Steady state and mixing of two run-and-tumble particles interacting through jamming and attractive forces

Leo Hahn (LMBP)

REVIEW 3 major objections 4 minor 46 references

Steady state and mixing of two run-and-tumble particles interacting through jamming and attractive forces

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper derives explicit invariant measures and exponential mixing rates for three one-dimensional models of two run-and-tumble particles whose attraction and jamming produce clustered, non-Boltzmann steady states, with sharp parameter…

desk verdict Solid extension of the jamming-RTP program with explicit invariant measures and sharp mixing bounds; the main fix needed is stating v > c in the linear-process theorems. read the letter →

arxiv 2501.11379 v1 pith:IBY7VJYD submitted 2025-01-20 math.PR cond-mat.stat-mech

Leo Hahn (LMBP) This is my paper · ORCID

classification math.PRcond-mat.stat-mech MSC 60J2560J75

keywords run-and-tumbleparticlesjamminginvariantmeasurepiecewisedeterministicMarkovprocessesexponentialergodicitycouplingactivematterclustering

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 studies two run-and-tumble particles on the real line that attract each other through a potential and are jammed so they cannot pass, in three model variants: linear attraction with instantaneous tumbling, linear attraction with finite-duration tumbling, and harmonic attraction with instantaneous tumbling. The paper's central claim is that each model has a unique steady-state probability measure given by an explicit formula, and that these measures are non-Boltzmann: they place positive probability on the exact glued configurations at the jamming boundary, so clusters are genuinely typical states. It further claims exponential relaxation to these steady states, with total-variation decay rates determined up to constant factors by the model parameters in the linear cases and with Wasserstein-type exponential decay in the harmonic case. This matters because explicit stationary statistics and relaxation times for interacting active particles are rare, and the results give a solvable picture of when clustering emerges and how fast.

What carries the argument

The load-bearing object is the extended generator of the processes viewed as piecewise-deterministic Markov processes — flows that evolve deterministically between random tumbling kicks. Stationarity is expressed through ∫Lf dπ=0, which encodes the jamming boundary in the domain of the generator rather than in a separate boundary condition. In the bulk x>0 the stationarity equations reduce to a linear ODE system Π′=AΠ, so the invariant density is a matrix exponential; the exponents are explicit roots of characteristic polynomials, and the Dirac weights are fixed by the invariant distribution of the velocity chain. For mixing, a synchronous coupling makes the velocities of two copies equal after a random time and then preserves their spatial order, so convergence to the steady state is controlled by the hitting time of the boundary; non-asymptotic large-deviation bounds for additive functionals of the velocity process bound the Laplace transform of that hitting time, and a separate large-separation argument gives matching upper bounds on the decay rate. The harmonic case instead exploits the deterministic flow's exponential contraction and uses a Wasserstein-type mixed distance.

What would settle it

Run a long simulation of the instantaneous linear process with v<c, say c=1, v=0.5, ω=1, and measure whether the occupation measure over x≥0 converges to any finite limit; the paper's formula is not normalizable in that regime, so a finite steady state there would disprove the claimed parameter dependence, while escape of x to large values would confirm the necessity of v>c.

Watch

Extended reading notes

Core claim

The central discovery is that adding jamming to two attracting run-and-tumble particles produces invariant measures with a singular boundary part: in the linear cases π = Σσ (dσ δ0 + aσ $e^{{ζx}}$ dx) ⊗ δσ, with positive Dirac weights at jammed relative-velocity states and an exponential bulk density whose exponent is ζ = −2cω/(v²−c²). The finite-tumble linear model changes character at v=2c: for c<v≤2c the bulk is a product measure and the jammed state (0,1) carries Dirac mass, while for v>2c an additional exponential term appears, the product form is lost, and that Dirac mass vanishes. The harmonic model has a compactly supported steady state on [0,v/µ]×Σ whose density can diverge or vanish at the boundary according to whether µ is larger or smaller than ω. The paper also establishes unique invariant probabilities and proves that convergence to them is exponential: the total-variation rate for the instantaneous linear process is between ωc²/(2v²) and 4ωc²/v², with analogous constant-factor bounds in the finite linear case, and the harmonic process converges in a Wasserstein-type distance at rate at least min(µ, ω/p).

Load-bearing premise

The linear-process results rest on the premise that the run speed v exceeds the attraction strength c, since the exponent ζ = −2cω/(v²−c²) is negative only then and otherwise the advertised exponential densities are not integrable, yet this condition appears only in a lemma, not in the main theorem statements.

Editorial extensions

If this is right

For the instantaneous linear process, the total-variation relaxation rate is sandwiched between ωc²/(2v²) and 4ωc²/v², so the dependence of the mixing time on all model parameters is known up to a constant factor.
For the finite linear process, the steady state undergoes a qualitative transition at v=2c: below it the bulk is a product measure and the jammed state (0,1) has positive probability, while above it an extra exponential term appears and that Dirac mass disappears.
For the instantaneous harmonic process, the invariant measure has compact support [0,v/µ], boundary Dirac masses at 0, and a density that either diverges or vanishes at the two boundaries depending on whether µ is larger or smaller than ω.
In all three processes the jammed boundary carries Dirac masses, so exactly clustered configurations are typical in steady state — a purely out-of-equilibrium phenomenon that cannot occur for a Boltzmann measure.
The non-asymptotic convergence bounds have prefactors growing exponentially in the initial separation, so widely separated particles take a time that grows like their separation before the steady-state description becomes accurate.

Reading between the lines

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

One step beyond the paper would be to use the same generator-plus-matrix-exponential route for other solvable tumbling mechanisms and power-law potentials; the paper notes that the finite-tumble harmonic potential already seems intractable this way, so the exact boundary weights might be special to linear and harmonic potentials.
The gap between the finite-linear upper and lower rates when α/β is large leaves which bound is sharp unresolved; a natural conjecture is that the true rate in the tumbling-dominated regime is controlled by rare long-running stretches rather than by the large-deviation rate used in the lower bound.
Because the two-particle setup reduces to a single relative-coordinate process, these exact Dirac-mass steady states are the cleanest available microscopic signature of motility-induced clustering; extending them to three or more jammed particles is open, and these formulas are the natural baseline for testing whether boundary masses persist.

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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

You should know two things about this paper. First, it does exactly what it says: it writes down explicit invariant measures and quantitative exponential mixing bounds for three two-particle run-and-tumble models with attraction plus jamming, including Dirac masses on the jamming boundary and a finite-tumble-duration zero-velocity sector. Second, the one real problem is a missing parameter condition in the main theorem statements: Theorem 4(i) and Theorem 5(i) omit the v > c hypothesis that the proof actually requires. For v ≤ c, the advertised exponential density is not integrable, and the true invariant measure is a Dirac mass at the wall. The stress-test note is right, and it is an oversight in the statement rather than a flaw in the computation for the intended regime.

What is genuinely new: the boundary Dirac masses, the zero-velocity tumble channel with its product-to-non-product transition, the threshold at v = 2c in the finite-linear case, and the total-variation decay rates that are sharp up to a constant factor in the instantaneous linear case. The bulk densities reweight known results from LDMS21 and BMR+20, and the paper says so openly. The credit is honest.

The derivations are mostly rigorous and detailed: the generator-based ODE system, the spectral analysis of the finite-linear matrix, the synchronous coupling, and the large-deviation bounds all hang together. The sharpness argument using the Laplace transform of the hitting time is a nice piece of work. The computer algebra optimization in the finite case is acceptable given the size of the expressions, though I would want the authors to double-check the resultant computations.

Where the soft spots are: first, the missing v > c condition in Theorems 4 and 5 is a genuine defect in presentation and should be fixed before publication. Second, the harmonic invariant measure imports the bulk density from BMR+20 and only sketches the boundary normalization; the formulas are plausible but a full derivation or a machine-checkable certificate would be better. Both are fixable without changing the substance. The harmonic mixing bound in Wasserstein distance is fine, though the statement gives a lower bound on the rate and does not claim sharpness there.

Who is this for? Anyone working on exact steady states of active particle systems with jamming, and people using PDMP couplings for mixing. It is an extension of an ongoing program rather than a breakthrough, but it is useful and mostly careful. I would send it to a serious referee. My recommendation: accept after minor revision, with the parameter domain stated correctly in the main theorems and the harmonic normalization either completed or marked as a lemma with a pointer to a verified appendix.

Referee Report

3 major / 4 minor

Summary. The paper studies the long-time behavior of two one-dimensional run-and-tumble particles interacting through an attractive potential and jamming, for three models: the instantaneous linear process (V(x)=c|x|, instantaneous tumbling), the finite linear process (V(x)=c|x|, finite tumble duration), and the instantaneous harmonic process (V(x)=\mu x^2/2). For each model the author derives an explicit invariant measure, typically a mixture of a bulk absolutely continuous part and Dirac masses on the jamming boundary x=0, and proves quantitative exponential convergence: total-variation bounds for the two linear processes and Wasserstein-type bounds for the harmonic process. The linear-process convergence rates are shown to be sharp up to constant factors. The methods are the extended generator of piecewise-deterministic Markov processes, reduction of the stationary Fokker-Planck equation to linear ODE systems, synchronous couplings, and non-asymptotic large-deviation estimates for additive functionals of Markov chains.

Significance. If the results are correct, they provide exact steady-state statistics and relaxation times for two interacting, jammed, attracting run-and-tumble particles in one dimension. The explicit invariant measures display boundary Dirac masses and parameter-dependent qualitative changes, which are relevant for the active-matter literature on clustering and on close-to-equilibrium versus strongly out-of-equilibrium universality classes. The paper's methods are a genuine strength: the invariant measures are derived from the generator equation by solving linear ODE systems without fitted parameters, the convergence bounds are non-asymptotic and explicit, and the sharpness analysis identifies concrete obstacles to mixing. The harmonic-process formulas import the bulk density from BMR+20 but remain self-contained enough to be checked. The central limitation is a missing parameter condition in several theorem statements, which is local and fixable.

major comments (3)

[Section 1.2, Theorem 4(i); Section 2.1, Proposition 6] Theorem 4(i) and Proposition 6 state the invariant measure of the instantaneous linear process without any restriction on v and c, but the proof uses the normalization 1/4 = a\int_0^\infty e^{\zeta x}dx = a/(-\zeta), which is valid only when \zeta<0, i.e. v>c. For v<c one has \zeta>0 and the displayed exponential density is not integrable; for v=c the quantity \zeta is undefined. In the regime v\le c the bulk drift v\sigma-2c is non-positive for every \sigma, so the relative position is driven to 0 and remains jammed, and the unique invariant probability is \delta_0\otimes\pi_Q rather than the mixture asserted in Theorem 4(i). The same missing hypothesis affects Theorem 5(i), where the rate I(c/v) is undefined for c/v>1, and the statements in Section 3.1 that use I(c/v). The theorems should be restricted to v>c and the v\le c case should be described separately; this is a local but load-bearing correction.
[Section 2.3, Proposition 12 and Theorem 4(iii)] The boundary normalization for the instantaneous harmonic process is only sketched. After importing the bulk density from [BMR+20], the text states that the three equations in (4) determine C1, C2, C3, but the algebra is not shown, and the final statement gives explicit formulas only for d0 and C, not for C3=\pi(\{(0,-2)\}) even though Theorem 4(iii) asserts d_{-2}>0. Please provide the explicit values of C1, C2, C3, or a verification that the displayed d0 and C satisfy (4), and give a separate argument for the asserted positivity of d_{-2}. This is needed because the boundary Dirac masses are one of the paper's central claims.
[Section 4.2, Lemma 37(iii)] The upper bound limsup - (1/t) log ||\delta_{(x,\sigma)}P_t - \pi||_{TV} \le 4\alpha(1+\alpha/\beta)c^2/v^2 is obtained by optimizing an explicit rational expression over r>0 and 0<R<1 'with the help of a computer algebra system', but no certificate, code, or derivative calculation is provided. Since this optimization is the only step producing the constant 4 in the finite-linear sharpness bound, please include a rigorous proof of the sup equals 4, or provide a verifiable computer-assisted proof in a supplementary file.

minor comments (4)

[Lemma 34, proof] In Case 1 and Case 2 the factors v are missing from the drift terms: the text reads '\partial_t x = -2c + \sigma_2(t_k) - \sigma_1(t_k)' and '\partial_t x > -2c + \sigma_2 - \sigma_1' instead of the v-scaled terms used in Definition 33.
[Theorem 25(ii)] The bound W_p(\eta P_t, \pi) \le (1+2^{1/s})^{1/p}(F_q(\eta)+v/\mu)\exp(...) uses F_q(\eta); the statement should explicitly assume that \eta has finite q-th moment for all q>p used in the bound, or define the right-hand side as +\infty otherwise.
[Lemma 15 vs. Lemma 30] The function \Lambda defined in Lemma 15 (the log-moment generating function used in the large-deviation bound) and the function \Lambda defined in Lemma 30 (the hitting-time exponent) are unrelated; please relabel one of them to avoid confusion.
[Proposition 10 and Notation] In the finite linear case the coefficient c_2 of the exponential term and the model parameter c (the attractive force strength) have very similar names; this is a readability issue, not a mathematical one, but it may confuse readers comparing the formula with the parameter c.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: invariant measures are solved from the generator equation and fixed by independent marginal normalizations; mixing rates follow from large-deviation and tail estimates, with no fitted input renamed as prediction.

full rationale

The paper's central derivations are self-contained and do not reduce to their own inputs. The invariant measures are obtained by writing the generator characterization ∫Lf dπ = 0, solving the induced linear ODE system in the bulk, and fixing the remaining constants by three independent ingredients: integrability (which discards positive eigenvalues), the invariant measure πQ of the one-dimensional velocity Markov chain (an elementary, independently solvable object), and an ergodicity argument showing the boundary Dirac mass at (0,2) is zero. For the instantaneous linear process, the normalization condition π([0,∞)×{2}) = 1/4 fixes the bulk coefficient a via the identity 1/4 = a∫_0^∞ e^{ζx}dx; this is a check against the velocity-marginal invariant measure, not a restatement of the target measure. The finite linear process uses the same scheme with the spectral decomposition of A = V^{-1}Q^t, with Lemmas 7 and 9 proved internally, and the constants c2, c3 fixed by the two or four marginal equations π(R_+×{σ}) = πQ({σ}). The harmonic process imports the bulk density P,Q,R from the external reference BMR+20 and then determines the boundary Dirac masses by the marginal system (4); this imports an external result as a tool, not the paper's own conclusion, and the boundary contribution is new content. The mixing-rate lower bounds are proved by an explicit synchronous coupling (Lemma 22) together with non-asymptotic large-deviation estimates quoted from Wu00, with the rate function I computed explicitly in Proposition 19; the upper bounds use a Laplace-transform computation for the hitting time of large x and the tail decay λπ of the already-computed invariant measure. None of these steps fits a fitted parameter renamed as a prediction. The paper does cite the author's prior work HGM23 and GHM24, but those citations supply the PDMP/jamming formalism and the synchronous-coupling idea; they are not the source of the exponential-rate estimates, which are derived here with the external results [Wu00], [MT93], and [Lez01]. The omission of the v>c hypothesis from the statements of Theorems 4(i) and 5(i), while mathematically important and noted in the paper's own proof (ζ = -2cω/(v²-c²) requires v>c for integrability), is a parameter-range oversight rather than circular reasoning. Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central derivations rest on standard PDMP and Markov-process tools plus the bulk density of BMR+20. The only parameter condition hidden in the main statements is v>c for the linear processes. There are no fitted free parameters; all constants are solved from normalization and generator equations.

assumptions (6)

standard math The interacting run-and-tumble process is a Feller PDMP with extended generator L; stationary measures are exactly those with integral Lf d pi = 0 for all domain functions.
Used throughout Section 2; Proposition 3 relies on Davis 1993 and GHM24.
standard math Foster-Lyapunov criteria imply existence, uniqueness, and exponential ergodicity for the linear processes.
Theorem 16(i) applies Meyn-Tweedie 1993 using the hitting-time Lyapunov function from Lemma 20.
standard math Non-asymptotic large-deviation bounds of Wu 2000 and Lezaud 2001 control additive functionals of the velocity process.
Lemma 15 and Proposition 19 transform tail probabilities of hitting times into exponential decay of the rate function I.
domain assumption The exact stationary bulk density of a three-state run-and-tumble particle in a harmonic trap, from BMR+20, is valid and is used to build the harmonic invariant measure.
Proposition 12 reweights this density and only sketches the boundary normalization.
domain assumption For the linear processes, the run speed exceeds the attractive force, v>c, so the exponential densities are integrable and the boundary escape rules are as stated.
Required for zeta=-2c omega/(v^2-c^2) to be negative; stated in Lemma 7 but not in Theorem 4.
domain assumption The jamming rule x(t)=max(0, phi^sigma_{t-T_n}(x(T_n))) correctly models collisions, with particles gluing at x=0 when their relative drift is non-positive.
This is the model definition; its Markovian well-posedness is quoted from GHM24.

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Cite this review

Pith. "Pith review of Steady state and mixing of two run-and-tumble particles interacting through jamming and attractive forces." pith.science (2026). https://pith.science/paper/IBY7VJYD

@misc{pith2026250111379,
  author       = {Pith},
  title        = {Pith review of: Steady state and mixing of two run-and-tumble particles interacting through jamming and attractive forces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IBY7VJYD}},
  note         = {Machine review of arXiv:2501.11379}
}

read the original abstract

We study the long-time behavior of two run-and-tumble particles on the real line subjected to an attractive interaction potential and jamming interactions, which prevent the particles from crossing. We provide the explicit invariant measure, a useful tool for studying clustering phenomena in out-ofequilibrium statistical mechanics, for different tumbling mechanisms and potentials. An important difference with invariant measures of equilibrium systems are Dirac masses on the boundary of the state space, due to the jamming interactions. Qualitative changes in the invariant measure depending on model parameters are also observed, suggesting, like a growing body of evidence, that run-andtumble particle systems can be classified into close-to-equilibrium and strongly out-of-equilibrium models. We also study the relaxation properties of the system, which are linked to the timescale at which clustering emerges from an arbitrary initial configuration. When the interaction potential is linear, we show that the total variation distance to the invariant measure decays exponentially and provide sharp bounds on the decay rate. When the interaction potential is harmonic, we give quantitative exponential bounds in a Wasserstein-type distance.

Figures

Figures reproduced from arXiv: 2501.11379 by the authors.

**Figure 2.** Relative velocity transition rates Note that if the σi follow figure 1b then σ2 − σ1 is not Markovian. Therefore, we split the state σ2 − σ1 = 0 into the states 0± (corresponding to σ1 = σ2 = ±1) and 00 (corresponding to σ1 = σ2 = 0) to recover a Markov jump process (see figure 2b). Assume without loss of generality that x(0) = x2(0) − x1(0) ≥ 0 and define the behavior of the process when the particles collide, i.e.… view at source ↗

**Figure 3.** Realization of the relative distance x The following notation is convenient when writing down the extended generator of the interacting run-and-tumble process. Notation 2. Denote Q = (qσ,σ˜ )σ,σ˜∈Σ the transition-rate matrix of the Σ-valued Markov jump process σ(t). In the instantaneous case (Figure 2a) Q =   σ˜ = 2 ˜σ = 0 ˜σ = −2 σ = 2 −2ω 2ω 0 σ = 0 ω −2ω ω σ = −2 0 2ω −2ω  , while in the finite case (Figure … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Single velocity coupling Let (σ1, σ˜1) and (σ2, σ˜2) be independent Markov jump processes with the transition rates of Figure 4a (resp. 4b) and initial states (σ1,0, σ˜1,0) and (σ2,0, σ˜2,0). It follows from [BY93, Theorem 2.4] 18 [PITH_FULL_IMAGE:figures/full_fig_p0…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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