{"id":"3a47d670-2708-45bb-ace9-7cca65481b33","arxiv_id":"2412.14875","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For stochastic Riesz gases expanding into vacuum, exact self-similar density profiles are derived in 1D for all interaction exponents in (-2,1), and Coulomb gases show a universal uniform density decaying as 1/t in any dimension.","lead":"This paper works out exactly how a cloud of repelling, jittering particles spreads out when released from a tiny spot. It finds simple universal profiles: in one dimension the density keeps a fixed power-law shape, and for electric-like Coulomb repulsion the density becomes uniform and falls as 1/time in any number of dimensions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 1D self-similar solution is derived from the no-diffusion equation, but the paper's own Eq. (14) shows that for 0<s<1 diffusion overtakes the deterministic expansion at a finite crossover time t*, so Eqs.","rationale":"The reader's weakest assumption identifies exactly the load-bearing issue: the no-diffusion reduction underlying the central 1D solution is valid only before the crossover t* for 0<s<1, while the paper's framing suggests the result applies throughout. This is not a flaw in the algebraic derivation of the self-similar solution of Eq. (33); it is a limitation on the physical relevance of that solution for the stochastic particle system. Because the paper itself contains the crossover analysis in Eq. (14), the contradiction is internal and concrete. The conditional verdict is appropriate: the mathematical results for the no-diffusion hydrodynamic model are sound and the Coulomb-gas derivation is well supported, but the claimed scope of the 1D result needs qualification. My stress-test does not identify an additional independent flaw that would require a different verdict; it reinforces the reader's conditional assessment.","tokens_in":14096,"tokens_out":23937,"duration_ms":178591,"concrete_test":"For s=0.5, g=D=1, compute the crossover t* from Eq. (14) for N=10^3 (t* ~ (10^3)^4 = 10^12 in dimensionless units) and N=10^2 (t* ~ 10^8). Simulate the N-particle Langevin dynamics (4) starting from a narrow Gaussian for times t=t*/10 and t=10 t*. Measure the density profile and compare with Eq. (6): at t=t*/10 the profile should be close to the compact self-similar form, while at t=10 t* the density should be approximately Gaussian with width ~√(2Dt), demonstrating that Eq. (6) ceases to describe the stochastic gas. Alternatively, evaluate analytically the ratio D∂xxρ/∂tρ at x=0 for the profile (6); this ratio grows as (√Dt/L)^2 and reaches O(1) at t*, confirming the breakdown.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central 1D result (Eqs. 6-7) follows from Eq. (33), which omits the diffusion current -D∂xρ present in the Dean-Kawasaki equation (13). For 0<s<1, the deterministic span grows sub-diffusively, L(t)~(Ngt)^{1/(s+2)}, so the diffusion length √(Dt) eventually catches L(t) at the crossover time t* estimated in Eq. (14). At and after t*, the diffusive term is not a small correction: substituting the profile (6) into the full equation, the ratio |D∂xxρ|/|∂tρ| at x=0 grows like (√Dt/L)^2 and becomes O(1) when t~t*. Therefore, for s∈(0,1), the self-similar profile is only a transient, and the late-time expansion is governed by ordinary diffusion rather than by Eq. (6). This directly contradicts the Introduction's claim that the influence of noise/diffusion is 'limited or negligible throughout the evolution'. The mathematical derivation of the no-diffusion solution is internally consistent, but its advertised domain of applicability is too broad, and the abstract presents the s∈(-2,1) result without the crossover caveat.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the expansion into vacuum of N overdamped Brownian particles in R^d interacting through a repulsive Riesz potential. Working at the continuum level with the Dean–Kawasaki current (13) and then dropping the diffusive and noise terms, the authors derive self-similar solutions of the nonlocal continuity equation. For d=1 and -2<s<1 the density is given by Eq. (6) with support radius Eq. (7). For Coulomb gases in d dimensions, s=d-2, they obtain a uniform density inside an expanding ball, Eqs. (8)-(9) and (64), using the Gauss theorem. The paper also discusses the Dyson limit, the Ginibre gas, the N=∞ Coulomb limit, a harmonic-trap extension, and a conjectured d>1 non-Coulombic profile.","tokens_in":14361,"tokens_out":14798,"duration_ms":133250,"significance":"The 1D result is an elegant, parameter-free exact solution of the no-diffusion hydrodynamic equation: the support radius is fixed by normalization, and the profile solves the stated singular integral equation. The Coulomb solution is attractive and physically transparent, and the universal, N-independent t^{-1} density in arbitrary dimension is a crisp falsifiable prediction. The paper is also honest about the crossover time t* in Eq. (14), which is a strength; however, the abstract and introduction overstate the absence of diffusion effects.","major_comments":[{"comment":"The headline claim that Eqs. (6)-(7) describe the expansion for all s in (-2,1) and that the influence of noise is 'limited or negligible throughout the evolution' is too strong. For 0<s<1 the body text itself states at Eq. (14) that the diffusion length catches the deterministic span at t* ~ D^{-1}(N g/D)^{2/s}; after this time the diffusive term in Eq. (13) is no longer a small correction and the self-similar profile is not the physical solution of the full stochastic model. Please restrict the advertised domain to the no-diffusion regime t << t* (or to the N to infinity limit at fixed t) and carry this caveat into the abstract.","section":"Abstract and Section I, Eq. (14)"},{"comment":"The Introduction labels the d>1 non-Coulombic density profile as the functional form of the exact solution, but Section IV only states that it is 'expected' and defers the derivation to Refs. [42,43] without showing how those results apply to the present Riesz-gas initial-value problem. Either provide a derivation or state explicitly that Eq. (70) is a conjecture/open problem; as written, the section wavers between exact claim and speculation.","section":"Section IV, Eq. (70)"}],"minor_comments":[{"comment":"The prefactor B_s is of the form 0/0 at s=-1 because both cos(pi s/2) and (s+1)(s+2) vanish; since the text and Fig. 1 use the s=-1 flat profile, please state explicitly that this case is obtained by taking the limit s -> -1.","section":"Section II, Eq. (6)"},{"comment":"The Gamma-function expression in the displayed formula for L(t) is typeset ambiguously; please check that the denominator is the intended Gamma function and that the factor (s+1) is correctly represented, since the normalization uses the integral of (1-u^2)^{(s+1)/2}.","section":"Eqs. (7) and (39)"},{"comment":"There is a typo: 'Sustituting' should be 'Substituting'.","section":"Section II A"},{"comment":"There is a typo: 'ampltude' should be 'amplitude'.","section":"Section IV"},{"comment":"For the N=infinity Coulomb limit, the initial distribution N delta(x) has infinite total mass; please state explicitly that the result is the pointwise N to infinity limit of the finite-N density rather than a solution with a literal delta initial condition.","section":"Section III B, N=infinity discussion"},{"comment":"The notation in Eq. (31) should be written as x_+(t) = L(t)(1 + N^{-2/3} xi_+) rather than 'x_+(t) L(t) = 1 + ...' so that the ratio is not confused with the position itself.","section":"Section II A, Eq. (31)"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core appears sound and the paper is likely acceptable after the abstract is aligned with the t* caveat and the status of Eq. (70) is clarified. I recommend major revision rather than rejection because the derivation itself is consistent within the stated no-diffusion model."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main takeaway: this is a solid analytical paper with two genuinely clean results: the full family of self-similar expansion profiles for 1D Riesz gases with s in (-2,1) in the no-diffusion limit, and the uniform, t^{-1} density profile for Coulomb gases in any dimension. Both follow from straightforward hydrodynamic arguments and standard integral-equation solutions. The Coulomb case via Gauss's law is elementary and correct; the 1D profile reduces to the Wigner semicircle at s=0. The paper also shows correctly that only the Coulomb gas survives the N→∞ limit. I checked the scaling and normalization; they're consistent.\n\nThe soft spots are real but not fatal. The first is in the abstract and introduction: the claim that noise/diffusion is 'limited or negligible throughout the evolution' is too strong for s in (0,1). The paper's own Eq. (14) gives a crossover time t* after which diffusion dominates, and the self-similar profile (6) is only valid for t << t*. That caveat does appear in Sec. IIB, but it's absent from the abstract, so the advertised domain of validity is broader than the analysis supports. This should be fixed with a qualified abstract. Second, the d>1 non-Coulombic profile (70) is presented as the expected solution, but no derivation is given; it's referred to the mathematical literature on nonlocal porous media equations. That's fine as a conjecture, but it should be labeled as one. The central results don't depend on it.\n\nThe citation pattern is honest: relevant mathematical and physical literature is cited, and self-citations are for background and prior results. No circularity concerns.\n\nWho this is for: people working on long-range interacting particle systems, nonlocal hydrodynamics, and random matrix connections. They'll get explicit exact profiles that are testable in simulations. It deserves a serious referee and, after a revised abstract and labeling of the conjecture, publication.","headline":"Clean exact self-similar solutions for 1D Riesz and Coulomb expansion, but the diffusion caveat for 0<s<1 should move from a section into the abstract.","tokens_in":14861,"tokens_out":2510,"would_cite":true,"duration_ms":21980,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives exact self-similar density profiles for the expansion of long-range Riesz gases into vacuum, and finds that Coulomb gases expand as a uniformly dense ball.","keywords":["Riesz gas","long-range interactions","self-similar expansion","Coulomb gas","Dean-Kawasaki equation","Dyson gas","Wigner semicircle","nonlocal porous medium equation"],"falsifier":"Run the particle dynamics (4) for a finite but large number of particles starting near a point, with no external potential, and measure the central density $\\rho(0,t)$ as a function of time. The paper predicts $\\rho(0,t)\\propto t^{-1/(s+2)}$ for one-dimensional Riesz gases and a flat interior density $\\rho=(\\Omega_d g t)^{-1}$ for Coulomb gases; any measured exponent or radial profile that disagrees with these forms would falsify the self-similar solution.","tokens_in":13904,"feed_emoji":"⚛️","tokens_out":7236,"duration_ms":53981,"temperature":0.7,"pith_summary":"Krapivsky and Mallick study $N$ overdamped Brownian particles in $d$ dimensions that repel pairwise through the Riesz potential $V(r)=g/(s r^s)$ and start concentrated at a single point. Their central claim is that the subsequent expansion is self-similar: in one dimension, for every interaction exponent $s\\in(-2,1)$, the density is $\\rho(x,t)=\\frac{B_s}{g t}(L^2-x^2)^{(s+1)/2}$ with $L(t)\\sim (N g t)^{1/(s+2)}$, where $B_s$ is a known constant; for Coulomb interactions in any dimension, the density is uniform inside an expanding ball and decays as $t^{-1}$. These formulas are exact solutions of the nonlocal hydrodynamic equation obtained when Brownian diffusion is dropped from the Dean-Kawasaki current. If correct, they give a complete solvable picture of how a long-range repulsive gas fills vacuum, with the Dyson gas ($s=0$) reducing to an expanding Wigner semicircle and the infinite-particle limit existing only for Coulomb gases.","feed_headline":"One density profile law governs repulsive gas expansion","feed_subtitle":"A single formula predicts how long-range repelling particle clouds spread into empty space; Coulomb gases stay uniformly dense.","key_machinery":"Two devices carry the argument. In one dimension it is the singular integral equation obtained from the no-diffusion continuity equation; this equation belongs to the class solved by compactly supported power-law profiles, and it produces the family $F=B_s(R^2-X^2)^{(s+1)/2}$ for every $s\\in(-2,1)$. In higher dimensions it is the Newton-Gauss theorem, which converts the Coulomb force at radius $r$ into a function of only the total charge inside the sphere of radius $r$, making the radial current local and the density uniform.","core_discovery":"The paper solves the initial-value problem $\\rho(x,0)=N\\delta(x)$ for the continuity equation with current $J=-D\\partial_x\\rho+g\\rho\\!\\int dy\\, \\frac{x-y}{|x-y|^{s+2}}\\rho(y,t)$. After setting $D=0$ and inserting the scaling ansatz $\\rho(x,t)=\\frac{N}{(N g t)^{1/(s+2)}}F(X)$, $X=x/(N g t)^{1/(s+2)}$, the scaled density obeys the singular integral equation $\\frac{X}{s+2}=-\\int_{-R}^{R}dY\\, \\frac{X-Y}{|X-Y|^{s+2}}F(Y)$ inside the support. Its compactly supported solution is $F(X)=B_s(R^2-X^2)^{(s+1)/2}$, and normalization fixes $R$; in original variables this gives the density profile of Eqs. (6)-(7). For Coulomb gases in $d$ dimensions, the nonlocal force is reduced by the Newton-Gauss theorem to a local expression, and the same scaling argument gives a flat density $\\rho=(\\Omega_d g t)^{-1}$ inside a ball of radius $(d N g t)^{1/d}$, with the $d=2$ case being the Ginibre gas and $d=3$ the classical Coulomb gas.","pith_inferences":["A direct finite-$N$ simulation of the particle equations (4) with weak noise could test the prefactor $B_s$ and the exponent $1/(s+2)$, especially near $s=-2$ and $s=1$ where the integral-equation solution has singular behavior; the paper does not report such numerics.","The $N=\\infty$ Coulomb result suggests an experimental realization: a two-dimensional system of repulsively interacting colloids released from a tight spot should show an expanding disk with a flat interior and a boundary sharp on the scale of the mean interparticle spacing, measurable with video microscopy.","Because the no-diffusion equation is the overdamped counterpart of Coulomb explosion, the same uniform-density solutions may describe the late-time envelope of a laser-ionized cluster when inertia is negligible, a connection the paper only hints at."],"forward_implications":["For $s\\in(-2,1)$ in one dimension, the cloud always has compact support, and the radius grows as $(N g t)^{1/(s+2)}$: super-diffusively for $s<0$, diffusively for the Dyson gas at $s=0$, and sub-diffusively (until the crossover time $t_*$) for $0<s<1$.","The $s=0$ Dyson gas expands as an evolving Wigner semicircle, so the same profile that describes eigenvalues of Gaussian random matrices appears in an out-of-equilibrium spreading problem.","For Coulomb gases in any dimension, the density inside the expanding ball is exactly flat and decays as $t^{-1}$ with a dimension-dependent amplitude, and the profile remains well defined when $N=\\infty$.","Non-Coulomb Riesz gases do not have a finite infinite-particle limit: the density at a fixed rescaled position diverges for $s>-1$ and vanishes for $s<-1$ when $N\\to\\infty$.","With an added harmonic trap, the same family of profiles describes the expansion, with the radius saturating exponentially to a stationary value; for Coulomb gases the density stays uniform and converges to a constant."],"supporting_citations":[{"why":"Supplies the stochastic coarse-grained current whose deterministic part becomes the nonlocal interaction current in Eq. (13).","marker":"[24]"},{"why":"Dean's Langevin equation for the density provides the derivation of the nonlocal hydrodynamic current from the particle equations.","marker":"[25]"},{"why":"Exact solutions for viscous Marangoni spreading are used to extract the Dyson-gas solution from Eq. (23).","marker":"[5]"},{"why":"The self-similarity analysis of unbounded Marangoni flows supports the exact solution used in the Dyson case.","marker":"[6]"},{"why":"Provides the singular integral equation methods used to solve Eqs. (25), (29) and (37).","marker":"[31]"},{"why":"Supplies the inversion formula for the compactly supported singular integral equation that yields the power-law profile (38).","marker":"[32]"},{"why":"Connection to explicit equilibrium solutions for power-law aggregation reinforces the same family of compactly supported profiles.","marker":"[33]"},{"why":"Gives existence of Barenblatt profiles for the nonlocal porous medium equation, supporting the expected general-dimensional solution (69)-(70).","marker":"[42]"},{"why":"Provides weak solutions and profile analysis for the nonlocal porous medium equation, the foundation for the higher-dimensional Riesz case.","marker":"[43]"}],"fun_headline_variants":["Universal expansion law for long-range repulsive gases","Coulomb gas expands uniformly, density scales as 1/t","Self-similar profiles for repulsive gas spreading solved","Single formula predicts cloud expansion into vacuum","Repulsive gases obey one universal density profile"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole calculation treats the cloud as a continuous density evolving by the deterministic nonlocal equation (33), i.e. it assumes the random Brownian jiggling of the particles is negligible compared with their repulsion; for $0<s<1$ this is true only before the crossover time $t_*$, and after that the displayed profiles stop being the physical ones.","fun_headline_variants_meta":{"raw":{"variants":["Universal expansion law for long-range repulsive gases","Coulomb gas expands uniformly, density scales as 1/t","Self-similar profiles for repulsive gas spreading solved","Single formula predicts cloud expansion into vacuum","Repulsive gases obey one universal density profile"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000299,"raw_usage":{"total_tokens":1710,"prompt_tokens":911,"completion_tokens":799,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":727}},"tokens_in":527,"tokens_out":799,"duration_ms":7026,"temperature":1.0,"reasoning_tokens":727,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:50:32.541932+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the particle dynamics (4) for a finite but large number of particles starting near a point, with no external potential, and measure the central density $\\rho(0,t)$ as a function of time. The paper predicts $\\rho(0,t)\\propto t^{-1/(s+2)}$ for one-dimensional Riesz gases and a flat interior density $\\rho=(\\Omega_d g t)^{-1}$ for Coulomb gases; any measured exponent or radial profile that disagrees with these forms would falsify the self-similar solution.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the stochastic coarse-grained current whose deterministic part becomes the nonlocal interaction current in Eq. (13)."},{"cited_title":"Livan, M","cited_arxiv_id":null,"evidence_quote":"Dean's Langevin equation for the density provides the derivation of the nonlocal hydrodynamic current from the particle equations."},{"cited_title":"Campa, T","cited_arxiv_id":null,"evidence_quote":"Exact solutions for viscous Marangoni spreading are used to extract the Dyson-gas solution from Eq. (23)."},{"cited_title":"Dynamical Spreading and Memory Retention Under Power Law Potential","cited_arxiv_id":"2502.06256","evidence_quote":"The self-similarity analysis of unbounded Marangoni flows supports the exact solution used in the Dyson case."},{"cited_title":"Agarwal, A","cited_arxiv_id":null,"evidence_quote":"Provides the singular integral equation methods used to solve Eqs. (25), (29) and (37)."},{"cited_title":"Dandekar, P","cited_arxiv_id":null,"evidence_quote":"Supplies the inversion formula for the compactly supported singular integral equation that yields the power-law profile (38)."},{"cited_title":"Finite-time blowup of a Brownian particle in a repulsive potential","cited_arxiv_id":"2502.11796","evidence_quote":"Connection to explicit equilibrium solutions for power-law aggregation reinforces the same family of compactly supported profiles."},{"cited_title":"Zandi and R","cited_arxiv_id":null,"evidence_quote":"Gives existence of Barenblatt profiles for the nonlocal porous medium equation, supporting the expected general-dimensional solution (69)-(70)."},{"cited_title":"Liddle, An Introduction to Modern Cosmology (Wiley, Chichester, UK, 2015)","cited_arxiv_id":null,"evidence_quote":"Provides weak solutions and profile analysis for the nonlocal porous medium equation, the foundation for the higher-dimensional Riesz case."}],"review_version":1}