General perturbative framework for kinetics of rare transitions in 1-dimensional active particle systems

Daan Crommelin; Michel Mandjes; Peter G. Bolhuis; Sara Jabbari Farouji; Vito Seinen

arxiv: 2604.15826 · v1 · submitted 2026-04-17 · ❄️ cond-mat.stat-mech

General perturbative framework for kinetics of rare transitions in 1-dimensional active particle systems

Vito Seinen , Peter G. Bolhuis , Daan Crommelin , Sara Jabbari Farouji , Michel Mandjes This is my paper

Pith reviewed 2026-05-10 07:46 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords active particlesrare transitionstransition ratespersistence timeAOUPprojection operatorperturbative expansiondriven systems

0 comments

The pith

A projection-operator approach yields an analytic transition rate for active particles valid across all persistence times.

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

The paper develops a perturbative framework to compute rates of rare transitions for an active particle moving in an external potential. It derives separate analytic expansions for the rate in the limits of very short and very long persistence times of the activity, then combines those expansions into a single rational approximation. The resulting closed-form expression covers the full range of persistence times and activity strengths in the rare-event regime and matches numerical simulations. This matters because escape rates control the long-time behavior of many driven systems, from biological swimmers to colloidal motors, where direct simulation of rare events is expensive.

Core claim

Using a projection-operator formalism, perturbative rates are obtained by integrating out the fast variable in each asymptotic regime: the activity for small persistence times and the position for large persistence times. These two expansions are combined into a rational approximation that supplies an analytic rate expression valid for all persistence times and activity strengths in the rare-event limit, in excellent agreement with simulations. The framework applies to a broad class of driven systems, with the thermal AOUP appearing as a special case.

What carries the argument

Projection-operator formalism that isolates the slow variable in each timescale limit and produces a rational approximation from the two resulting perturbative expansions.

If this is right

The rate formula reproduces known results for small persistence times by integrating out the activity.
New analytic rates are obtained for large persistence times by integrating out the position instead.
The approximation supplies a parameter-free expression that holds for any activity strength in the rare-event regime.
The same construction applies to general driven systems beyond the specific AOUP model.

Where Pith is reading between the lines

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

The rational-approximation strategy may extend to other driven systems that possess two well-separated timescales.
If the form remains accurate, it could be used to predict how activity strength and persistence together control escape rates without running separate simulations for each parameter set.

Load-bearing premise

The two asymptotic expansions can be combined into a rational approximation that remains accurate at intermediate persistence times without additional fitting or higher-order corrections.

What would settle it

Direct numerical simulations of the particle dynamics that show large deviations from the rational approximation at moderate persistence times would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.15826 by Daan Crommelin, Michel Mandjes, Peter G. Bolhuis, Sara Jabbari Farouji, Vito Seinen.

**Figure 3.** Figure 3: FIG. 3. First Passage Time Distribution (FPTD) for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 1.** Figure 1: In the intermediate-to-large persistence-time [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Comparison between analytically predicted transi [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

read the original abstract

We present a theoretical framework that enables investigating rare transitions in a general model of an active particle in an external potential, with the thermal Active Ornstein-Uhlenbeck Particle (AOUP) appearing as a special case. Using a projection-operator formalism, we compute transition rates perturbatively in two distinct asymptotic regimes. In the regime of small persistence times-where the activity evolves much faster than the particle's position-integrating out the activity reproduces the rates previously reported in the literature. In the opposite regime of large persistence times, we instead integrate out the position and obtain the corresponding rates analytically. Together, these asymptotic expansions uniquely specify a rational approximation that remains accurate across intermediate persistence times. As a result, we obtain an analytic expression for the rate valid across all persistence times and activity strengths in the rare-event limit, which are in excellent agreement with numerical simulations. The presented framework applies to rare transitions in a broad class of driven systems.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

This paper gives a projection-operator way to compute analytic rare-transition rates in 1D active particles across all persistence times by rationally matching the small-tau and large-tau limits, and the match to simulations looks decent but the interpolation step is the part that needs scrutiny.

read the letter

The main advance is a perturbative framework that recovers known small-persistence rates by eliminating fast activity and derives new large-persistence rates by eliminating slow position, then stitches them with a rational function that the authors say works through the intermediate regime. This is practical for people who need closed-form expressions instead of always running long simulations for rare events in active systems. The generality to a broad class of driven particles, not just AOUP, is a clear plus, and the fact that it reproduces prior literature in one limit shows they are building on solid ground rather than starting from scratch. The numerical agreement they report is the kind of check that makes the claim testable. On the soft spots, the rational approximant is doing the heavy lifting for the 'valid across all' statement. The abstract presents the two expansions as uniquely fixing the function, but without an explicit remainder or demonstration that higher-order terms stay small in the crossover, it is possible the fit drifts where neither asymptotic dominates. If the full paper only shows a few intermediate points without systematic error analysis or comparison to exact solvable cases, that leaves the central claim a bit under-supported. The work stays in one dimension and the rare-event limit, so readers should not expect immediate carry-over to higher dimensions or moderate barriers. This is for specialists in non-equilibrium statistical mechanics and active matter who already use projection methods and want an analytic handle on rates. It is grounded enough and has enough new pieces that it deserves a serious referee rather than a desk reject; the referees can press on the interpolation details and the simulation setup for rare events. I would send it out with that in mind.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a projection-operator formalism for perturbative computation of rare transition rates in a general class of one-dimensional active particles in an external potential (with AOUP as a special case). Small-persistence rates are obtained by integrating out activity and recover prior literature; large-persistence rates are obtained by integrating out position. These two leading-order expansions are asserted to uniquely determine a rational approximant that remains accurate at all intermediate persistence times in the rare-event limit, yielding an analytic rate expression that agrees excellently with simulations and applies broadly to driven systems.

Significance. If the rational interpolation is rigorously controlled, the work would supply a useful analytic tool for rare-event kinetics in active matter across the full persistence range, extending beyond the usual asymptotic limits. The systematic projection-operator approach and recovery of known small-tau results are strengths; the framework's generality to other driven systems is also positive.

major comments (2)

[Abstract] Abstract: the central claim that the small- and large-persistence expansions 'uniquely specify a rational approximation that remains accurate across intermediate persistence times' is load-bearing yet unsupported by a controlled error bound, remainder estimate, or demonstration that the projection-operator kernels produce a meromorphic structure free of non-analytic contributions at crossover. Without this, systematic deviation remains possible where neither asymptotic applies.
[Numerical validation] The numerical validation section: 'excellent agreement with numerical simulations' is stated without quantitative error metrics, the precise range of persistence times and activity strengths tested, or confirmation that the rational form was not adjusted post-hoc; this leaves the accuracy claim unverifiable from the given information.

minor comments (2)

The explicit algebraic form of the rational approximant should be written out in the main text (rather than described) so that readers can reproduce the expression without reconstructing it from the asymptotics.
Notation for the projection operators and the resulting kernels could be introduced more explicitly at first use to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comments below and will revise the manuscript to incorporate improvements where appropriate.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the small- and large-persistence expansions 'uniquely specify a rational approximation that remains accurate across intermediate persistence times' is load-bearing yet unsupported by a controlled error bound, remainder estimate, or demonstration that the projection-operator kernels produce a meromorphic structure free of non-analytic contributions at crossover. Without this, systematic deviation remains possible where neither asymptotic applies.

Authors: We acknowledge that the manuscript does not provide a rigorous controlled error bound or a proof that the kernels lead to a meromorphic structure without non-analytic terms at the crossover. The rational approximation is constructed to match the leading-order asymptotics from the projection-operator formalism in both limits, and its form is motivated by the structure of the effective rate equations. While we cannot rule out systematic deviations in principle, extensive numerical tests (detailed in the manuscript) show close agreement. In the revision, we will add a section discussing the approximation's limitations and the absence of a full error estimate, and we will moderate the claim in the abstract to reflect that it is an interpolation supported by asymptotics and numerics rather than a rigorously controlled result. revision: yes
Referee: [Numerical validation] The numerical validation section: 'excellent agreement with numerical simulations' is stated without quantitative error metrics, the precise range of persistence times and activity strengths tested, or confirmation that the rational form was not adjusted post-hoc; this leaves the accuracy claim unverifiable from the given information.

Authors: The referee is correct that the current presentation lacks quantitative metrics and precise parameter ranges. The rational approximant is uniquely determined by matching the two asymptotic expansions and was not fitted to intermediate data. In the revised version, we will include a table summarizing the relative errors for various persistence times and activity strengths, specify the tested ranges (e.g., persistence times from 10^{-2} to 10^2 and activity parameters up to the rare-event regime), and add a statement confirming the a priori determination of the approximant. This will make the validation section more transparent and verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; asymptotics independently derived and combined via interpolation validated externally

full rationale

The paper derives small-persistence rates via projection-operator integration of activity, reproducing external prior literature. Large-persistence rates follow from integrating out position in the same formalism. The rational approximation is explicitly constructed to match the two leading-order asymptotic expansions at their respective limits. Because both expansions are obtained from the projection-operator framework (one matching known results, one newly analytic) and the final expression is cross-checked against independent numerical simulations rather than being tautological with the inputs, the derivation chain does not reduce to self-definition or fitted inputs by construction. No load-bearing self-citations or uniqueness theorems from the authors' prior work are invoked.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the standard projection-operator formalism of non-equilibrium statistical mechanics and the assumption that the rare-event limit permits a perturbative treatment in the two asymptotic regimes. No free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Projection-operator formalism can be applied to separate fast and slow variables in the active-particle dynamics.
Invoked to compute the perturbative rates in both small- and large-persistence regimes.
domain assumption The rare-event limit allows a well-defined perturbative expansion for transition rates.
Central to obtaining analytic expressions valid in the two asymptotic regimes.

pith-pipeline@v0.9.0 · 5478 in / 1307 out tokens · 46945 ms · 2026-05-10T07:46:39.127429+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

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A. Militaru, M. Innerbichler, M. Frimmer, F. Tebbenjo- hanns, L. Novotny, and C. Dellago, Escape dynamics of active particles in multistable potentials, Nature Com- munications12, 10.1038/s41467-021-22647-6 (2021)

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Caprini, F

L. Caprini, F. Cecconi, and U. Marini Bettolo Marconi, Correlated escape of active particles across a poten- tial barrier, The Journal of Chemical Physics155, 234902 (2021), https://pubs.aip.org/aip/jcp/article- pdf/doi/10.1063/5.0074072/13625527/234902 1 online.pdf

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J. O’Byrne, A. Solon, J. Tailleur, and Y. Zhao, An intro- duction to motility-induced phase separation, inOut-of- equilibrium Soft Matter, edited by C. Kurzthaler, L. Gen- tile, and H. A. Stone (The Royal Society of Chemistry, 2023)

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L. Angelani, R. Leonardo, and M. Paoluzzi, First-passage time of run-and-tumble particles, The European physical journal. E, Soft matter37, 15 (2014)

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Woillez, Y

E. Woillez, Y. Zhao, Y. Kafri, V. Lecomte, and J. Tailleur, Activated escape of a self-propelled particle from a metastable state, Phys. Rev. Lett.122, 258001 (2019)

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G. V. Vinci and M. Mattia, Escape time in bistable neu- ronal populations driven by colored synaptic noise, Phys. Rev. Res.7, 023172 (2025)

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Assaf, E

M. Assaf, E. Roberts, Z. Luthey-Schulten, and N. Gold- enfeld, Extrinsic noise driven phenotype switching in a self-regulating gene, Phys. Rev. Lett.111, 058102 (2013)

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J. F. Mejias, H. J. Kappen, and J. J. Torres, Irregular dynamics in up and down cortical states, PLOS ONE5, 1 (2010)

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S. Faetti, P. Grigolini, and R. Mannella, The projection approach to the fokker-planck equation. i. colored gaus- sian noise, Journal of Statistical Physics52, 951 (1988)

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R. F. Fox, Mean first-passage times and colored noise, Phys. Rev. A37, 911 (1988). 7 NON-DIMENSIONALIZATION We start with the dimensional version of the generalized active particle Langevin equations: dx∗ dt∗ =− k∗ γ F(x ∗)− λ∗ γ Fa(x∗, a) + s 2kBT γ ηx(t∗),(14) da dt∗ =− 1 τ ∗ v′(a) + r 2 τ ∗ ηa(t∗),(15) where⟨η x(t∗ 1)ηx(t∗ 2)⟩=δ(t ∗ 1 −t ∗

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[25]

We have chosenato be dimensionless, such that the non-dimensionalization will not affect it

and⟨η a(t∗ 1)ηa(t∗ 2)⟩=δ(t ∗ 1 −t ∗ 2). We have chosenato be dimensionless, such that the non-dimensionalization will not affect it. The corresponding Fokker-Planck equation takes the form ∂t∗ p∗ =∂ x∗ k∗ γ F(x ∗) + λ∗ γ Fa(x∗, a) p∗ + kBT γ ∂2 x∗ p∗ + 1 τ ∗ ∂a[v′(a)p∗] + 1 τ ∗ ∂2 ap∗.(16) Here,p ∗ denotes the dimensionful density. To render the equations...

work page
[26]

terms. We can write theA ′ term in terms ofAthrough integration by parts Z ∞ −∞ 2A′e−v(a)da= 2Ae −v(a) ∞ −∞| {z } =0 − Z ∞ −∞ 2A(a)[−v′(a)e−v(a)]da,(66) such that we finally obtain r≈ Z ∞ −∞ 1 Za e−v(a) R0(a) + 1 τ v′(a)A(a)−B(a) da,(67) as mentioned in the main text in Eqs. (10) and (11). We also could have obtained Eq. (67) by simply holding that 1 τ La...

work page

[1] [1]

Fodor, C

E. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. van Wijland, How far from equilibrium is active matter?, Phys. Rev. Lett.117, 038103 (2016). 6

work page 2016

[2] [2]

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)

work page 2010

[3] [3]

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)

work page 2013

[4] [4]

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)

work page 2016

[5] [5]

Needleman and Z

D. Needleman and Z. Dogic, Active matter at the inter- face between biology and physics, Nature Reviews Mate- rials2, 17048 (2017)

work page 2017

[6] [6]

Martin, J

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

work page 2021

[7] [7]

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

work page 2015

[8] [8]

H¨ anggi, P

P. H¨ anggi, P. Talkner, and M. Borkovec, Reaction-rate theory: fifty years after kramers, Rev. Mod. Phys.62, 251 (1990)

work page 1990

[9] [9]

Woillez, Y

E. Woillez, Y. Kafri, and V. Lecomte, Nonlocal stationary probability distributions and escape rates for an active ornstein–uhlenbeck particle, Journal of Statistical Me- chanics: Theory and Experiment2020, 063204 (2020)

work page 2020

[10] [10]

Martin and T

D. Martin and T. Arnoulx de Pirey, Aoup in the pres- ence of brownian noise: a perturbative approach, Journal of Statistical Mechanics: Theory and Experiment2021, 043205 (2021)

work page 2021

[11] [11]

Militaru, M

A. Militaru, M. Innerbichler, M. Frimmer, F. Tebbenjo- hanns, L. Novotny, and C. Dellago, Escape dynamics of active particles in multistable potentials, Nature Com- munications12, 10.1038/s41467-021-22647-6 (2021)

work page doi:10.1038/s41467-021-22647-6 2021

[12] [12]

Caprini, F

L. Caprini, F. Cecconi, and U. Marini Bettolo Marconi, Correlated escape of active particles across a poten- tial barrier, The Journal of Chemical Physics155, 234902 (2021), https://pubs.aip.org/aip/jcp/article- pdf/doi/10.1063/5.0074072/13625527/234902 1 online.pdf

work page doi:10.1063/5.0074072/13625527/234902 2021

[13] [13]

O’Byrne, A

J. O’Byrne, A. Solon, J. Tailleur, and Y. Zhao, An intro- duction to motility-induced phase separation, inOut-of- equilibrium Soft Matter, edited by C. Kurzthaler, L. Gen- tile, and H. A. Stone (The Royal Society of Chemistry, 2023)

work page 2023

[14] [14]

Angelani, R

L. Angelani, R. Leonardo, and M. Paoluzzi, First-passage time of run-and-tumble particles, The European physical journal. E, Soft matter37, 15 (2014)

work page 2014

[15] [15]

Woillez, Y

E. Woillez, Y. Zhao, Y. Kafri, V. Lecomte, and J. Tailleur, Activated escape of a self-propelled particle from a metastable state, Phys. Rev. Lett.122, 258001 (2019)

work page 2019

[16] [16]

V. Bach, J. Fr¨ ohlich, and I. M. Sigal, Quantum electro- dynamics of confined nonrelativistic particles, Advances in Mathematics137, 299 (1998)

work page 1998

[17] [17]

C. J. Pethick and H. Smith,Bose–Einstein Condensation in Dilute Gases, 2nd ed. (Cambridge University Press, 2008)

work page 2008

[18] [18]

Ashwin and A

P. Ashwin and A. S. von der Heydt, Extreme sensitivity and climate tipping points, Journal of Statistical Physics 179, 1531 (2020)

work page 2020

[19] [19]

G. V. Vinci and M. Mattia, Escape time in bistable neu- ronal populations driven by colored synaptic noise, Phys. Rev. Res.7, 023172 (2025)

work page 2025

[20] [20]

C. W. Gardiner,Handbook of stochastic methods for physics, chemistry and the natural sciences, 3rd ed., Springer Series in Synergetics, Vol. 13 (Springer-Verlag, Berlin, 2004) pp. xviii+415

work page 2004

[21] [21]

Assaf, E

M. Assaf, E. Roberts, Z. Luthey-Schulten, and N. Gold- enfeld, Extrinsic noise driven phenotype switching in a self-regulating gene, Phys. Rev. Lett.111, 058102 (2013)

work page 2013

[22] [22]

J. F. Mejias, H. J. Kappen, and J. J. Torres, Irregular dynamics in up and down cortical states, PLOS ONE5, 1 (2010)

work page 2010

[23] [23]

Faetti, P

S. Faetti, P. Grigolini, and R. Mannella, The projection approach to the fokker-planck equation. i. colored gaus- sian noise, Journal of Statistical Physics52, 951 (1988)

work page 1988

[24] [24]

R. F. Fox, Mean first-passage times and colored noise, Phys. Rev. A37, 911 (1988). 7 NON-DIMENSIONALIZATION We start with the dimensional version of the generalized active particle Langevin equations: dx∗ dt∗ =− k∗ γ F(x ∗)− λ∗ γ Fa(x∗, a) + s 2kBT γ ηx(t∗),(14) da dt∗ =− 1 τ ∗ v′(a) + r 2 τ ∗ ηa(t∗),(15) where⟨η x(t∗ 1)ηx(t∗ 2)⟩=δ(t ∗ 1 −t ∗

work page 1988

[25] [25]

We have chosenato be dimensionless, such that the non-dimensionalization will not affect it

and⟨η a(t∗ 1)ηa(t∗ 2)⟩=δ(t ∗ 1 −t ∗ 2). We have chosenato be dimensionless, such that the non-dimensionalization will not affect it. The corresponding Fokker-Planck equation takes the form ∂t∗ p∗ =∂ x∗ k∗ γ F(x ∗) + λ∗ γ Fa(x∗, a) p∗ + kBT γ ∂2 x∗ p∗ + 1 τ ∗ ∂a[v′(a)p∗] + 1 τ ∗ ∂2 ap∗.(16) Here,p ∗ denotes the dimensionful density. To render the equations...

work page

[26] [26]

terms. We can write theA ′ term in terms ofAthrough integration by parts Z ∞ −∞ 2A′e−v(a)da= 2Ae −v(a) ∞ −∞| {z } =0 − Z ∞ −∞ 2A(a)[−v′(a)e−v(a)]da,(66) such that we finally obtain r≈ Z ∞ −∞ 1 Za e−v(a) R0(a) + 1 τ v′(a)A(a)−B(a) da,(67) as mentioned in the main text in Eqs. (10) and (11). We also could have obtained Eq. (67) by simply holding that 1 τ La...

work page