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REVIEW 2 major objections 2 minor 104 references

Run-and-tumble particles with 1D Coulomb interaction: the active jellium model and the non-reciprocal self-gravitating gas

T0 review · 2 major / 2 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2502.09466 v1 pith:OYA6A2RW submitted 2025-02-13 cond-mat.stat-mech cond-mat.softmath-phmath.MP

Léo Touzo , Pierre Le Doussal This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.softmath-phmath.MP
keywords run-and-tumbleparticlesrankinteraction1DCoulombgasactivematternon-reciprocalinteractionsshocksjelliummodelLambertWfunction
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 $N$ 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.

What carries the argument

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 $x=0$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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 $N\to\infty$.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

2 major / 2 minor

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.

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 (2)
  1. [III C, Eqs. (52) and (55)] 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.
  2. [III F 3, Eqs. (125)–(128) and Eq. (12)] 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.
minor comments (2)
  1. [II B, Eq. (26) and surrounding text] 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.
  2. [I and III C] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stationary densities are derived from the rank-field equations with constants fixed by boundary conditions and normalization, then checked against independent finite-N simulations.

full rationale

The derivation chain is self-contained. The active-jellium density follows from the stationary Dean-Kawasaki rank-field equations (31), reduced to the nonlinear ODE (40) for U(r); the constants C and u0 are fixed by the boundary and normalization conditions G(U(0))=0 and 1/2=I(u0) in (43)-(48), and the densities (50)-(56) are then obtained by differentiating the parametric solution. No parameter is fitted to the finite-N simulations of Figs. 2 and 6; those simulations are independent checks of the N-to-infinity formulas. The non-reciprocal densities (21), (24), and (26) are explicit solutions of the first-order ODEs (162)-(168), with the integration constants C+ and C- fixed by the no-shock continuity conditions (172), by the phase-II shock weight (187)-(190), or by h- continuity plus f-(0)=0 in phase III (191)-(202); the phase boundaries follow from positivity of A+ and A- and from the signs of the forces, not from data. Self-citations [48, 66, 68] supply the Dean-Kawasaki/rank-field method and earlier linear-potential results, but the equations are restated or rederived here, and the harmonic-trap and non-reciprocal results are new; the cited method is not an unverified premise that forces the final densities. The edge-logarithm concern raised in the review (Eqs. (125)-(128), where the leading edge decay is 1/|ln(xe-x)| rather than 1/[ln(xe-x)]^2) is an internal algebraic/calculus consistency issue in an asymptotic expansion, not an equivalence between an input and a prediction, so it does not constitute circularity and does not change this score.

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

No fitted parameters: u0 and C are integration constants fixed by normalization and boundary conditions; model parameters κ, v0, γ, μ, a, b are inputs. No new particles, forces, or conserved quantities are postulated beyond the existing interaction matrix.

assumptions (5)
  • domain assumption The Dean-Kawasaki equation (27) for the rank interaction is valid with noise of order N^{-1/2}, which is neglected in the N to infinity limit.
    Invoked in Sec. III A, Eq. (27), citing [66,48,68]. All subsequent stationary equations are deterministic and exact only in this limit.
  • domain assumption The stationary rank fields satisfy r(±∞)=±1/2 and s(±∞)=0, with r odd and no shock at x=0 for the even harmonic potential.
    Used to derive the parametric representation (36)-(37) and to fix constants; see Sec. III B-C.
  • domain assumption Shocks are described by the jump conditions (88)-(89), obtained by integrating the equations across a discontinuity, and the edge shock condition (92).
    Introduced in Sec. III E 2 for the attractive case and used to determine shock weights and edge positions.
  • domain assumption For the non-reciprocal model, the fractions of + and - particles remain fixed at 1/2, giving boundary conditions h_±(±∞)=±1/4; in phase III the additional selection is h_- continuous at x=0 and f_-(0)=0.
    Stated in Sec. IV A-B; the phase III selection is justified by force-sign arguments and checked numerically.
  • standard math Properties of the Lambert W function and hypergeometric evaluations used to invert parametric representations.
    Used in Sec. II B and IV B for the Lambert W function and in Sec. III D for the hypergeometric 2F1.

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Pith. "Pith review of Run-and-tumble particles with 1D Coulomb interaction: the active jellium model and the non-reciprocal self-gravitating gas." pith.science (2026). https://pith.science/paper/OYA6A2RW

@misc{pith2026250209466,
  author       = {Pith},
  title        = {Pith review of: Run-and-tumble particles with 1D Coulomb interaction: the active jellium model and the non-reciprocal self-gravitating gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OYA6A2RW}},
  note         = {Machine review of arXiv:2502.09466}
}
abstract

Recently we studied $N$ run-and-tumble particles in one dimension - which switch with rate $\gamma$ between driving velocities $\pm v_0$ - interacting via the long range 1D Coulomb potential (also called rank interaction), both in the attractive and in the repulsive case, with and without a confining potential. We extend this study in two directions. First we consider the same system, but inside a harmonic confining potential, which we call "active jellium". We obtain a parametric representation of the particle density in the stationary state at large $N$, which we analyze in detail. Contrary to the linear potential, there is always a steady-state where the density has a bounded support. However, we find that the model still exhibits transitions between phases with different behaviors of the density at the edges, ranging from a continuous decay to a jump, or even a shock (i.e. a cluster of particles, which manifests as a delta peak in the density). Notably, the interactions forbid a divergent density at the edges, which may occur in the non-interacting case. In the second part, we consider a non-reciprocal version of the rank interaction: the $+$ particles (of velocity $+v_0$) are attracted towards the $-$ particles (of velocity $-v_0$) with a constant force $b/N$, while the $-$ particles are repelled by the $+$ particles with a force of same amplitude. In order for a stationary state to exist we add a linear confining potential. We derive an explicit expression for the stationary density at large $N$, which exhibits an explicit breaking of the mirror symmetry with respect to $x=0$. This again shows the existence of several phases, which differ by the presence or absence of a shock at $x=0$, with one phase even exhibiting a vanishing density on the whole region $x>0$. Our analytical results are complemented by numerical simulations for finite $N$.

Figures

Figures reproduced from arXiv: 2502.09466 by the authors.

Figure 1
Figure 1. FIG. 1. Left panel: phase diagram of the active jellium model, (i.e. the active rank diffusion in a harmonic potential [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Left: Phase diagram of the non-reciprocal active rank diffusions. The phase diagram is symmetric upon [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Rank field [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Sign of the integration constant [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Plots of the stationary density [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]

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Works this paper leans on

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    Presence of shocks Although in general we do not have an explicit relation between r(x) and x, it is possible to know whether or not shocks are present in the stationary state. Let us recall the general relation (46) µx(r) = 2κr − U ′(r) . (99) Shocks will appear if x(r) is a non-monotonous function of r ∈ [−1/2, 1/2]. Taking the derivative w.r.t. r leads...

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