{"id":"f3f95795-289c-4765-aa14-f00790afe68a","arxiv_id":"2502.09466","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Run-and-tumble particles with 1D rank interactions have exactly computable large-N stationary densities in harmonic and linear traps, with shock and symmetry-breaking phases.","lead":"This paper finds exact formulas for the stationary density of run-and-tumble particles in one dimension that interact through a force proportional to their rank, both inside a harmonic trap and with a non-reciprocal version of the interaction. It identifies phases with edge jumps, delta-peak clusters or shocks, and a symmetry-broken state where particles accumulate on one side of the origin.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Edge density at γ=μ is misderived: Eqs. (126)-(128) yield ρs(x) ~ 1/|ln(xe−x)|, not 1/[ln(xe−x)]^2.","rationale":"The reader's weakest assumption was about the validity of dropping noise in the Dean–Kawasaki equation and selecting the physical branch. That is a generic mean-field concern. My review identifies a more concrete and checkable defect: the derivation of the edge density on the γ=μ line contains a calculus error. Since the paper's central claim explicitly includes the edge behavior—and the abstract and Eq. (12) advertise the 1/(ln ε)^2 law—this error makes the statement 'exact in each phase' inaccurate for that boundary. The parametric representation, phase diagram, and non-reciprocal densities appear unaffected, and the numerical comparisons in Figs. 2 and 4 support those parts. Therefore the appropriate verdict is CONDITIONAL: accept after correcting the edge asymptotics, or alternatively revise the claim to state the correct 1/|ln ε| behavior. The concern is not an attack on the authors' integrity; it is a precise mathematical inconsistency that can be fixed.","tokens_in":38217,"tokens_out":17647,"duration_ms":153357,"concrete_test":"Re-derive the asymptotics symbolically: insert z = −K ε/ln ε into −z ln z and compute ρs = −dz/dx; verify that ρs ≈ K/|ln ε|. Alternatively, simulate Eq. (1) at μ=γ=1, κ=1, v0=1 with N=10^4–10^5, extract ρs(x) near the edge, and fit to A/|ln(xe−x)| vs B/[ln(xe−x)]^2; the former should match with A=μ/(2κ).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Section III F 3, Eqs. (125)–(128) derive the edge behavior at a=1 (γ=μ). Let z=1/2−r and ε=xe−x. Eq. (126) is −z ln z ≃ K ε with K=2γ/b. The stated solution (127) is z ≃ −K ε/ln ε. Differentiating this expression gives dz/dx = K(ln ε − 1)/(ln ε)^2, so ρs = r′ = −dz/dx ≃ K/|ln ε|, not K/(ln ε)^2 as written in Eq. (128). The 1/(ln ε)^2 term is a subleading correction; the leading behavior is 1/|ln ε|. Thus Eq. (12) and Eq. (128) do not follow from the preceding equations. This is a concrete internal inconsistency in a claimed exact prediction, and it directly affects the edge law highlighted in the abstract and main results.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies N run-and-tumble particles with 1D Coulomb (rank) interactions in two settings: (i) the \"active jellium\" with harmonic confinement, for which it derives a parametric representation of the stationary density in the N→∞ limit and analyzes the phase diagram with smooth, jump, and shock edges; and (ii) a non-reciprocal rank interaction with linear confinement, for which it obtains explicit stationary densities in Lambert-function form and a four-phase diagram. The analytical results are compared with finite-N simulations. The paper claims exact large-N stationary densities in each phase, including edge exponents, shock weights, and broken mirror symmetry.","tokens_in":38366,"tokens_out":18540,"duration_ms":157283,"significance":"If the results are correct, the paper provides rare exact stationary-state information for non-equilibrium many-body systems with long-range and non-reciprocal interactions. The derivations are systematic and parameter-free: constants are fixed by boundary conditions, normalization, and force balance, and the analytical densities are checked against independent finite-N simulations. The non-reciprocal model's explicit Lambert-function densities and the phase diagram with one-sided support are particularly valuable. However, the two algebraic issues detailed below affect central displayed formulas and must be fixed; they are local and do not, in my assessment, invalidate the overall framework.","major_comments":[{"comment":"The parametric density formula is inconsistent with the preceding equations. Differentiating Eq. (38), μx = 2κr − U′(r), with respect to x gives μ = (2κ − U″)r′, hence ρs = r′ = μ/(2κ − U″) = μ/(2κ − ½G′(U)) by Eq. (53). Eq. (52) instead writes ρs = μ/(2κ) − U″ and Eq. (55) writes ρs = μ/(2κ) − ½G′(U). These are not equivalent. A concrete check is the C=0 line of Section III F 2, where U″ = −2v0: Eq. (52) would yield ρs = μ/(2κ) + 2v0, whereas the explicit solution Eq. (116) and the correct formula both give ρs = μ/[2(κ+v0)]. The correct parametric density is ρs = μ/(2κ − ½G′(U)) (equivalently Eq. (117)); Eqs. (52) and (55) must be corrected.","section":"III C, Eqs. (52) and (55)"},{"comment":"The claimed edge law ρs(x) ≃ (μ/2κ)/[ln(xe−x)]^2 does not follow from Eq. (126). Setting z = 1/2 − r and ε = xe − x, Eq. (126) reads −z ln z ≃ (2γ/b)ε. The leading solution is z ≃ (2γ/b)ε/|ln ε|, so dz/dx ≃ −(2γ/b)/|ln ε| and ρs = r′ = −dz/dx ≃ (2γ/b)/|ln ε| = (μ/2κ)/|ln ε|. The 1/(ln ε)^2 term is a subleading correction. Consequently Eq. (12) and the corresponding main-result statement are incorrect as written; Eq. (135) for ρ−(x) should similarly contain (xe−x)/|ln(xe−x)| rather than (xe−x)/[ln(xe−x)]^2.","section":"III F 3, Eqs. (125)–(128) and Eq. (12)"}],"minor_comments":[{"comment":"Phase III is defined by the condition a < v0 − b/2 in the main results, but this is the same condition as phase I and is impossible for a > 0 when b > 2v0; the correct condition, used in Section IV B 3 and in Fig. 3, is a < b/2 − v0.","section":"II B, Eq. (26) and surrounding text"},{"comment":"There are several typographical slips (\"the the\" in Section I, \"It it\" in the introduction, \"one the other hand\" in Section III D) that should be cleaned up.","section":"I and III C"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is valid, and I found an additional inconsistency in the parametric density formula (52)/(55). Both are local algebraic errors: the subsequent shock and edge analyses use the correct relations (100) and (117), so the phase diagram conclusions are largely unaffected, but the central parametric representation must be corrected before publication. The phase III condition typo in Section II B should also be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper genuinely extends the active rank diffusion program: the harmonic-trap parametric representation (50)-(56) and the non-reciprocal Lambert-W densities (21), (24), (26) are new, have no fitted parameters, and the finite-N simulations back them up. Second, there is a concrete error in the edge asymptotics at γ=μ. From Eq. (126), -z ln z ≃ K ε with K=2γ/b, the asymptotic solution is z ≃ -K ε/ln ε. Differentiating gives ρs(x) ≃ K/|ln ε|, not K/(ln ε)^2 as stated in Eq. (128) and in the abstract's Eq. (12). So the highlighted inverse-log-squared divergence is wrong; the correct divergence is slower. And the abstract's claim that interactions forbid a divergent edge density is false in this marginal case—the interaction actually produces a divergence where the non-interacting system is uniform.\n\nWhat is solid: the parametric construction via the ODE (39)-(43); the shock analysis and edge exponents in phases I-III; the non-reciprocal model's explicit solution with force-consistency checks; the phase diagrams; and the simulations. The self-citation to [68] is appropriate because the method is inherited but the harmonic-trap and non-reciprocal results are not in the earlier paper. No fitted parameters; constants are fixed by normalization and boundary conditions.\n\nThe main flaw is the edge asymptotics above. It is localized to the a=1 limit and to Eq. (12), so it is correctable, but it needs fixing before the abstract and main text can stand. Minor issues: no code or data deposition, and phase III's slow convergence is noted but not quantified. The reduction from Dean-Kawasaki to deterministic rank-field equations is standard and reasonable.\n\nThis paper deserves a serious referee; the error is localized and likely repairable. A referee should check the a=1 asymptotic line carefully. Once corrected, the non-reciprocal Lambert-W results alone are worth citing. Right now I would be cautious about citing the harmonic edge claim.","headline":"Strong new results on two solvable active rank models, but the edge law at γ=μ is misderived: it should be ~ 1/|ln|, not ~ 1/(ln)^2.","tokens_in":38889,"tokens_out":5432,"would_cite":false,"duration_ms":49355,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Run-and-tumble particles with 1D Coulomb forces have exactly computable stationary densities in two new settings, with phases distinguished by edge jumps, shocks, and broken mirror symmetry.","keywords":["run-and-tumble particles","rank interaction","1D Coulomb gas","active matter","non-reciprocal interactions","shocks","jellium model","Lambert W function"],"falsifier":"Simulate the active jellium at $\\mu=2\\gamma$ with $\\kappa$ slightly below $v_0$ and measure the edge density for increasing $N$: the paper predicts a finite jump $(\\mu-\\gamma)/(2\\kappa)$ in phase II and a delta-peak shock in phase III, while a smooth crossover whose width does not shrink as $N\\to\\infty$, or a jump of different height, would refute the claim. For the non-reciprocal model, simulate in phase II, $v_0-b/2<a<v_0+b/2$, and count particles at $x>0$: the paper predicts exactly zero such particles (besides a $+$ cluster at $x=0$), so any finite fraction on the right half-line at large $N$ would falsify the phase-II density.","tokens_in":37995,"feed_emoji":"⚡","tokens_out":6784,"duration_ms":58397,"temperature":0.7,"pith_summary":"This paper studies $N$ run-and-tumble particles in one dimension whose pair force is the linear Coulomb interaction (the rank interaction), and claims that in the $N\\to\\infty$ limit the stationary density can be obtained exactly in two previously unsolved settings. In a harmonic trap, the 'active jellium', the density is given by a parametric representation that produces a phase diagram with smooth edges, finite edge jumps, and shocks (delta-peak clusters), with interactions suppressing the edge divergence of the non-interacting gas. For a non-reciprocal version in a linear trap, where $+$ particles attract $-$ particles while $-$ particles repel $+$ particles, the density is given explicitly in terms of the Lambert $W$ function, with phases that break mirror symmetry and one phase in which the density vanishes on the whole half-line $x>0$. The paper checks both sets of predictions against finite-$N$ simulations.","feed_headline":"Run-and-tumble gas phases include edge jumps and delta-peak shocks","feed_subtitle":"Harmonic-trap 'active jellium' and a non-reciprocal self-gravitating gas both admit exact stationary densities at large N.","key_machinery":"The argument is carried by the rank fields $r(x,t)$ and $s(x,t)$---the cumulative densities of the sum and difference of the $+$ and $-$ densities---whose large-$N$ evolution reduces to the deterministic coupled equations (29)--(30) once the order-$N^{-1/2}$ noise in the Dean-Kawasaki equation (27) is dropped. For the harmonic trap, introducing $U(r)=-v_0 s$ and the function $G(U)=(U')^2$ converts the stationary equations into the parametric representation (50)--(56), with the constant $C$ fixed by boundary conditions; shocks appear when the map $x(r)$ is non-monotonic. For the non-reciprocal model, the same rank-field reduction leads to first-order ODEs solved by the Lambert $W$ function, with phases selected by force-balance conditions at $x=0$.","core_discovery":"The central claim is that two active rank-diffusion models have exactly computable stationary densities at large $N$. For the active jellium, the total density $\\rho_s(x)$ is captured by the parametric representation (50)--(56), which yields four phases as functions of $\\mu/\\gamma$ and $\\kappa/v_0$: a smooth-edge phase, two jump-edge phases distinguished by convexity, a two-edge-shock phase, and a fully clustered phase; the explicit special case $\\mu=2\\gamma$ gives the rank field in closed trigonometric/hyperbolic form. For the non-reciprocal model with a linear potential, the density is written explicitly via the Lambert $W$ function in phase I, Eq. (21), phase II, Eq. (24), and phase III, Eq. (26), with a delta-peak shock at $x=0$ in phases II and III and complete absence of particles for $x>0$ in phase II. The paper also shows that a 'vision cone' non-reciprocal variant maps back to the reciprocal model with a shifted velocity.","pith_inferences":["The same parametric scheme could be applied to other convex confining potentials; the condition for shocks will again be non-monotonicity of $x(r)$, so the phase boundaries should follow from the sign of $G'(u)-4\\kappa$.","The finite-$N$ convergence visible in the simulations suggests a systematic $1/N$ expansion of the shock weights and edge jumps could be obtained from the Dean-Kawasaki noise term that the paper drops.","The explicit Lambert-$W$ densities provide a benchmark against which coarse-grained hydrodynamic theories of non-reciprocal active matter can be tested, for example by measuring the asymmetry ratio $\\rho_s(0^-)/\\rho_s(0^+)$.","Adding a reciprocal rank interaction on top of the non-reciprocal one is a natural next step; the exact solution here gives the zero-reciprocity limit that such an extension must reproduce."],"forward_implications":["In the repulsive active jellium with $\\mu>\\gamma$, the stationary density has a finite jump $(\\mu-\\gamma)/(2\\kappa)$ at the edge, and on the line $\\mu=\\gamma$ it vanishes as $[\\ln(x_e-x)]^{-2}$; both features disappear without interactions.","In the attractive case, shocks (delta peaks of same-sign particles) appear at the two edges, and the shock presence extends the support beyond the shock-free edge.","For $\\bar\\kappa\\ge 2v_0$, all particles collapse into a single delta peak at $x=0$ in the harmonic trap.","In the non-reciprocal model, phase II has a cluster of $+$ particles at $x=0$ and an empty half-line $x>0$, while phase III has non-zero density on both sides plus a shock at $x=0$.","The non-reciprocal 'vision cone' model is exactly the reciprocal model with $v_0\\to v_0-\\kappa$, so its phase diagram follows from the reciprocal one."],"supporting_citations":[{"why":"Introduced the active rank diffusion model and the rank-field method that the present paper extends to the harmonic trap.","marker":"[68]"},{"why":"Supplies the non-interacting run-and-tumble stationary densities in a harmonic potential, which are the $\\kappa=0$ baseline and the origin of the edge exponents in phase I.","marker":"[12]"},{"why":"Provides the passive ranked-diffusion/Burgers framework used to write the deterministic equations (29)--(30).","marker":"[66]"},{"why":"Gives the Dean-Kawasaki derivation and the small-$\\gamma$ fixed-point analysis technique reused here for the active jellium.","marker":"[48]"},{"why":"Underlies the exact Dean-Kawasaki stochastic evolution equation for the density fields used at the start of both derivations.","marker":"[89]"},{"why":"Underlies the Dean-Kawasaki approach in its many-body form, as invoked at the start of Section III.","marker":"[90]"}],"fun_headline_variants":["Active jellium: exact densities with edge jumps and shocks","Non-reciprocal self-gravitating gas solved exactly","Run-and-tumble particles show shock phases in Coulomb gas","Lambert W reveals exact densities for active gas phases","Inhomogeneous active gas: delta-peak shocks and mirror symmetry breaking"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exact densities are derived from deterministic rank-field equations obtained by dropping the order-$N^{-1/2}$ noise, so the whole phase diagram depends on the assumption that this noise does not change the selected stationary branch or shock weights as $N\\to\\infty$.","fun_headline_variants_meta":{"raw":{"variants":["Active jellium: exact densities with edge jumps and shocks","Non-reciprocal self-gravitating gas solved exactly","Run-and-tumble particles show shock phases in Coulomb gas","Lambert W reveals exact densities for active gas phases","Inhomogeneous active gas: delta-peak shocks and mirror symmetry breaking"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00085,"raw_usage":{"total_tokens":3804,"prompt_tokens":1161,"completion_tokens":2643,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":777,"completion_tokens_details":{"reasoning_tokens":2559}},"tokens_in":777,"tokens_out":2643,"duration_ms":40874,"temperature":1.0,"reasoning_tokens":2559,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T21:22:04.927856+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the active jellium at $\\mu=2\\gamma$ with $\\kappa$ slightly below $v_0$ and measure the edge density for increasing $N$: the paper predicts a finite jump $(\\mu-\\gamma)/(2\\kappa)$ in phase II and a delta-peak shock in phase III, while a smooth crossover whose width does not shrink as $N\\to\\infty$, or a jump of different height, would refute the claim. For the non-reciprocal model, simulate in phase II, $v_0-b/2<a<v_0+b/2$, and count particles at $x>0$: the paper predicts exactly zero such particles (besides a $+$ cluster at $x=0$), so any finite fraction on the right half-line at large $N$ would falsify the phase-II density.","supporting_citations":[{"cited_title":"Kumar, B","cited_arxiv_id":null,"evidence_quote":"Introduced the active rank diffusion model and the rank-field method that the present paper extends to the harmonic trap."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the passive ranked-diffusion/Burgers framework used to write the deterministic equations (29)--(30)."},{"cited_title":"Le Doussal, S","cited_arxiv_id":null,"evidence_quote":"Gives the Dean-Kawasaki derivation and the small-$\\gamma$ fixed-point analysis technique reused here for the active jellium."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Underlies the exact Dean-Kawasaki stochastic evolution equation for the density fields used at the start of both derivations."},{"cited_title":"Knezevic, T","cited_arxiv_id":null,"evidence_quote":"Underlies the Dean-Kawasaki approach in its many-body form, as invoked at the start of Section III."}],"review_version":1}