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Long-time analysis of a pair of on-lattice and continuous run-and-tumble particles with jamming interactions

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that two run-and-tumble particles with jamming have a well-defined continuous limit, and pins down the time they take to reach their stationary regime.

desk verdict Genuine discrete-to-continuous convergence proof for RTP jamming models, but the CITP lower bound and two lemmas are left unproved. read the letter →

arxiv 2411.13964 v2 pith:SEDEVCT3 submitted 2024-11-21 math.PR cond-mat.stat-mech

Arnaud Guillin (LMBP) , Leo Hahn (LMBP) , Manon Michel (LMBP) This is my paper · ORCID
classification math.PRcond-mat.stat-mech MSC 60J2560J2760B1082C22
keywords run-and-tumbleparticlesjammingpiecewisedeterministicMarkovprocessesmixingtimeinvariantmeasurescalinglimitactivemattermotility-inducedphaseseparation
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 establishes a rigorous bridge between two ways of describing a pair of run-and-tumble particles (RTPs) that cannot pass through each other: a discrete lattice model with blocking and a continuous-space model on a ring. The discrete instantaneous and finite tumble processes DITP and DFTP are shown to converge, under the scaling $\gamma_L=(L-1)/\ell$ and with explicit coupling bounds, to continuous piecewise deterministic Markov processes CITP and CFTP. The same convergence transfers to their invariant measures in Wasserstein distance, and the continuous invariant measure has positive mass at the jammed configurations plus exponential decay away from them, which the authors read as effective attraction consistent with motility-induced phase separation. In addition, the paper proves matching upper and lower bounds for the mixing times, of order $(1/\omega)(1+\omega^2\ell^2)$ for the instantaneous-tumble process and $(1/\alpha+1/\beta)(1+\alpha^2\ell^2)$ for the finite-tumble process, exposing persistent and diffusive regimes.

What carries the argument

The argument is carried by the PDMP construction of Davis, in which the continuous process is described on a state space that adds explicit boundary states representing jammed contacts, so the generator has transport terms in the bulk and pure jump terms at jamming. For the discrete-to-continuous limit, the central device is a coupling that maps the jamming dynamics back to a jamming-free random walk: when both velocities are equal ($\sigma_1=\sigma_2=\pm 1$), the separation process is represented through the periodic map $p_L$, whose lumping property lets the authors transfer random-walk fluctuation estimates to the blocked process. Long-time behavior is controlled by an order-preserving coupling between two copies of the process, reducing the mixing time to hitting times of the jammed states; those hitting times are computed from the generator through systems of differential equations, while matching lower bounds come from identifying slow observables and applying concentration inequalities such as Paley-Zygmund.

What would settle it

Simulate the DITP and DFTP on a ring of $L$ sites with $\gamma_L=(L-1)/\ell$ together with the coupled continuous processes CITP and CFTP, and check the coupling bound of Theorem 2: if the supremum distance $\sup_{t\le T}|i_L(y_L(t))-x(t)|$ fails to tend to zero at the stated rate, or if the empirical invariant measure does not approach the continuous measure in Wasserstein distance as $L\to\infty$, the discrete-to-continuous claim would be false. For the mixing-time claim, measure the total-variation distance to stationarity for large $\omega\ell$: if the relaxation time does not grow like $\omega\ell^2$ (respectively $\alpha^2\ell^2$) up to constants, the matching bounds would be contradicted.

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Extended reading notes

Core claim

The central claim is that the on-lattice jump processes DITP and DFTP, obtained by putting two RTPs on a periodic chain with jamming exclusion, have a well-defined continuous limit: under $\gamma_L=(L-1)/\ell$, the rescaled processes converge in the Skorokhod topology to the CITP and CFTP, which are PDMPs on the torus whose generator is constructed through Davis's formalism with explicit boundary states for jamming. The paper proves this convergence by an explicit coupling rather than by generator methods, which gives quantitative control and also yields Wasserstein convergence of the discrete invariant measures to the continuous one. It further establishes that the continuous invariant measures put positive mass at the jammed separations $0$ and $\ell$, with density decaying exponentially in the bulk, and that the mixing times of the continuous processes are, up to constants depending only on the target accuracy, $(1/\omega)(1+\omega^2\ell^2)$ and $(1/\alpha+1/\beta)(1+\alpha^2\ell^2)$; the lower bounds match the upper bounds, so the dependence on $\omega, \alpha, \beta, \ell$ is optimal.

Load-bearing premise

The proof leans on the special structure of the two velocity mechanisms: when $\sigma_1=\sigma_2=\pm 1$, the separation process can be written through the periodic map $p_L$, and without such a map the quantitative coupling bound does not go through, so the convergence theorem is established only for the instantaneous-tumble and finite-tumble mechanisms of figures 2a and 2b, not for general tumble rates.

Editorial extensions

If this is right

  • The lattice-to-continuum approximation used in earlier exact solutions of jamming RTPs is valid: as the lattice spacing vanishes, the discrete processes converge in Skorokhod topology and their invariant measures converge in Wasserstein distance.
  • The continuous invariant measure has positive mass at jammed separations $0$ and $\ell$ and decays exponentially away from them, so a pair of RTPs exhibits effective attraction without any interaction energy, matching the microscopic picture of motility-induced phase separation.
  • The CITP mixing time is of order $(1/\omega)(1+\omega^2\ell^2)$: linear in $1/\omega$ in the persistent regime $\omega\ell\ll 1$, and diffusive $\omega\ell^2$ in the regime $\omega\ell\gg 1$.
  • The CFTP mixing time is of order $(1/\alpha+1/\beta)(1+\alpha^2\ell^2)$, and it reduces to the CITP result when $\beta\to\infty$ with $\omega=\alpha/2$.
  • The upper and lower bounds on mixing times match, so the parameter dependence is optimal and the accuracy-dependent constant behaves like $\log(1/\epsilon)$ for small $\epsilon$.
  • The quantitative coupling bounds give non-asymptotic control on the approach to stationarity, going beyond spectral asymptotics for these non-reversible processes.

Reading between the lines

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

  • A generator-based proof would likely extend the scaling limit to the general tumble mechanisms treated in the companion PDMP framework, but at the cost of quantitative rates; the boundary-jamming states make the generator domain delicate, so this is a natural next step rather than an immediate consequence.
  • The $p_L$ mapping suggests a lumpability criterion: any tumble mechanism whose conditioned separation process is lumpable to the interval process should admit the same style of quantitative convergence, a property that could be checked by verifying the associated transition-rate identity.
  • The mixing-time formula predicts that in the diffusive regime the position is the slowest observable and behaves diffusively, so the $\omega\ell^2$ term should be recoverable from a spectral-gap calculation for large $\ell$; a direct check on the generator's eigenvalues would sharpen the constant.
  • The singular nature of the jamming mass means the effective attraction is not a smooth potential; extending the result to $N>2$ particles will require handling multiple simultaneous jammed contacts, which the pair analysis leaves open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper provides a rigorous connection between two discrete-space run-and-tumble particle models with jamming (DITP and DFTP) and their continuous-space PDMP limits (CITP and CFTP). Under the scaling gamma_L=(L-1)/ell, the authors construct an explicit coupling and prove a quantitative Skorokhod-convergence bound (Theorem 2), use a uniform exponential ergodicity result for the discrete models (Proposition 5) to deduce Wasserstein convergence of invariant measures (Corollary 1), derive exact stationary measures with Dirac masses at jammed configurations and exponential bulk densities (Propositions 7-8), and give matching upper and lower bounds for the mixing times of the CITP and CFTP (Theorem 3), of orders (1/omega)(1+omega^2 ell^2) and (1/alpha+1/beta)(1+alpha^2 ell^2). The proofs are largely self-contained and use couplings, Doeblin minorization, deterministic hitting lemmas, first-step analysis, and explicit ODE solutions.

Significance. If the gaps identified below are repaired, this is a useful contribution. It justifies the previously formal continuous limits in [SEB16, SEB17], provides explicit invariant measures for a non-reversible interacting active-particle system, and gives non-asymptotic mixing-time estimates with the correct dependence on the model parameters. The paper's self-contained coupling method, including the lumping construction of Definition 5 and the hitting-time ODE calculations, is a strength; it avoids spectral assumptions and yields explicit probability bounds. The explicit scaling predictions for the persistent and diffusive regimes are falsifiable and should be of interest to the active-matter community. The candid discussion of the limitations of the coupling approach for general tumble mechanisms (Remark 2) is also a positive feature.

major comments (3)
  1. [Section 4.1, Theorem 3(i)] Theorem 3(i) is not proven as stated. Section 4.1 explicitly says that "only the lower bound of assertion (ii) is proven" and then proves Lemmas 5-7 for the CFTP. The lower bound for the CITP, t^i_mix(epsilon) >= C(epsilon)(1/omega)(1+omega^2 ell^2), requires a two-state analogue of Lemma 6, whose excursion decomposition uses the zero-velocity state and does not transfer verbatim to the +/-1-only CITP velocity chain, and also requires an invariant-measure estimate pi(M^c) >= 1-a for the CITP analogous to the one used in Lemma 7. Since Theorem 3(i) is the paper's optimality claim for the CITP, this is a load-bearing gap rather than a presentational shortcut. The gap appears repairable by a parallel argument, but the proof needs to be written.
  2. [Appendix A.1, Lemma 12] Lemma 12 is stated without proof ("the proof ... omitted for the sake of brevity") and is then used in Proposition 6 to establish the symmetries pi = rho_i#pi of the invariant measure. Those symmetries are used in Appendix A.2 to fix the constants in the explicit invariant measures (c2 = e^{kappa ell} c1, c4 = 0, and the relations d^ell_{(sigma1,sigma2)} = d^0_{(sigma2,sigma1)}), which in turn support Theorem 1(3) and the lower bound Lemma 7. As written, the appendix's derivation of the invariant measure is therefore incomplete. Please provide the proof of Lemma 12 or give a precise reference to the corresponding statement in [HGM25] and explain how it applies to the present construction.
  3. [Section 3.2, Lemma 4 / Proposition 5] The uniformity result for the DFTP is not proved in the manuscript. Lemma 4(i)-(ii) is proved only for the DITP; the text says the same arguments apply "up to some slight modifications" to the DFTP. Proposition 5, which supplies the t -> infinity and L -> infinity interchange for both processes and hence Corollary 1 for the DFTP, depends on Lemma 4(ii) for the DFTP. Please include the DFTP calculation (the generating function for the three-state velocity chain and the resulting fourth-moment ratio) or provide a reference where this is done.
minor comments (4)
  1. [Section 4.1, Lemma 7] The conclusion of Lemma 7 reads "ti_mix(epsilon) >= ..." but the proof is for the CFTP; it should read t^f_mix(epsilon).
  2. [Proposition 4] The minorization state is written as xi = epsilon delta_{(0,(-1,1))}, while the statement and proof use (0,(1,-1)); the two are distinct states, so the Dirac should be at (0,(1,-1)).
  3. [Throughout] The notation for the factor (-1)^i sigma_i is typeset without the exponent in several displays (e.g., in Definition 5 and Lemma 1), which makes expressions such as "(-1)i sigma_i" hard to parse; please fix the typesetting.
  4. [Various locations] There are several typos, including "proccess" (Proposition 4), "identfies" (Lemma 2), "unnormalizedinvariant" (Proposition 8), and "a pairs" (Section 1.2.1).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's main theorems are derived from explicit couplings, PDMP generator computations, and independent invariant-measure derivations; self-citations are contextual, not load-bearing.

full rationale

The central derivation chain is self-contained. The discrete DITP/DFTP and continuous CITP/CFTP are defined independently, and the scaling limit (Theorem 2) is proved by an explicit coupling with a martingale estimate (Lemma 1, Lemma 2), not by assuming the target convergence. The invariant measures are not imported as inputs: the appendix solves the generator equation Lur f dpi = 0 directly for the CITP and CFTP (Propositions 7 and 8), with the leading reduction to a linear ODE system and boundary conditions performed in the paper. Convergence of discrete invariant measures (Corollary 1) uses the uniform ergodicity of Proposition 5, whose proof relies on coupling and Paley–Zygmund type estimates on velocity integrals, not on the continuous invariant measure. The mixing-time upper bounds are obtained from explicit hitting-time equations (Lemma 8) and couplings with velocity-coupling estimates (Lemmas 9–11); the lower bounds use slow observables and explicit mass of the invariant measure, which is derived in the appendix. Citations to the authors' own [HGM25] are used for framing, terminology, and universality classes, but the load-bearing statements here do not reduce to that citation: the PDMP construction follows Davis and the invariant measures are re-derived. One flagged limitation, relevant to completeness but not to circularity, is the missing CITP lower-bound proof in Section 4.1: the text says 'only the lower bound of assertion (ii) is proven' and then states the lower bound of Theorem 3(i) without an explicit two-state analogue of Lemmas 6–7. This is a proof gap, not a case of a prediction being equivalent to its input. Since no load-bearing argument reduces to a fitted input, a definitional identity, or an unverified self-citation, the circularity score is 0.

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

No new free parameters are fitted to data; the only numbers are model parameters (ω, α, β, ℓ) and existential constants C(ε), C'(ε) whose dependence on ε is not optimized. The paper postulates no new particles, forces, or conserved quantities. The assumptions are the standard PDMP framework, the specific tumble rates, and the jamming rule, all of which are model definitions rather than hidden inputs.

assumptions (5)
  • domain assumption The velocity processes σ1 and σ2 are independent Markov jump processes with transition rates of Figure 2a (instantaneous tumble) or Figure 2b (finite tumble).
    This defines the DITP/DFTP and CITP/CFTP models in Definitions 1 and 2; all results are for these two tumble mechanisms.
  • domain assumption Particles cannot pass through each other: the separation x is clamped to [0,ℓ] and jammed states are absorbing for the deterministic flow until a velocity jump occurs.
    This is the jamming interaction that generates the boundary states ∂K in Section 2 and the max/min definition of x in Definition 2.
  • standard math The standard PDMP theory of Davis [Dav93] applies: strong Markov property and the generator characterization (Theorem 26.14).
    Used in Section 2 to construct the continuous processes and to compute hitting times in Lemma 8 and invariant measures in the appendix.
  • standard math Standard probabilistic inequalities are used: Doob's martingale inequality, Paley-Zygmund, Kolmogorov's inequality, Chebyshev's inequality.
    These underpin the coupling estimates in Theorem 2, Lemma 4, Lemma 6 and Lemma 11.
  • domain assumption The scaling γL = (L-1)/ℓ keeps the physical velocity constant as L→∞.
    This is the natural discretization scaling stated in equation (*) in Section 3.1; it fixes the relationship between lattice spacing and jump rate that makes the limit nontrivial.

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Cite this review

Pith. "Pith review of Long-time analysis of a pair of on-lattice and continuous run-and-tumble particles with jamming interactions." pith.science (2026). https://pith.science/paper/SEDEVCT3

@misc{pith2026241113964,
  author       = {Pith},
  title        = {Pith review of: Long-time analysis of a pair of on-lattice and continuous run-and-tumble particles with jamming interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SEDEVCT3}},
  note         = {Machine review of arXiv:2411.13964}
}
read the original abstract

Run-and-Tumble Particles (RTPs) are a key model of active matter. They are characterized by alternating phases of linear travel and random direction reshuffling. By this dynamic behavior, they break time reversibility and energy conservation at the microscopic level. It leads to complex out-of-equilibrium phenomena such as collective motion, pattern formation, and motility-induced phase separation (MIPS). In this work, we study two fundamental dynamical models of a pair of RTPs with jamming interactions and provide a rigorous link between their discrete- and continuous-space descriptions. We demonstrate that as the lattice spacing vanishes, the discrete models converge to a continuous RTP model on the torus, described by a Piecewise Deterministic Markov Process (PDMP). This establishes that the invariant measures of the discrete models converge to that of the continuous model, which reveals finite mass at jamming configurations and exponential decay away from them. This indicates effective attraction, which is consistent with MIPS. Furthermore, we quantitatively explore the convergence towards the invariant measure. Such convergence study is critical for understanding and characterizing how MIPS emerges over time. Because RTP systems are non-reversible, usual methods may fail or are limited to qualitative results. Instead, we adopt a coupling approach to obtain more accurate, non-asymptotic bounds on mixing times. The findings thus provide deeper theoretical insights into the mixing times of these RTP systems, revealing the presence of both persistent and diffusive regimes.

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Cited by 1 Pith paper

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