Pith. sign in

REVIEW 3 major objections 3 minor 48 references

Non-monotonic diffusion from nonequilibrium driving

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

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

Pith's one-line read A passive particle driven by an active or hot particle always diffuses at long times, but its effective diffusivity first rises, then falls, with drive strength.

desk verdict Nonmonotonic diffusion under active/thermal drive is likely real and the weak-drive formula is clean and parameter-free, but the strong-drive 'prediction' is a semi-empirical fit with fitted force parameters, so the paper deserves a serious referee but needs honest reframing. read the letter →

arxiv 2607.25902 v1 pith:3JEX6NO6 submitted 2026-07-28 cond-mat.stat-mech

Manish Patel , Ritwick Sarkar , Urna Basu , Debasish Chaudhuri This is my paper · ORCID
classification cond-mat.stat-mech
keywords nonmonotonicdiffusionactivematterrun-and-tumbleparticleeffectivediffusivitynonreciprocalinteractionssingle-filenonequilibriumdrivingfirst-passagetime
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 how a passive particle's long-time diffusion changes when it is pushed by a second particle that is out of equilibrium—either an active self-propelled particle or a passive particle held at a higher temperature. On a one-dimensional ring, the authors show analytically that the driven particle always becomes diffusive at long times, regardless of whether the interaction is reciprocal or nonreciprocal. The effective diffusion constant first increases with the drive strength, then reaches a maximum, and finally decreases, scaling as 1/(v0^2 tau) for active driving and 1/D0 for thermal driving. The same two-regime structure—bound-pair motion at weak drive, intermittent collision kicks at strong drive—accounts for both types of driving. If correct, this unifies transport under active and passive nonequilibrium driving and identifies a general mechanism for both enhanced and suppressed diffusion.

What carries the argument

The central objects are two coupled Langevin equations for a passive particle x and a driving particle y on a ring of circumference L, with a short-range repulsive exponential potential. For weak drive, the key device is the harmonic approximation of the interaction, which yields a bound pair whose center-of-mass combination (mu x + y)/(1+mu) controls the diffusivity. For strong drive, the key device is the effective stochastic force model: the interaction force is approximated as a Gaussian of strength f0 and width sigma0, and the two-time force correlation is computed from the free-flight propagator of the relative coordinate on the ring. The identity that carries the strong-drive argument

What would settle it

Measure (numerically or experimentally) the two-time force correlation between the particles directly without fitting, and check whether it obeys Eq. (15) with f0 and sigma0 constant as v0, tau, and L vary. Alternatively, if Deff in the large-v0 regime scales with an exponent different from -2—or if the fitted f0 changes systematically with v0 or L—the strong-drive mechanism fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that on a periodic ring a passive Brownian particle driven by an interacting active particle is always diffusive at long times, and its effective diffusivity Deff depends nonmonotonically on the active particle's self-propulsion speed v0 (or, for thermal driving, on the driving particle's diffusivity D0). In the weak-drive regime the particles stay bound, and a harmonic approximation gives Deff = (mu^2 D + v0^2 tau)/(1+mu)^2, where mu is the reciprocity parameter. In the strong-drive regime the active particle repeatedly crosses the driven particle, delivering correlated stochastic kicks; modeling these kicks with an effective Gaussian force gives Deff ≈ D + f0^2

Load-bearing premise

The strong-driving branch assumes that collisions can be captured by an effective Gaussian force whose strength f0 and range sigma0 are independent of v0, tau, and L (they are fitted from the MSD), and that the relative-coordinate propagator can be computed by neglecting the interaction force entirely; if f0 or sigma0 vary with drive parameters, the predicted 1/v0^2 tail is not an independent derivation.

Editorial extensions

If this is right

  • If the central claim is correct, a passive tracer coupled to an active particle on a ring will exhibit normal diffusion at all drive strengths, with no anomalous scaling at long times regardless of reciprocity.
  • The effective diffusivity is predicted to peak at an intermediate v0 (or D0) and then fall off as v0^{-2} (or D0^{-1}); this is a testable, quantitative signature of the mechanism.
  • The weak-drive result Deff = (mu^2 D + v0^2 tau)/(1+mu)^2 holds independent of the interaction strength k, so tuning the potential stiffness should not affect the long-time diffusivity in the bound regime.
  • Because the force-correlation form is common to run-and-tumble, active Brownian, and active Ornstein–Uhlenbeck particles, the nonmonotonic diffusivity should appear across these active-matter classes, not just for RTPs.
  • The same framework predicts that an 'equilibrium' hot particle driving a cold particle shows the same enhancement-then-suppression, so the effect is not specific to self-propulsion.

Reading between the lines

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

  • A natural extension is to a many-body ring of passive particles driven by a single active particle: the single-file intermediate scaling observed at large L suggests that in finite systems the crossover to normal diffusion may be controlled by L^2, which could be measurable in colloidal experiments.
  • One could test the universality claim by replacing the exponential potential with a different short-range potential (e.g., Lennard-Jones-like); if the nonmonotonic shape survives but the fitted f0 changes, the mechanism is robust, while a breakdown would reveal that the Gaussian-kick approximation is not self-similar.
  • The strong-drive result implies that the diffusivity of the driven particle is independent of its own bare diffusivity D at large v0 (D drops out in the leading term); this could be checked by varying the temperature of the passive particle's bath.
  • If the effective force parameters f0 and sigma0 are fitted at one v0 and then used to predict Deff at another v0, agreement would establish the effective force as a transferable quantity; the paper does not report such a cross-validation, so this remains an inference.
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.

Referee Report

3 major / 3 minor

Summary. The paper studies a passive Brownian particle driven by either a run-and-tumble active particle or a higher-diffusivity passive particle on a one-dimensional ring, with reciprocal (mu=1) and nonreciprocal (mu<1) couplings. The central claim is that, on a periodic ring, the driven particle always diffuses at long times, with an effective diffusivity D_eff that depends nonmonotonically on the driving strength: D_eff ~ (mu^2 D + v0^2 tau)/(1+mu)^2 in the weak-activity regime (Eq. 9) and D_eff ~ D + f0^2/(12 v0^2 tau) in the strong-activity regime (Eq. 20), with analogous expressions for thermal driving (Eqs. 26 and 30). The paper develops a harmonic bound-pair theory for weak drive and an effective Gaussian-collision-force theory for strong drive, and supports the results with numerical simulations. The authors further argue that the same mechanism underlies active and thermal driving.

Significance. If the results hold, the paper provides a minimal analytically tractable example in which nonreciprocal interactions do not destroy long-time diffusion, and in which transport is both enhanced and suppressed by the same nonequilibrium drive. The weak-drive prediction Eq. (9) is parameter-free, depends only on the nonreciprocity parameter mu, and matches simulations well; this is a clean and useful result. The strong-drive scaling D_eff ~ v0^{-2} is plausible and supported by simulations and by a first-passage-time argument, and the unification of active and thermal driving is conceptually appealing. However, the quantitative large-drive branch is semi-empirical: the force parameters f0 and sigma0 are fitted from the same MSD that the theory then reproduces, and the relative-coordinate propagator neglects the interaction force. The significance would be higher if the strong-drive branch were an independent derivation or a direct prediction across the full v0 range with fixed parameters.

major comments (3)
  1. [§II A 2, Eqs. (18)–(20) and Figs. 1–2] The strong-driving prediction is not independent. f0 and sigma0 are 'extracted by fitting the short- and long-time regimes of the MSD' (Eqs. 18–19), and then the same MSD is reproduced through Eq. (17) and D_eff through Eq. (20). Thus Eq. (20) is a parameterization fitted to the quantity it is claimed to predict, not a genuinely predictive derivation. Moreover, Figs. 1 and 2 show theory curves only for the weak-drive Eq. (9); no Eq. (20) curve with fixed f0, sigma0 is plotted against the large-v0 simulation data. To substantiate the nonmonotonic curve, the authors should demonstrate that f0 and sigma0 obtained at one (v0, tau, L) value yield Eq. (20) across the full v0/tau range, or should derive f0 and sigma0 from the microscopic potential parameters. The v0^{-2} scaling alone is not enough.
  2. [Appendix B and Eq. (10)] The strong-drive theory rests on two ad hoc modeling choices: (i) a Gaussian functional form for the collision force with two free parameters f0 and sigma0 (Eq. 10), and (ii) neglecting the interaction force in the relative-coordinate propagator (Appendix B). These are reasonable as a first approximation, but the paper does not quantify their error or derive them from the original V(|x-y|). In particular, the fitted values of f0 and sigma0 change strongly with system size (Fig. 4), so they are not universal constants. The first-passage argument in §IV is a useful physical justification for the v0^{-2} scaling, but it is not developed into a microscopic calculation. The authors should either derive f0/sigma0 from the potential or present an explicit test of the free-propagation assumption, and should clearly state the range of validity of the strong-drive branch.
  3. [§II B 1, Fig. 4] The strong dependence of the fitted parameters on L (f0 = 2.44, 1.22, 0.66 and sigma0 = 1.82, 0.58, 0.17 for L = 32, 64, 128) means the amplitude of the asymptotic D_eff in Eq. (20) is not predicted by the theory; it is an input obtained from the MSD at each system size. This is a limitation that should be acknowledged explicitly, and it undermines any claim that Eq. (20) is a universal closed-form prediction. The v0^{-2} scaling itself is unaffected, but the quantitative nonmonotonic curve is system-size dependent in a way that the current derivation does not capture.
minor comments (3)
  1. [Figs. 1–2] The theoretical lines in Figs. 1 and 2 are drawn only for Eq. (9). Adding the large-v0 prediction Eq. (20) with fixed fitted parameters (e.g., from one representative v0) would help the reader see the crossover and the claimed nonmonotonicity.
  2. [Eq. (20)] The derivation of the approximation D_eff ≈ f0^2/(12 v0^2 tau) from the series uses sigma0/L << 1. For the parameters in Fig. 4, sigma0/L ranges from 0.057 (L=32) to 0.0013 (L=128); this condition is only marginally satisfied for the smallest system. A brief comment on this would help.
  3. [General] The paper uses v0^2 tau as the effective diffusion coefficient of the active particle and notes that the results extend to ABP and AOUP. This is correct, but the claim that the framework is 'applicable to the principal classes of active matter' could be softened in the abstract, since the derivation is explicitly for a two-particle ring system and the strong-drive branch is semi-empirical.

Circularity Check

2 steps flagged · score 6.0 of 10

Large-drive branch of the nonmonotonic prediction is calibrated to the very MSD it predicts; the v0^-2 and D0^-1 asymptotes are semi-empirical.

  1. fitted input called prediction [Sec. II A 2, Eqs. (10), (17), (19)-(20), Fig. 3 caption]
    "The analytical expression provides an excellent description of the numerical simulation results for the MSD, as demonstrated in Figs. 3 and 4. In evaluating the expression, we include n=300 modes to achieve convergence of the series, while the parameters f0 and σ0 are extracted by fitting the short- and long-time regimes of the MSD, as presented below. ... Deff = D + 2 f0^2/L^2 sum ... ≈ f0^2/(12 v0^2 τ)."

    The strong-activity effective diffusivity, Eq. (19), and its asymptotic form Eq. (20), depend on f0 and σ0, which are explicitly 'extracted by fitting the short- and long-time regimes of the MSD.' Since the long-time slope of the MSD is by definition 2 Deff, fitting the long-time MSD fixes Deff; Eq. (20) then restates that fitted value in the model's notation. The v0^-2 scaling is presented as an analytical prediction, but no demonstration that f0 and σ0 remain fixed as v0, τ, or L vary is given. The simulation data independently show the v0^-2 trend, but the analytical large-drive branch is a calibrated fit, not a parameter-free derivation.

  2. fitted input called prediction [Sec. III B, Eqs. (29)-(30), Fig. 6 caption]
    "We compare the above expression with numerical simulations using the fitted values of f0 and σ0, finding good agreement [see Fig. 6]. ... At late times, the MSD becomes diffusive, with effective diffusivity of the driven particle given by Deff = D + 2 f0^2/[L^2(D0+D)] sum ... ≈ D + f0^2/[12(D0+D)]."

    The same fitting procedure is used for passive driving: f0 and σ0 are fitted to the MSD, and then used in Eq. (29) to reproduce that MSD. The late-time diffusive coefficient in Eq. (30) is therefore tied to the fitted long-time MSD slope. The D0^-1 scaling is the model's functional form evaluated with fitted parameters, not an independent prediction of Deff across D0 without recalibration. The weak-driving branch Eq. (26) is parameter-free, but the strong-driving branch inherits the circularity.

full rationale

The central nonmonotonic claim is not entirely circular: the weak-driving branch, Eq. (9) for active driving and Eq. (26) for thermal driving, is derived from a harmonic approximation with no fitted parameters and agrees with simulations. Simulations themselves independently exhibit the nonmonotonic behavior, including the large-drive v0^-2 and D0^-1 trends. However, the analytical strong-driving branch, which the paper presents as the prediction closing the nonmonotonic curve, relies on f0 and σ0 fitted to the short- and long-time MSD of the same system. Therefore the large-drive asymptotes are partly a semi-empirical parameterization rather than a first-principles derivation. The self-citations (Refs. [27], [44]) are not load-bearing for the main claim: Ref. [44] supports the shared force-correlation structure, and Ref. [27] is used for comparison of an intermediate single-file regime. Since the primary nonmonotonic result has independent support but the strong-drive analytical prediction reduces, in part, to a fit of the quantity it predicts, a score of 6 is appropriate.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central result is supported by a parameter-free weak-driving formula and by simulations, but the strong-driving quantitative branch relies on two fitted effective parameters (f0, sigma0). The harmonic bound-state parameters and the Gaussian collision force are modeling constructs; no new physical particle or conserved quantity is introduced.

free parameters (3)
  • f0 (collision force strength) = 0.66 (L=128, v0=5, tau=10); 2.44 (L=32); 1.22 (L=64); 1.086 (passive D0=0.8); 0.971 (passive D0=4)
    Fitted from short- and long-time MSD in the strong-driving branch (Sec. II A 2, Fig. 3/4); controls the prefactor of Deff in Eq. (20).
  • sigma0 (collision force range) = 0.17 (L=128, v0=5, tau=10); 1.82 (L=32); 0.58 (L=64); 1.162 (passive D0=0.8); 0.681 (passive D0=4)
    Fitted together with f0 from the MSD; sets the exponential cutoff in the force correlation, Eqs. (15) and (28).
  • a and keff (harmonic bound-state parameters)
    Introduced in Eqs. (6)-(7) as effective mean separation and repulsion strength; they cancel in Deff, Eq. (9), so they do not affect the central result.
assumptions (6)
  • domain assumption Dichotomous-orientation correlation <sigma(t) sigma(t')> = exp(-|t-t'|/tau) with tau=1/(2 alpha) captures RTP, ABP, and AOUP forcing.
    Invoked in Sec. II after Eq. (4) to generalize the result across active-matter models.
  • ad hoc to paper Weak drive: the pair remains in contact and the interaction can be replaced by a harmonic spring with strength keff and mean separation a (Eqs. (6)-(7)).
    Used throughout Sec. II A 1; keff and a cancel in Deff.
  • ad hoc to paper Strong drive: intermittent collisions are represented by a Gaussian force f(r)=f0/(sqrt(2 pi) sigma0) exp(-r^2/(2 sigma0^2)) with fitted f0 and sigma0 (Eq. (10)).
    Load-bearing modeling assumption for the high-activity branch; parameters are fitted to the MSD.
  • ad hoc to paper Strong drive: the relative-coordinate propagator neglects the interaction force (Appendix B).
    Used to derive Eqs. (11)-(15); questionable at intermediate v0 where collisions are not instantaneous.
  • domain assumption In the long-time limit on a ring, the relative separation is uniformly distributed (Pst=1/L) and the mean collision force vanishes after subtracting fbar.
    Used in Eq. (14) and the surrounding text; consistent with periodic boundary conditions.
  • standard math The mode sum sum_{n>=1} exp(-sigma0^2 q_n^2/2)/q_n^2 approx L^2/24 for sigma0/L << 1.
    Approximation used to obtain Eqs. (20) and (30).
invented entities (1)
  • Effective Gaussian collision force (delta f) with fitted strength f0 and range sigma0
    purpose: Models the fluctuating force on the passive particle from intermittent crossings by the active or hot particle in the strong-driving regime; Eq. (10) and Eq. (15).
    The form and parameters are chosen to reproduce the simulated MSD; the parameters are not measured or predicted by an independent theory, so this is a phenomenological entity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-monotonic diffusion from nonequilibrium driving." pith.science (2026). https://pith.science/paper/3JEX6NO6

@misc{pith2026260725902,
  author       = {Pith},
  title        = {Pith review of: Non-monotonic diffusion from nonequilibrium driving},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3JEX6NO6}},
  note         = {Machine review of arXiv:2607.25902}
}
read the original abstract

The stochastic dynamics of interacting particles far from equilibrium remains a fundamental challenge in statistical physics. While reciprocal interactions often permit effective one-body descriptions, such reductions generally fail for nonreciprocal interactions, which are ubiquitous in driven and active systems. We develop a unified theoretical framework for interacting particles with reciprocal and nonreciprocal couplings, applicable to the principal classes of active matter, including run-and-tumble, active Brownian, and active Ornstein-Uhlenbeck particles. As a minimal example, we study a passive particle driven by an active particle. On a periodic ring, we show analytically that the driven particle is always diffusive at long times, independent of the microscopic driving mechanism. Analytical predictions and numerical simulations reveal a nonmonotonic dependence of the effective diffusivity on the driving activity, giving rise to both enhanced and suppressed transport. Remarkably, the same behavior occurs in an equilibrium system driven out of equilibrium by coupling the driving particle to a higher local temperature. Our framework quantitatively captures both systems, identifies the common mechanism underlying the nonmonotonic transport, and establishes a unified description of transport under active and passive nonequilibrium driving.

Figures

Figures reproduced from arXiv: 2607.25902 by the authors.

Figure 1
Figure 1. FIG. 1. Behaviour of effective diffusion coefficient for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Mean-squared displacement and two-time force corre [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Mean-squared displacement and two-time force cor [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Active driving: Mean-squared displacement [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 1 linked inside Pith

  1. [27]

    Ginot, J

    F. Ginot, J. Caspers, M. Kr¨ uger, and C. Bechinger, Phys. Rev. Lett.128, 028001 (2022)

    2022

  2. [1]

    Small activity In this regime, the persistent active force enhances the passive-particle diffusivity by an amount proportional to v2 0τ. This behavior is captured by a harmonic approxima- tion of the interparticle interaction, yielding the Langevin dynamics ˙x(t) =k eff [a−(x−y)] + √ 2D η(t),(6) ˙y(t) =−µ keff [a−(x−y)] +v 0σ(t),(7) where,adenotes the mea...

  3. [2]

    In this regime, collisions can be viewed as an effective stochastic force acting on the driven particle

    Large activity Beyond a critical self-propulsion speed, set by the in- teraction strengthk, the active particle acquires sufficient energy to overcome the interaction barrier and repeatedly cross the passive particle. In this regime, collisions can be viewed as an effective stochastic force acting on the driven particle. We model the interaction force as ...

  4. [3]

    Consistent with this, the analytical MSD in Eq

    Effect of system size In the large-v 0 regime, the effective kick picture be- comes increasingly accurate as the system size exceeds the interaction range and collisions become infrequent. Consistent with this, the analytical MSD in Eq. (17) shows progressively better agreement with simulations asLincreases from 32 to 128 [Fig. 4(a)]. The longer time betw...

  5. [4]

    At large activity, however, the late-time dynamics be- come boundary dependent

    Single-file diffusion as intermediate scaling in large systems At short times, persistent pushing by the active particle drives ballistic motion of the passive particle, ⟨∆x2⟩(t)∼t 2, independent of the boundary conditions. At large activity, however, the late-time dynamics be- come boundary dependent. Under open boundaries, the active particle delivers i...

    2021

  6. [5]

    T. M. Liggett,Interacting particle systems(Springer New York, 1985)

    1985

  7. [6]

    Spohn,Large scale dynamics of interacting particles (Springer Berlin, Heidelberg, 2012)

    H. Spohn,Large scale dynamics of interacting particles (Springer Berlin, Heidelberg, 2012)

    2012

  8. [7]

    V. S. Dotsenko, A. Imparato, P. Viot, and G. Oshanin, SciPost Phys. Core6, 056 (2023)

    2023

Show all 48 references
  1. [8]

    Abdoli and H

    I. Abdoli and H. L¨ owen, (2026), arXiv:2606.27012

    2026 arXiv

  2. [9]

    B´ erut, A

    A. B´ erut, A. Imparato, A. Petrosyan, and S. Ciliberto, Phys. Rev. Lett.116, 068301 (2016)

    2016

  3. [10]

    B´ erut, A

    A. B´ erut, A. Imparato, A. Petrosyan, and S. Ciliberto, Phys. Rev. E94, 052148 (2016)

    2016

  4. [11]

    T. L. and S. Ciliberto, Phys. Rev. Lett.134, 227101 (2025)

    2025

  5. [12]

    Bechinger, R

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

    2016

  6. [13]

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

    2015

  7. [14]

    A. B. Slowman, M. R. Evans, and R. A. Blythe, Phys. Rev. Lett.116, 218101 (2016)

    2016

  8. [15]

    Slowman, M

    A. Slowman, M. Evans, and R. Blythe, J. Phys. A: Math. Theor.50, 375601 (2017)

    2017

  9. [16]

    Mallmin, R

    E. Mallmin, R. A. Blythe, and M. R. Evans, J. Stat. Mech.2019, 013204 (2019)

    2019

  10. [17]

    Le Doussal, S

    P. Le Doussal, S. N. Majumdar, and G. Schehr, Phys. Rev. E104, 044103 (2021)

    2021

  11. [18]

    Chaudhuri and A

    D. Chaudhuri and A. Dhar, J. Stat. Mech.2021, 013207 (2021)

    2021

  12. [19]

    Patel and D

    M. Patel and D. Chaudhuri, New J. Phys.26, 073048 (2024)

    2024

  13. [20]

    S. C. Takatori, R. De Dier, J. Vermant, and J. F. Brady, Nat. Commun.7, 10694 (2016). 8

    2016

  14. [21]

    Sarkar and U

    R. Sarkar and U. Basu, Soft Matter21, 3595 (2025)

    2025

  15. [22]

    Sarkar, S

    R. Sarkar, S. Chatterjee, and U. Basu, J. Phys. A: Math. Theor.58, 415001 (2025)

    2025

  16. [23]

    Goswami, H.-Y

    K. Goswami, H.-Y. Shih, C.-H. Chang, H.-Y. Chen, and P.-Y. Lai, Phys. Rev. Res.8, 023248 (2026)

    2026

  17. [24]

    Ginot and C

    F. Ginot and C. Bechinger, Nat. Commun.16, 10114 (2025)

    2025

  18. [25]

    Dzubiella, H

    J. Dzubiella, H. L¨ owen, and C. N. Likos, Phys. Rev. Lett. 91, 248301 (2003)

    2003

  19. [26]

    Hayashi and S.-i

    K. Hayashi and S.-i. Sasa, J. Phys.: Condens. Matter18, 2825 (2006)

    2006

  20. [28]

    N. S. Mandal, A. Sen, and R. D. Astumian, Chem10, 1147 (2024)

    2024

  21. [29]

    Golestanian, Europhys

    R. Golestanian, Europhys. News55, 12 (2024)

    2024

  22. [30]

    Knotz, T

    G. Knotz, T. M. Muenker, T. Betz, and M. Kr¨ uger, J. Chem. Phys.163, 124118 (2025)

    2025

  23. [31]

    U. Basu, P. L. Krapivsky, and S. N. Majumdar, Phys. Rev. E110, 064105 (2024)

    2024

  24. [32]

    Paul and D

    S. Paul and D. Chaudhuri, (2026), arXiv:2601.03591

    2026

  25. [33]

    F. A. Lavergne, H. Wendehenne, T. B¨ auerle, and C. Bechinger, Science364, 70 (2019)

    2019

  26. [34]

    S. Saha, S. Ramaswamy, and R. Golestanian, New J. Phys.21, 063006 (2019)

    2019

  27. [35]

    Agudo-Canalejo and R

    J. Agudo-Canalejo and R. Golestanian, Phys. Rev. Lett. 123, 018101 (2019)

    2019

  28. [36]

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

    2020

  29. [37]

    Ganguly and D

    C. Ganguly and D. Chaudhuri, Phys. Rev. E88, 032102 (2013)

    2013

  30. [38]

    Chaudhuri, Phys

    D. Chaudhuri, Phys. Rev. E90, 022131 (2014)

    2014

  31. [39]

    Fodor, R

    ´E. Fodor, R. L. Jack, and M. E. Cates, Annu. Rev. Con- dens. Matter Phys.13, 215 (2021)

    2021

  32. [40]

    Szamel, Phys

    G. Szamel, Phys. Rev. E100, 050603(R) (2019)

    2019

  33. [41]

    Fernandez, S

    L. Fernandez, S. Hess, and S. H. L. Klapp, Phys. Rev. E 109, 054129 (2024)

    2024

  34. [42]

    Fodor and M

    ´E. Fodor and M. C. Marchetti, Physica A504, 106 (2018)

    2018

  35. [43]

    A. P. Solon, M. E. Cates, and J. Tailleur, Eur. Phys. J.: Spec. Top.224, 1231 (2015)

    2015

  36. [44]

    Malakar, V

    K. Malakar, V. Jemseena, A. Kundu, K. Vijay Ku- mar, S. Sabhapandit, S. N. Majumdar, S. Redner, and A. Dhar, J. Stat. Mech.2018, 043215 (2018)

    2018

  37. [45]

    A. Shee, A. Dhar, and D. Chaudhuri, Soft Matter16, 4776 (2020)

    2020

  38. [46]

    Fodor, C

    ´E. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. van Wijland, Phys. Rev. Lett.117, 038103 (2016)

    2016

  39. [47]

    Martin, J

    D. Martin, J. O’Byrne, M. E. Cates, ´E. Fodor, C. Nar- dini, J. Tailleur, and F. van Wijland, Phys. Rev. E103, 032607 (2021)

    2021

  40. [48]

    Patel, S

    M. Patel, S. Paul, and D. Chaudhuri, J. Chem. Phys. 164, 134110 (2026)

    2026

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.