, and M. Khan, Optical micromanipulation of soft materials: Applications in devices and technologies, inSoft Materials for Functional Applications(Springer Nature Singapore,

  49. [57]

    Chandrasekhar, Stochastic problems in physics and astronomy, Reviews of Modern Physics15, 1 (1943)

    S. Chandrasekhar, Stochastic problems in physics and astronomy, Reviews of Modern Physics15, 1 (1943)

    1943

  50. [58]

    Khan and T

    M. Khan and T. G. Mason, Random walks of colloidal probes in viscoelastic materials, Physical Review E89, 042309 (2014)

    2014

  51. [59]

    Khan and T

    M. Khan and T. G. Mason, Trajectories of probe spheres in generalized linear viscoelastic complex fluids, Soft Mat- ter10, 9073 (2014)

    2014

  52. [60]

    Szamel, Self-propelled particle in an external potential: Existence of an effective temperature, Physical Review E 90, 012111 (2014)

    G. Szamel, Self-propelled particle in an external potential: Existence of an effective temperature, Physical Review E 90, 012111 (2014)

    2014

  53. [61]

    Garcia-Millan and G

    R. Garcia-Millan and G. Pruessner, Run-and-tumble mo- tion in a harmonic potential: field theory and entropy production, Journal of Statistical Mechanics: Theory and Experiment2021, 063203 (2021)

    2021

  54. [62]

    Semeraro, G

    M. Semeraro, G. Gonnella, A. Suma, and M. Zamparo, Work fluctuations for a harmonically confined active ornstein-uhlenbeck particle, Physical Review Letters131, 158302 (2023)

    2023

  55. [63]

    Dutta, A

    D. Dutta, A. Kundu, S. Sabhapandit, and U. Basu, Har- monically trapped inertial run-and-tumble particle in one dimension, Physical Review E110, 044107 (2024)

    2024

  56. [64]

    A. Sun, F. Ye, and R. Podgornik, Exact moments for a run-and-tumble particle with a finite tumble time in a harmonic trap, Physical Review E111, 044136 (2025)

    2025

  57. [65]

    R. Ni, M. A. C. Stuart, and M. Dijkstra, Pushing the glass transition towards random close packing using self-propelled hard spheres, Nature Communications4, 10.1038/ncomms3704 (2013)

    2013 doi

  58. [66]

    Berthier and J

    L. Berthier and J. Kurchan, Non-equilibrium glass transi- tions in driven and active matter, Nature Physics9, 310 (2013)

    2013

  59. [67]

    L. M. C. Janssen, Active glasses, Journal of Physics: Condensed Matter31, 503002 (2019)

    2019

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