Gigantic dynamical spreading and anomalous diffusion of jerky active particles

Hartmut L\"owen

REVIEW 3 major objections 6 minor 1 cited by

Gigantic dynamical spreading and anomalous diffusion of jerky active particles

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A third-order jerk term in a particle's equation of motion produces superballistic spreading with MSD scaling as t^6, t^5, t^4, or t^3, and can eject the particle from a harmonic trap.

desk verdict Novel model and careful algebra, but the headline superballistic exponents come from a gamma=0 limit that kills the noise, so the central numbers are currently artifacts. read the letter →

arxiv 2507.08910 v1 pith:ETCYSZYV submitted 2025-07-11 cond-mat.soft

Hartmut Löwen This is my paper · ORCID

classification cond-mat.soft

keywords jerkactiveOrnstein-Uhlenbeckprocessanomalousdiffusionmean-squaredisplacementsuperballisticspreadinglocalization-delocalizationtransitionfeedback-controlledparticlesmemoryfriction

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

The paper proposes a minimal model of a 'jerky active particle': a self-propelled Brownian particle whose equation of motion contains the third derivative of position, the jerk. Because activity is modeled by an active Ornstein-Uhlenbeck process, the full system is linear, and the author derives the mean-square displacement exactly from the Green's function. The central result is that for small damping and small spring constants the MSD grows superballistically with time exponents 6, 5, 4, or 3, depending on the competition between jerk, inertia, friction, and persistence time; these are far above the exponents of ordinary or ballistic diffusion. In a harmonic trap the same mechanism produces a localization-delocalization transition, so a sufficiently jerky particle escapes confinement, with a first- or second-order transition line in the dimensionless parameter plane. The paper also spells out two experimental realizations: feedback-controlled macroscopic particles and active colloids with memory friction, so the predictions are directly testable.

What carries the argument

The central object is the jerk coefficient $\lambda$, which enters the third-order equation of motion $\lambda \dddot x + m\ddot x + \gamma \dot x + kx = \gamma u(t)$. The calculation is carried by the Green's function $G(t)$, the response of this third-order system to a $\delta$-kick, whose Fourier transform is $1/(i\lambda\omega^3 - m\omega^2 - i\gamma\omega + k)$, and by the characteristic cubic $(\omega-\omega_1)(\omega-\omega_2)(\omega-\omega_3) = \omega^3 + i(m/\lambda)\omega^2 - (\gamma/\lambda)\omega - ik/\lambda$, whose roots come from Cardano's formula. The MSD is the double convolution of $G$ with the activity memory kernel $M(t) = \gamma^2 v_0^2 \exp(-|t|/\tau_p)$, and the stability condition that all three roots have negative imaginary parts, stated in Appendix A, separates the localized from the delocalized phase. All results reduce to two dimensionless control parameters: the scaled jerk $\tilde\lambda = \lambda\gamma^3/k^2$ and the scaled mass $\tilde m = \gamma^2/k$.

What would settle it

Take a single particle on a tilted plane with camera feedback that sets the tilt angle to a measured function of the particle's acceleration with a known small delay $\delta t$, and measure the mean-square displacement; if the short-time exponent is not close to 6 (or the later exponents are not 5, 4, or 3 in the predicted crossover windows), the linear jerk model is falsified. In the trapped case, increasing the jerk strength while measuring the steady-state spread $a^2$ should show $1/a^2$ dropping to zero exactly on the stability boundary given by condition (A2); failure to find a transition where the boundary predicts one would also falsify the delocalization claim.

Watch

Extended reading notes

Core claim

The central claim is that including jerk, the third time derivative of position arising from a force that depends on the delayed measurement of acceleration, fundamentally changes how an active particle spreads. Starting from the linearized feedback force $F(\ddot z(t-\delta t)) \approx F(z_0) + F'(z_0)(\ddot z(t) - \delta t \dddot z(t))$, the paper solves the force balance $\lambda \dddot x + m\ddot x + \gamma \dot x + kx = \gamma u(t)$, with $u(t)$ an active Ornstein-Uhlenbeck noise of persistence time $\tau_p$. The MSD is obtained as a double convolution of the Green's function with the activity memory kernel, and it scales as $t^6$ at short times and as $t^5$, $t^4$, or $t^3$ at later times in the regimes of small damping and small spring constants. For a harmonically trapped particle, the three eigenfrequencies of the characteristic cubic determine stability; when the jerk coefficient $\lambda$ is large enough, the inverse localization length $1/a^2$ drops to zero along a first-order line ($\tilde\lambda = 0$ for $\tilde m > 0$) and a second-order line $\tilde\lambda_c(\tilde m)$, meaning the particle leaves the trap. The kinetic and spreading temperatures diverge at the transition.

Load-bearing premise

The load-bearing premise is the linearized feedback expansion $F(\ddot z(t-\delta t)) \approx F(z_0) + F'(z_0)(\ddot z(t) - \delta t \dddot z(t))$, which drops all nonlinear acceleration dependence and all delay corrections of order $\delta t^2$ and higher; the paper itself notes that Coulomb friction is relevant in experiments, and either nonlinearity could remove the predicted $t^6$ scaling and the delocalization transition.

Editorial extensions

If this is right

Any linear feedback-controlled particle whose force depends on delayed acceleration obeys the same equation, so the predicted exponents 6, 5, 4, and 3 should show up in macroscopic feedback experiments with small damping and weak springs.
In a harmonic trap, jerk reverses the usual effect of the spring: increasing the spring constant $k$ favors delocalization, and the particle can escape confinement along first- or second-order transition lines.
The model predicts that effective kinetic and spreading temperatures diverge at the delocalization transition, so the trapped jerky particle is no longer thermalized by the potential.
The same third-order structure emerges from the expansion of a memory-friction kernel in Fourier space, making active colloids in viscoelastic solvents a second, microscopic experimental realization.
For the pure-jerk limit without inertia or friction the long-time MSD scales as $t^5$, and adding inertia changes the long-time exponent to $t^3$ (active noise) or $t^4$ (persistent noise), with crossovers at the persistence and inertial times.

Reading between the lines

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

A testable extension: measuring the short-time MSD exponent of a feedback-controlled particle provides a direct estimate of the jerk coefficient $\lambda$ and the delay time $\delta t$, since the $t^6$ regime sets in before any crossover time.
Because the full process is Gaussian, the model also predicts closed-form joint distributions of position, velocity, acceleration, and noise; comparing these to experimental histograms would test the linearization more stringently than the MSD alone.
The first- versus second-order character of the delocalization transition should be distinguishable in a colloid experiment by scanning the memory kernel's second moment $\Gamma_2$ (via solvent composition) and watching whether $1/a^2$ jumps or vanishes continuously.
The stability criterion of Appendix A can serve as a design rule for feedback-controlled robotic particles: choose $\tilde\lambda$ and $\tilde m$ to either keep the particle localized or trigger a controlled release from a trap.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the one-sentence take: the model is novel and the analytic machinery is serious, but the headline exponents 6 and 5 sit on a self-contradictory limit, because Eq. (2) drives the particle with gamma*u(t) and the special cases that produce those exponents set gamma=0.

What is actually new: the combination of a jerk term with AOUP activity is new relative to the cited literature, and the paper solves the linear model exactly via Green's functions. The general formalism in Sections II and IV, the crossover tables, and the localization-delocalization transition for the full damped system (gamma>0) look internally consistent. Known limits such as the overdamped Brownian oscillator are recovered, and the experimental proposals are creative, if speculative.

The soft spot, and it is a load-bearing one, is the pure jerky (Section III.A) and inertial jerky (Section III.B) limits. Eq. (2) has gamma on both sides: the Stokes friction gamma*x' on the left and the activity gamma*u(t) on the right. The pure jerky particle is defined by m=0, gamma=0, k=0. Then for t>0 the equation reduces to lambda*x''' = 0, so a particle starting from rest stays at rest. The Green's function G(t)=t^2/(2*lambda) describes the response to a delta-kick, but the noise term in the MSD is gamma^2*v0^2 times a convolution; with gamma=0 that MSD is identically zero, not t^6 or t^5. Eq. (23) is proportional to gamma^2*v0^2, so it vanishes in the limit the section defines. The same problem hits Eq. (27). The abstract's exponents 6,5,4,3 are taken directly from these sections. The paper never specifies a double limit gamma->0, v0->infinity with gamma^2*v0^2 fixed, and if you try to rescue the case by keeping gamma>0 and setting m=k=0, the characteristic polynomial develops an oscillatory pair and the advertised t^5 long-time tail disappears. So the central new numbers are, as written, an artifact of an inconsistent limit.

Secondary issues: several lengthy formulas (e.g., Eq. (27), (34)) are stated without derivation; the first-order transition line is asserted more than derived; and no numerical or experimental verification is provided. Those are minor compared to the gamma problem.

Who this is for: an active-matter theorist interested in higher-order linear stochastic models. The general gamma>0 framework may be salvageable and the paper deserves a serious referee, but the referee should send it back for major revision. I would not cite the headline exponents in their current form.

Referee Report

3 major / 6 minor

Summary. The paper introduces a one-dimensional linear 'jerky active particle' model, Eq. (2): λx''' + mx'' + γx' + kx = γu(t), driven by an active Ornstein-Uhlenbeck process. It solves the linear system via Green's functions and obtains closed-form mean-square displacements for several special cases, reporting superballistic scaling exponents 6, 5, 4, and 3 (Table I) and a localization-delocalization transition for a harmonic trap, with first- and second-order branches. It also sketches experimental realizations via feedback-controlled macroscopic particles and colloids with memory friction.

Significance. If correct, the model is a useful exactly solvable extension of inertial active matter, and the Green's-function approach transparently recovers known limits such as the overdamped Brownian oscillator. The crossover tables and the phase diagram in the dimensionless jerk-mass plane are potentially valuable benchmarks. However, the headline exponents and the accompanying kinetic-energy exponents are currently derived in a parameter limit that is internally inconsistent with the stated equation of motion, so the central numerical claims cannot be accepted in the present form.

major comments (3)

[III.A, III.B; Eqs. (2), (23), (27); Table I] The pure-jerky and inertial-jerky special cases set γ=0 on the left-hand side of Eq. (2) while retaining the active force γu(t) on the right-hand side. Under Eq. (2) as written, γ=0 makes the stochastic drive vanish identically, so with zero initial conditions x(t)≡0 and MSD(t)=0. Equations (23) and (27), and the exponents 6→5 and 6→3 in Table I and Figs. 3–4, are therefore not solutions of the stated equation in that limit. The paper never defines a singular double limit γ→0, v0→∞ with γ²v0² fixed, nor does it introduce a noise amplitude independent of γ. This is load-bearing because these limits are the source of the headline exponents 6, 5, 4, and 3; the model needs an independent activity amplitude A, e.g. M(t)=A²v0²exp(−|t|/τp), or an explicit statement of the limit under which the formulas are asserted.
[III.A, Eq. (20), Table III] For the pure-jerky particle, m=0 by definition. Equation (20) then gives Wkin(t)=0 identically, yet Table III and the text of Section III.A report finite dynamical exponents for the mean kinetic energy of this case (4 and 3 for active noise, 4 and 4 for persistent, 3 and 3 for passive). Either the kinetic energy must be defined with a finite test mass, or these entries and the accompanying discussion must be removed; as written, the table is internally inconsistent with the model definition.
[IV.A, text before Eq. (37); Eqs. (37), (46)–(48); Fig. 6] The parameters printed as dimensionless jerk λ~=λγ³/k² and dimensionless mass m~=γ²/k are not dimensionless, and m~ does not contain the mass m. Yet these parameters define the phase diagram in Fig. 6 and the transition lines in Eqs. (37) and (46)–(48). The subsequent asymptotics, e.g. λ~c(m~)=m~+O(m~²) for small mass, are consistent instead with λ~=λk²/γ³ and m~=mk/γ², suggesting a typographical error. Please correct the definitions and re-derive or re-state the phase diagram accordingly, since the localization-delocalization claim depends on this parameter plane.

minor comments (6)

[III.A, after Eq. (23)] The persistent-noise limit of the pure-jerky MSD should be γ²v0²t⁶/(36λ²), without a factor τp in the denominator; the printed expression appears to contain a spurious τp. Please check and correct.
[II.B.2, Eq. (11)] With the Fourier convention used in Eqs. (13) and (15), the expansion Γ0+iωΓ1−Γ2ω²/2 leads to m=−Γ1 in Eq. (2), not m=Γ1. Please verify the sign and state the convention explicitly, since the experimental mapping for memory-friction colloids is affected.
[III.B, Eq. (27)] The notation BI1 and BI0 should be defined explicitly as B I1 and B I0, and the factors of B in the subsequent terms are hard to follow; a short derivation or a more transparent notation would help.
[Abstract and Table II] The abstract's phrase 'for small damping and small spring constants' may mislead: for any nonzero γ with k=0, the long-time exponent in Table I is 1, and the high exponents 5, 4, and 3 survive only in intermediate time windows (Table II). Please clarify that the long-time superballistic exponents are specific to the γ=0 limits.
[II.B.1] The tilted-plane feedback experiment realizes Eq. (2) only in the small-angle linearization of sin φ; the text says it 'represents exactly' the model. Please soften this to 'approximately' or state the linearization assumptions.
[III.D] The word 'desastrous' is a typo; it should be 'disastrous'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the model equation is postulated and all MSD exponents, crossover scalings, and the localization-delocalization transition are derived from it analytically without fitting parameters to the predicted outputs.

full rationale

The paper proposes the linear jerk equation of motion, Eq. (2), and then solves it by Green's functions and residue calculus. The key outputs — the MSD scaling exponents 6, 5, 4, and 3, the crossover time scales in Tables I and II, and the localization-delocalization transition in Sec. IV.C — follow from explicit analytic expressions such as Eqs. (22), (23), (26), (27), and (38). No parameter is fitted to the target MSD or to the transition line, and no output is inserted back into the input equation. The linearization in Eq. (1) is a stated modeling assumption, not a quantity claimed to be derived from later results. The active Ornstein-Uhlenbeck correlation, Eq. (6), is a standard textbook result cited to the original literature, and the cited previous jerk-dynamics works are not used as substitutes for the derivations performed here. Self-citations appear mainly as contextual references (e.g., [54], [66], [84]), and none of them supplies the central claim. A separate concern, noted by a skeptical reader, is that setting gamma=0 in Secs. III.A and III.B makes the noise term in Eq. (2) vanish while the reported MSDs remain proportional to gamma^2; this is an internal-consistency or limiting-procedure issue about the validity of the parameter regime, not a circularity in which the predicted quantity is equivalent to an input by construction. The derivation chain is self-contained, so no circularity steps are identified.

Assumptions & free parameters 6 free parameters · 6 assumptions · 2 invented entities

The central claims rest on a postulated linear third-order equation with AOUP noise, not on fitted data. The model parameters are inputs, the stability condition is imposed, and the only invented concepts are the model class itself and the associated negative effective mass.

free parameters (6)

jerk coefficient λ
Coefficient of the third-order term in Eq (2); a model input, not fitted to data.
effective inertial mass m
Renormalized mass from feedback in Eq (3); input, can be negative.
Stokes friction γ
Friction coefficient in Eq (2); input.
spring constant k
Harmonic trap stiffness; input.
persistence time τp
AOUP correlation time in Eq (5); input.
self-propulsion speed v0
Activity amplitude in Eq (5); input.

assumptions (6)

ad hoc to paper Feedback force linearization, Eq (1): F(z¨(t-δt)) ≈ F(z0)+F'(z0)(z¨(t)-δt z⃛(t)), neglecting nonlinear and higher-order delay terms.
Defines the linear third-order model that generates all results.
domain assumption Activity is an AOUP process, Eq (5), with Gaussian white noise ζ and correlation time τp.
Standard active particle noise model; used in Eq (14) for MSD.
domain assumption Stability condition (A2): all three characteristic roots have non-positive imaginary parts.
Restricts to the regime where steady state exists; delocalization is defined by violating this condition.
domain assumption Initial conditions x(0)=ẋ(0)=ẍ(0)=0.
Used to set up the Green's function solution; steady state is independent of initial conditions for k>0.
domain assumption No fluctuation-dissipation relation: active noise amplitude γv0 is independent of γ.
Separates active from thermal noise; allows the reported scaling.
standard math Cardano's formula and the residue theorem are valid and applied to the cubic and Green's function integrals.
Technical tools used in Appendix A and Eq (15).

invented entities (2)

negative effective inertial mass
purpose: Explains how acceleration-dependent feedback can make the inertia term change sign, Eq (3).
No direct observable is proposed to isolate negative mass separately from the jerk term; the predicted MSD reflects both effects combined.
jerky active particle model class independent evidence
purpose: New model combining jerk with activity, Eq (2).
Predicts MSD exponents 6,5,4,3 and a trap-escape transition that can be tested in feedback experiments or simulations.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Gigantic dynamical spreading and anomalous diffusion of jerky active particles." pith.science (2026). https://pith.science/paper/ETCYSZYV

@misc{pith2026250708910,
  author       = {Pith},
  title        = {Pith review of: Gigantic dynamical spreading and anomalous diffusion of jerky active particles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ETCYSZYV}},
  note         = {Machine review of arXiv:2507.08910}
}

read the original abstract

Jerky active particles are Brownian self-propelled particles which are dominated by ``jerk'', the change in acceleration. They represent a generalization of inertial active particles. In order to describe jerky active particles, a linear jerk equation of motion which involves a third-order derivative in time, Stokes friction and a spring force is combined with activity modeled by an active Ornstein-Uhlenbeck process. This equation of motion is solved analytically and the associated mean-square displacement (MSD) is extracted as a function of time. For small damping and small spring constants, the MSD shows an enormous superballistic spreading with different scaling regimes characterized by anomalous high dynamical exponents 6, 5, 4 or 3 arising from a competition between jerk, inertia and activity. When exposed to a harmonic potential, the gigantic spreading tendency induced by jerk gives rise to an enormous increase of the kinetic temperature and even to a sharp localization-delocalization transition, i.e. a jerky particle can escape from harmonic confinement. The transition can be either first or second order as a function of jerkiness. Finally it is shown that self-propelled jerky particles governed by the basic equation of motion can be realized experimentally both in feedback-controlled macroscopic particles and in active colloids governed by friction with memory.

Figures

Figures reproduced from arXiv: 2507.08910 by the authors.

**Figure 2.** FIG. 2. Green’s function of a jerky particle in units of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. a) Mean-square displacement MSD( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. a) Mean-square displacement MSD( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. a) Mean-square jerk MSJ( [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. State diagram and relaxational dynamics of a jerky harmonic oscillator as a function of the dimensionless jerk [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: FIG. 7. a) Order parameter [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

Discussion (0). Continue with ORCID to comment.

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

Active Brownian particles in power-law viscoelastic media
cond-mat.soft 2025-12 unverdicted novelty 6.0 of 10

An active Brownian particle in a power-law viscoelastic medium exhibits a stretched superdiffusive persistence phase (MSD ~ t^{2-alpha_R}) and a modified persistence-diffusion relation.

Reference graph

Works this paper leans on

132 extracted references · 72 canonical work pages · cited by 1 Pith paper

[1]

(9) We refer to this noise as passive noise and correspond- ingly the particle will be called “passive”

(8) In the opposite limit of a very small persistence time τp, taken such that D=v2 0τp stays constant, we obtain 3 (Markovian) white noise characterized by the force mem- ory kernel Mw(t)=2γ2Dδ(t). (9) We refer to this noise as passive noise and correspond- ingly the particle will be called “passive”. This special case was treated previously with jerk-li...
[2]

all the past particle velocities contribute to its total friction

Colloids with memory-dominated friction Here it is shown that the basic equation (2) is also re- alized in overdamped active colloids exposed to friction that includes memory, i.e. all the past particle velocities contribute to its total friction. Such effects do occur for active colloidal particles in non-Newtonian, viscoelastic solvents. For an active c...
[3]

active jerks

Macroscopic feedback experiment A macroscopic realization of an active jerky particle is shown in Figure 1. This demonstrates that “active jerks” can conveniently be implemeted in experiments. Consider the one-dimensional motion of an inertial par- ticle on a tilted plane where the tilt angle is controlled by external feedback: a camera measures the veloc...
[4]

A. V. Ivlev, J. Bartnick, M. Heinen, C.-R. Du, V. Nosenko, and H. L¨ owen, Phys. Rev. X 5, 011035 (2015)

2015
[5]

underdamped

In the limit of large persistence, the MSD reduces to γ2v2 0t6/36λ2τp which is the short-time behavior intro- duced above while in the opposite limit of white noise, MSD(t)=γ2Dt5/10λ2 reflecting the long-time behavior from above. These results are included in Figure 3a for comparison. In Figure 3b, we have included the dynam- ical exponent α(t) defined as...
[6]

in the steady state as a function of ˜λ for τp=τI and three different values for ˜m: ˜m=1/5 (solid line), ˜m=1/4 (dashed line), ˜m=1/3 (dotted line). b) Effective tem- perature ratios Ts/T (solid line) and Tkin/T (dashed line) for ˜m=1/5 as a function of the dimensionless jerk parameter ˜λ for white noise (red) and active noise (blue) with τp=τI . ics on ...
[7]

Newton, Philosophiae Naturalis Principia Mathemat- ica (Royal Society Press, London, 1687)

I. Newton, Philosophiae Naturalis Principia Mathemat- ica (Royal Society Press, London, 1687)
[8]

Rebhan, Theoretische Physik: Mechanik , Vol

E. Rebhan, Theoretische Physik: Mechanik , Vol. 1 (Spektrum Akademischer Verlag, 2006)

2006

Show all 132 references

[9]

M. W. McCall, Classical mechanics: from Newton to Einstein: a modern introduction , Vol. 2 (Chichester, West Sussex, UK: Wiley, 2011)

2011
[10]

Gompper et al., J

G. Gompper et al., J. Phys. Condens. Matter 37, 143501 (2025)

2025
[11]

Dinelli, J

A. Dinelli, J. O’Byrne, A. Curatolo, Y. Zhao, P. Sollich, and J. Tailleur, Nat. Commun. 14, 7035 (2023)

2023
[12]

Kneˇ zevi´ c, T

M. Kneˇ zevi´ c, T. Welker, and H. Stark, Sci. Rep. 12, 19437 (2022)

2022
[13]

S. A. M. Loos and S. H. L. Klapp, New J. Phys. 22, 123051 (2020)

2020
[14]

Osat and R

S. Osat and R. Golestanian, Nat. Nanotechnol. 18, 79 (2023)

2023
[15]

S. Osat, J. Metson, M. Kardar, and R. Golestanian, Phys. Rev. Lett. 133, 028301 (2024)

2024
[16]

S. H. Schot, Am. J. Phys. 46, 1090 (1978)

1978
[17]

Bechhoefer, Rev

J. Bechhoefer, Rev. Mod. Phys. 77, 783 (2005)

2005
[18]

van Heerden, N

B. van Heerden, N. A. Vickers, T. P. J. Kr¨ uger, and S. B. Andersson, Small 18, 2107024 (2022)

2022
[19]

D. Saha, S. Tarama, H. L¨ owen, and S. U. Egelhaaf, Soft Matter 20, 8112 (2024)

2024
[20]

Z. Hou, Z. Zhang, J. Li, K. Yasuda, and S. Komura, EPL 150, 27001 (2025)

2025
[21]

J. Li, Z. Zhang, Z. Hou, Y. Hosaka, K. Yasuda, L. He, and S. Komura, arXiv:2502.20752 (2025)

2025 arXiv
[22]

Rajagopal, S

K. Rajagopal, S. Takougang Kingni, G. Fautso, V. Kamdoum Tamba, and V.-T. Pham, Adv. Theor. Math. Phys. 2018, 1 (2018)

2018
[23]

S. J. Linz, Am. J. Phys. 66, 1109 (1998)

1998
[24]

S. J. Linz, Am. J. Phys. 65, 523 (1997)

1997
[25]

Eichhorn, S

R. Eichhorn, S. J. Linz, and P. H¨ anggi, Phys. Rev. E 58, 7151 (1998)

1998
[26]

Eichhorn, S

R. Eichhorn, S. J. Linz, and P. H¨ anggi, Chaos Solit. Fractals 13, 1 (2002)

2002
[27]

Umut and S

O. Umut and S. Ya¸ sar, Int. J. Mod. Nonlinear Theory Appl. 2, 60 (2013)

2013
[28]

Patidar and K

V. Patidar and K. K. Sud, Pramana 64, 75 (2005)

2005
[29]

B. Wu, C. Lim, and W. Sun, Phys. Lett. A 354, 95 (2006)

2006
[30]

Gottlieb, J

H. Gottlieb, J. Sound Vib. 297, 243 (2006)

2006
[31]

Gottlieb, J

H. Gottlieb, J. Sound Vib. 271, 671 (2004)

2004
[32]

Gottlieb, Am

H. Gottlieb, Am. J. Phys. 66, 893 (1998)

1998
[33]

S. J. Linz, Phys. Lett. A 275, 204 (2000)

2000
[34]

N. J. Poplawski, Phys. Lett. B 640, 135 (2006)

2006
[35]

Njitacke, J

Z. Njitacke, J. Kengne, and L. Kengne, Chaos. Solitons. Fract. 105, 77 (2017)

2017
[36]

Baeyer, Sciences 38, 12 (1998)

H. Baeyer, Sciences 38, 12 (1998)

1998
[37]

F. X. Liu, R. J. Cheng, and H. X. Ge, Nonlinear Dyn. 83, 793 (2016)

2016
[38]

Zhai and W

C. Zhai and W. T. Wu, Nonlinear Dyn. 93, 2185 (2018)

2018
[39]

R. K. Tiwari, D. Sofuoglu, and A. Beesham, Gravit. Cosmol. 28, 196 (2022)

2022
[40]

te Vrugt and R

M. te Vrugt and R. Wittkowski, Eur. Phys. J. E 48, 12 (2025)

2025
[41]

Ramaswamy, Annu

S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, 323 (2010)

2010
[42]

Marchetti, J

M. Marchetti, J. Joanny, S. Ramaswamy, T. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, 1143 (2013)

2013
[43]

Elgeti, R

J. Elgeti, R. Winkler, and G. Gompper, Rep. Prog. Phys. 78, 056601 (2015)

2015
[44]

Bechinger, R

C. Bechinger, R. di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, Rev. Mod. Phys. 88, 045006 (2016)

2016
[45]

Gompper et al., J

G. Gompper et al., J. Phys.: Condens. Matter 32, 193001 (2020)

2020
[46]

Dabelow and R

L. Dabelow and R. Eichhorn, Front. Phys. 8, 2020 (2021)

2021
[47]

F¨ urth, Z

R. F¨ urth, Z. Phys.2, 244 (1920)

1920
[48]

G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930)

1930
[49]

Szamel, Phys

G. Szamel, Phys. Rev. E 90, 012111 (2014)

2014
[50]

Bonilla, Phys

L. Bonilla, Phys. Rev. E 100, 022601 (2019)

2019
[51]

Martin, J

D. Martin, J. O’Byrne, M. E. Cates, E. Fodor, C. Nar- dini, J. Tailleur, and F. van Wijland, Phys. Rev. E 103, 032607 (2021)

2021
[52]

te Vrugt, J

M. te Vrugt, J. Jeggle, and R. Wittkowski, New J. Phys. 23, 063023 (2021)

2021
[53]

Wittmann, J

R. Wittmann, J. M. Brader, A. Sharma, and U. M. B. Marconi, Phys. Rev. E 97, 012601 (2018)

2018
[54]

T. J. H. Fritz and U. Seifert, J. Stat. Mech. Theory Exp. 2023, 093204 (2023)

2023
[55]

Gupta, S

D. Gupta, S. Klapp, and D. Sivak, Phys. Rev. E 108, 024117 (2023)

2023
[56]

Crisanti and M

A. Crisanti and M. Paoluzzi, Phys. Rev. E 107, 034110 (2023)

2023
[57]

Pacheco-Pozo, I

A. Pacheco-Pozo, I. M. Sokolov, R. Metzler, and 13 D. Krapf, arXiv:2505.13363 (2025)

2025 arXiv
[58]

H. Li, G. He, Y. Peng, and H. Cheng, Chin. J. Phys. 77, 1997 (2022)

2022
[59]

Caprini and U

L. Caprini and U. M. B. Marconi, J. Chem. Phys. 154, 024902 (2021)

2021
[60]

G. H. P. Nguyen, R. Wittmann, and H. L¨ owen, J. Phys.: Condens. Matter 34, 035101 (2022)

2022
[61]

L¨ owen, J

H. L¨ owen, J. Chem. Phys.152, 040901 (2020)

2020
[62]

Sandoval, Phys

M. Sandoval, Phys. Rev. E 101, 012606 (2020)

2020
[63]

L. L. Gutierrez-Martinez and M. Sandoval, J. Chem. Phys. 153, 044906 (2020)

2020
[64]

S. Ye, P. Liu, F. Ye, K. Chen, and M. Yang, Soft Matter 16, 4655 (2020)

2020
[65]

Howse, R

J. Howse, R. Jones, A. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Phys. Rev. Lett. 99, 048102 (2007)

2007
[66]

Metzler and J

R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000)

2000
[67]

Metzler, J

R. Metzler, J. Jeon, A. Cherstvy, and E. Barkai, Phys. Chem. Chem. Phys. 16, 24128 (2014)

2014
[68]

Babel, B

S. Babel, B. ten Hagen, and H. L¨ owen, J. Stat. Mech. Theory Exp. 2024, P02011 (2014)

2014
[69]

Masoliver and J

J. Masoliver and J. M. Porr` a, Phys. Rev. E 48, 4309 (1993)

1993
[70]

A. R. Sprenger, M. A. Fernandez-Rodriguez, L. Alvarez, L. Isa, R. Wittkowski, and H. L¨ owen, Langmuir36, 7066 (2020)

2020
[71]

H¨ anggi and P

P. H¨ anggi and P. Jung,Colored Noise in Dynamical Sys- tems (John Wiley & Sons, Ltd, 1994)

1994
[72]

Caprini and H

L. Caprini and H. L¨ owen, Phys. Rev. Letters 130, 148202 (2023)

2023
[73]

will be included as a special case. B. Experimental realizations
[74]

Pavsic, Phys

M. Pavsic, Phys. Rev. D 87, 107502 (2013)

2013
[75]

Argun, J

A. Argun, J. Soni, L. Dabelow, S. Bo, G. Pesce, R. Eich- horn, and G. Volpe, Phys. Rev. E 96, 052106 (2017)

2017
[76]

M. A. Fernandez-Rodriguez, F. Grillo, L. Alvarez, M. Rathlef, I. Buttinoni, G. Volpe, and L. Isa, Nat. Commun. 11, 4223 (2020)

2020
[77]

Scholz, S

C. Scholz, S. Jahanshahi, A. Ldov, and H. L¨ owen, Nat. Commun. 9, 5156 (2018)

2018
[78]

Caprini, D

L. Caprini, D. Breoni, A. Ldov, C. Scholz, and H. L¨ owen, Communications Physics7, 343 (2024)

2024
[79]

Nosenko, F

V. Nosenko, F. Luoni, A. Kaouk, M. Rubin-Zuzic, and H. Thomas, Phys. Rev. Res. 2, 033226 (2020)

2020
[80]

Holubec, G

V. Holubec, G. Volpe, and F. Cichos, arXiv:2505.11042 (2025)

2025 arXiv
[81]

Narinder, C

N. Narinder, C. Bechinger, and J. R. Gomez-Solano, Phys. Rev. Lett. 121, 078003 (2018)

2018
[82]

F. J. Sevilla, R. F. Rodr ´ ıguez, and J. R. Gomez-Solano, Phys. Rev. E 100, 032123 (2019)

2019
[83]

A. R. Sprenger, C. Bair, and H. L¨ owen, Phys. Rev. E 105, 044610 (2022)

2022
[84]

Ruiz-Silva, B

A. Ruiz-Silva, B. B. Cassal-Quiroga, R. d. J. Escalante- Gonzalez, J. A. Del-Puerto-Flores, H. E. Gilardi- Velazquez, and E. Campos, Mathematics 13, 804 (2025)

2025
[85]

Huerta-Cuellar, E

G. Huerta-Cuellar, E. Jim´ enez-L´ opez, E. Campos- Cant´ on, and A. N. Pisarchik, Commun. Nonlinear Sci. Numer. Simul. 19, 2740 (2014)

2014
[86]

M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Vol. ISBN 0-07-060224-7 (Schaum, 1968)

1968
[87]

Risken, The Fokker-Planck Equation, Methods of So- lution and Applications, Vol

H. Risken, The Fokker-Planck Equation, Methods of So- lution and Applications, Vol. 2 (Springer Berlin, Heidel- berg, 1996)

1996
[88]

J. K. G. Dhont, An introduction to dynamics of colloids, Vol. 2 (Elsevier, 1996)

1996
[89]

Doi and S

M. Doi and S. Edwards, Theory of polymer dynamics (Oxford Univ. Press, 1986)

1986
[90]

L¨ owen, J

H. L¨ owen, J. Phys. Condens. Matter21, 474203 (2009)

2009
[91]

Berner, B

J. Berner, B. M¨ uller, J. R. Gomez-Solano, M. Kr¨ uger, and C. Bechinger, Nat. Commun. 9, 999 (2018)

2018
[92]

D. S. Dean, S. N. Majumdar, and H. Schawe, Phys. Rev. E 103, 012130 (2021)

2021
[93]

ten Hagen, R

B. ten Hagen, R. Wittkowski, and H. L¨ owen, Phys. Rev. E 84, 031105 (2011)

2011
[94]

ten Hagen, S

B. ten Hagen, S. van Teeffelen, and H. L¨ owen, J. Phys. Condens. Matter 23, 194119 (2011)

2011
[95]

A. P. Antonov, L. Caprini, A. Ldov, C. Scholz, and H. L¨ owen, Phys. Rev. Lett.133, 198301 (2024)

2024
[96]

Breoni, M

D. Breoni, M. Schmiedeberg, and H. L¨ owen, Phys. Rev. E 102, 062604 (2020)

2020
[97]

Antonov, Y

A. Antonov, Y. Zheng, B. Liebchen, and H. L¨ owen, arXiv:2305.16131 (2025)

2025 arXiv
[98]

M. V. Ostrogradski, Mem. Acad. Sci. St. Petersbourg VI 4, 385 (1850)
[99]

V. V. Nesterenko, Phys. Rev. D 75, 087703 (2007)

2007
[100]

Buttinoni, L

I. Buttinoni, L. Caprini, L. Alvarez, F. Schwarzendahl, and H. L¨ owen, EPL140, 27001 (2022)

2022
[101]

Szamel, Phys

G. Szamel, Phys. Rev. E 107, 054602 (2023)

2023
[102]

Hecht, L

L. Hecht, L. Caprini, H. L¨ owen, and B. Liebchen, J. Chem. Phys. 161, 224904 (2024)

2024
[103]

F. A. Lavergne, H. Wendehenne, T. B¨ auerle, and C. Bechinger, Science 364, 70 (2019)

2019
[104]

Nasiri, H

M. Nasiri, H. L¨ owen, and B. Liebchen, EPL142, 17001 (2023)

2023
[105]

S. Goh, R. G. Winkler, and G. Gompper, Commun. Phys. 6, 310 (2023)

2023
[106]

S. Goh, R. G. Winkler, and G. Gompper, New J. Phys. 24, 093039 (2022)

2022
[107]

S. M. J. Khadem and S. H. L. Klapp, Phys. Chem. Chem. Phys. 21, 13776 (2019)

2019
[108]

P.-C. Chen, K. Kroy, F. Cichos, X. Wang, and V. Hol- ubec, EPL 142, 67003 (2023)

2023
[109]

T¨ opfer, M

U. T¨ opfer, M. R. Bailey, S. Schreiber, F. Paratore, and L. Isa, arXiv:2505.15396 (2025)

2025 arXiv
[110]

C. R. Packard and D. M. Sussman, arXiv:2505.10657 (2025)

2025 arXiv
[111]

R. A. Kopp and S. H. L. Klapp, Phys. Rev. E 110, 054126 (2024)

2024
[112]

S. Dago, J. Pereda, S. Ciliberto, and L. Bellon, J. Stat. Mech. 2022, 053209 (2022)

2022
[113]

Chatterjee, J

S. Chatterjee, J. Sound Vib. 330, 1860 (2011)

2011
[114]

van Leeuwen, D

R. van Leeuwen, D. M. Karabacak, H. S. J. van der Zant, and W. J. Venstra, Phys. Lett. B 88, 214301 (2013)

2013
[115]

Agoritsas and P

E. Agoritsas and P. K. Morse, arXiv:2403.11701 (2025)

2025 arXiv
[116]

K¨ ursten, V

R. K¨ ursten, V. Sushkov, and T. Ihle, Phys. Rev. Lett. 119, 188001 (2017)

2017
[117]

B. A. Dalton, A. Klimek, H. Kiefer, F. N. Br¨ unig, H. Co- linet, L. Tepper, A. Abbasi, and R. R. Netz, Annu. Rev. Phys. Chem. 76, 431 (2025)

2025
[118]

Sandford and A

C. Sandford and A. Y. Grosberg, Phys. Rev. E 97, 012602 (2018)

2018
[119]

Mungan, E

M. Mungan, E. Clement, D. Vandembroucq, and S. Sas- try, Annual Reviews in Condensed Matter Physics ???, to be published (2025)

2025
[120]

Khadka, V

U. Khadka, V. Holubec, H. Yang, and F. Cichos, Nat. Commun. 9, 3864 (2018)

2018
[121]

Huang, M

C. Huang, M. Ding, and X. Xing, Phys. Rev. Res. 2, 14 043222 (2020)

2020
[122]

Fischer, F

A. Fischer, F. Schmid, and T. Speck, Phys. Rev. E 101, 012601 (2020)

2020
[123]

Caprini, A

L. Caprini, A. R. Sprenger, H. L¨ owen, and R. Wittmann, J. Chem. Phys. 156, 071102 (2022)

2022
[124]

de Gennes, J

P.-G. de Gennes, J. Stat. Phys. 119, 953 (2005)

2005
[125]

Hayakawa, Physica 205D, 48 (2005)

H. Hayakawa, Physica 205D, 48 (2005)

2005
[126]

Daniel, M

S. Daniel, M. K. Chaudhury, and P.-G. de Gennes, Langmuir 21, 4240 (2005)

2005
[127]

Chen and W

Y. Chen and W. Just, Phys. Rev. E 89, 022103 (2014)

2014
[128]

Lequy and A

T. Lequy and A. M. Menzel, Phys. Rev. E 108, 064606 (2023)

2023
[129]

Cates and J

M. Cates and J. Tailleur, Annu. Rev. Condens. Matter Phys. 6, 219 (2015)

2015
[130]

Vicsek, A

T. Vicsek, A. Czir´ ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 (1995)

1995
[131]

Toner, The Physics of Flocking: Birth, Death, and Flight in Active Matter (Cambridge University Press, 2024)

J. Toner, The Physics of Flocking: Birth, Death, and Flight in Active Matter (Cambridge University Press, 2024)

2024
[132]

H. H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher, R. E. Goldstein, H. L¨ owen, and J. M. Yeomans, PNAS 109, 14308 (2012)

2012

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

(9) We refer to this noise as passive noise and correspond- ingly the particle will be called “passive”

(8) In the opposite limit of a very small persistence time τp, taken such that D=v2 0τp stays constant, we obtain 3 (Markovian) white noise characterized by the force mem- ory kernel Mw(t)=2γ2Dδ(t). (9) We refer to this noise as passive noise and correspond- ingly the particle will be called “passive”. This special case was treated previously with jerk-li...

[2] [2]

all the past particle velocities contribute to its total friction

Colloids with memory-dominated friction Here it is shown that the basic equation (2) is also re- alized in overdamped active colloids exposed to friction that includes memory, i.e. all the past particle velocities contribute to its total friction. Such effects do occur for active colloidal particles in non-Newtonian, viscoelastic solvents. For an active c...

[3] [3]

active jerks

Macroscopic feedback experiment A macroscopic realization of an active jerky particle is shown in Figure 1. This demonstrates that “active jerks” can conveniently be implemeted in experiments. Consider the one-dimensional motion of an inertial par- ticle on a tilted plane where the tilt angle is controlled by external feedback: a camera measures the veloc...

[4] [4]

A. V. Ivlev, J. Bartnick, M. Heinen, C.-R. Du, V. Nosenko, and H. L¨ owen, Phys. Rev. X 5, 011035 (2015)

2015

[5] [5]

underdamped

In the limit of large persistence, the MSD reduces to γ2v2 0t6/36λ2τp which is the short-time behavior intro- duced above while in the opposite limit of white noise, MSD(t)=γ2Dt5/10λ2 reflecting the long-time behavior from above. These results are included in Figure 3a for comparison. In Figure 3b, we have included the dynam- ical exponent α(t) defined as...

[6] [6]

in the steady state as a function of ˜λ for τp=τI and three different values for ˜m: ˜m=1/5 (solid line), ˜m=1/4 (dashed line), ˜m=1/3 (dotted line). b) Effective tem- perature ratios Ts/T (solid line) and Tkin/T (dashed line) for ˜m=1/5 as a function of the dimensionless jerk parameter ˜λ for white noise (red) and active noise (blue) with τp=τI . ics on ...

[7] [7]

Newton, Philosophiae Naturalis Principia Mathemat- ica (Royal Society Press, London, 1687)

I. Newton, Philosophiae Naturalis Principia Mathemat- ica (Royal Society Press, London, 1687)

[8] [8]

Rebhan, Theoretische Physik: Mechanik , Vol

E. Rebhan, Theoretische Physik: Mechanik , Vol. 1 (Spektrum Akademischer Verlag, 2006)

2006