{"id":"d57d40c3-9d04-43c9-90e8-dbc5a34f6650","arxiv_id":"2501.11379","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two attracting, non-crossing run-and-tumble particles on a line have explicit steady-state measures with boundary Dirac masses and quantified exponential relaxation rates.","lead":"This paper derives exact long-time probability laws for two run-and-tumble particles that attract each other but cannot cross, in one dimension. It also gives explicit estimates for how quickly the two-particle system forgets its starting configuration.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4(i) omits the v>c hypothesis that its own proof requires; for v≤c the advertised exponential density is non-integrable and the true invariant measure is a Dirac mass at the jamming wall.","rationale":"The reader's weakest assumption is exactly the missing v>c condition, and this is the most load-bearing issue: the invariant-measure formulas are the central deliverable, and Theorem 4(i) is false as stated for v≤c. This is not a matter of disagreement with an outside consensus; it is an internal inconsistency between a main theorem and its proof, which explicitly integrates e^{ζx} against -1/ζ. The finite-linear theorem already states v>c explicitly, and the harmonic boundary normalization is only sketched, but that is a completeness concern rather than a demonstrable false statement. I do not see an equally concrete false statement elsewhere in the paper. Therefore the reader's conditional verdict should stand, with the v>c hypothesis added to the instantaneous linear statements.","tokens_in":32561,"tokens_out":12010,"duration_ms":127004,"concrete_test":"Evaluate the proof of Proposition 6 at v = 1, c = 2, ω = 1: ζ = 4/3 > 0, so the step 1/4 = a∫₀^∞ e^{ζx}dx = a/(-ζ) is invalid, and the formula's RHS has infinite total mass while a₂ and a₋₂ change sign. Independently, solve the bulk Fokker-Planck system in this parameter range: every drift -2c + vσ is negative, so the only candidate invariant probability is δ₀ ⊗ πQ, which differs from the theorem's expression. If the manuscript intends to assume v > c throughout, adding that hypothesis to Theorems 4(i) and 5(i) resolves the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 6 defines ζ = -2cω/(v²-c²) and fixes the bulk coefficient by 1/4 = a∫₀^∞ e^{ζx}dx = a/(-ζ), an identity that is valid only when ζ < 0, i.e. v > c. Theorem 4(i) states the resulting invariant measure without the v>c restriction. If v < c, then ζ > 0 and e^{ζx} is not integrable on R₊, so the displayed expression is not a finite measure; if v = c, ζ is undefined. Moreover, in this regime every bulk drift is -2c + vσ ≤ -2c + 2v < 0 for all σ, so the relative distance is pushed to 0 and remains jammed; the unique invariant probability is δ₀ ⊗ πQ, not the exponential mixture asserted in Theorem 4(i). The same missing hypothesis affects Theorem 5(i), whose mixing-rate statement is tied to this invariant measure. The finite-linear theorem explicitly restricts to c < v ≤ 2c and v > 2c, so the omission in the instantaneous case appears to be an oversight in the theorem statement rather than a flaw in the computation for the intended parameter range.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":32773,"tokens_out":11484,"duration_ms":113382,"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":[{"comment":"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":"Section 1.2, Theorem 4(i); Section 2.1, Proposition 6"},{"comment":"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":"Section 2.3, Proposition 12 and Theorem 4(iii)"},{"comment":"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.","section":"Section 4.2, Lemma 37(iii)"}],"minor_comments":[{"comment":"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.","section":"Lemma 34, proof"},{"comment":"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.","section":"Theorem 25(ii)"},{"comment":"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.","section":"Lemma 15 vs. Lemma 30"},{"comment":"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.","section":"Proposition 10 and Notation"}],"recommendation":"major_revision","confidential_remarks":"The missing v>c hypothesis in Theorem 4(i), Proposition 6, and Theorem 5(i) appears to be an oversight in the theorem statements rather than a flaw in the computations for the intended parameter range; the derivations for v>c are detailed and self-consistent. The harmonic boundary normalization and the computer-algebra optimization in Lemma 37(iii) should receive more support before publication. The paper is within the scope of the journal and builds appropriately on prior work; I see no citation or attribution concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nWhere 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.\n\nWho 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.","headline":"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.","tokens_in":33314,"tokens_out":1464,"would_cite":true,"duration_ms":18272,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J25","60J75"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["run-and-tumble particles","jamming","invariant measure","piecewise deterministic Markov processes","exponential ergodicity","coupling","active matter","clustering"],"falsifier":"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.","tokens_in":32337,"feed_emoji":"🧫","tokens_out":12519,"duration_ms":124364,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact steady states and mixing rates for jammed active pairs","feed_subtitle":"Explicit formulas for clustering probabilities and relaxation times of two attracting, non-crossing run-and-tumble particles.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"gives the attractive two-RTP model without jamming whose bulk invariant measure is reweighted here, fixing the exponential bulk form and highlighting the new boundary Dirac masses","marker":"[LDMS21]"},{"why":"supplies the exact bulk density of a run-and-tumble particle with three internal states in a harmonic trap, which the harmonic-process invariant measure imports for its density part","marker":"[BMR+20]"},{"why":"introduces the PDMP-generator treatment of jammed run-and-tumble pairs and the observation that bulk Fokker-Planck reduces to a linear ODE system","marker":"[HGM23]"},{"why":"provides the synchronous coupling that preserves order between two copies, the basis for the total-variation lower bounds","marker":"[GHM24]"},{"why":"gives the non-asymptotic large-deviation inequality for additive functionals used to control the hitting-time Laplace transform","marker":"[Wu00]"},{"why":"supplies the Chernoff-type Markov-chain bound that yields the explicit finite-linear lower decay rate","marker":"[Lez01]"},{"why":"provides the contracting-coupling framework and Wasserstein-type distance used for the harmonic process","marker":"[BLBMZ12]"},{"why":"establishes the PDMP formalism and the explicit generator characterization on which all invariant-measure computations rest","marker":"[Dav93]"},{"why":"used together with the constructed Lyapunov function to obtain existence and uniqueness of the invariant probability","marker":"[MT93]"}],"fun_headline_variants":["Jammed particles yield exact steady states and fast mixing","Exact invariant measures for two jammed active particles","Precise clustering and relaxation in jammed run-and-tumble pairs","Sharp bounds on mixing for jammed attractive particles","Dirac masses and exponential mixing in jammed active pairs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Jammed particles yield exact steady states and fast mixing","Exact invariant measures for two jammed active particles","Precise clustering and relaxation in jammed run-and-tumble pairs","Sharp bounds on mixing for jammed attractive particles","Dirac masses and exponential mixing in jammed active pairs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000823,"raw_usage":{"total_tokens":3628,"prompt_tokens":999,"completion_tokens":2629,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":2549}},"tokens_in":615,"tokens_out":2629,"duration_ms":17922,"temperature":1.0,"reasoning_tokens":2549,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:19:46.853178+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}