Very long-term relaxation of harmonic 1D self-gravitating systems

Christophe Pichon; Jean-Baptiste Fouvry; Kerwann Tep

REVIEW 3 major objections 6 minor 68 references

Very long-term relaxation of harmonic 1D self-gravitating systems

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read For a 1D self-gravitating system in a harmonic potential, the relaxation time scales as the square of the particle number, t_rel ∝ N², not linearly as Balescu–Lenard theory would predict.

desk verdict A clean numerical study that plausibly identifies N² relaxation for degenerate harmonic 1D systems, but the asymptotic claim outruns the data; deserves refereeing with requests for larger-N checks. read the letter →

arxiv 2603.11238 v2 pith:ORBR2A2B submitted 2026-03-11 astro-ph.GA cond-mat.stat-mech

Kerwann Tep , Jean-Baptiste Fouvry , Christophe Pichon This is my paper · ORCID

classification astro-ph.GAcond-mat.stat-mech

keywords one-dimensionalself-gravitatingsystemsrelaxationtimedynamicaldegeneracyharmonicpotentialBalescu-Lenardequationthermodynamicequilibrium

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

The paper asks how long a one-dimensional self-gravitating system takes to relax when all its orbits share exactly the same orbital frequency, a regime in which standard kinetic theory is ill-defined. Using an exact collision-driven N-body integrator, it reports that a fully degenerate harmonic system relaxes on a timescale proportional to N² (the square of the particle number), whereas non-degenerate systems relax proportionally to N. It also shows that partially degenerate systems start with the N² behavior at low N and cross over to the linear N scaling once N is large enough, with the crossover happening at larger N when the degenerate fraction is larger. If correct, this means harmonic density cores relax much more slowly than conventional estimates, a result directly relevant to the survival of shallow cores in dwarf galaxies.

What carries the argument

The load-bearing object is the dynamically degenerate harmonic potential, a constant-density slab whose quadratic mean-field potential gives every orbit the same frequency Ω, making the Balescu–Lenard resonance condition degenerate. The measurements rely on an exact collision-driven 1D integrator (forces are constant between crossings, collisions are processed with a heap-based event scheduler) and on a relaxation metric: the ensemble-averaged Kolmogorov–Smirnov distance between the evolving cumulative mass profile and the Rybicki finite-N thermodynamic equilibrium. The relaxation time is defined implicitly as the time when this distance crosses an ad hoc threshold D0 = 0.015.

What would settle it

Run the same relaxation measurement for the harmonic system at substantially larger N (e.g., N = 300 to 1000) using a faster, approximate integrator and check whether log(t_rel) versus log(N) remains a clean power law with slope 2. If the local slope flattens toward 1 as N grows, the quadratic law is a finite-N artifact. A second check is to repeat the measurement at a much lower threshold (e.g., D0 = 0.005) with much longer integration times and see whether the same N² slope emerges from the late-time relaxation.

Watch

Extended reading notes

Core claim

For a 1D self-gravitating system with a harmonic mean-field potential, every particle has the same orbital frequency Ω, so the resonance condition δ(kΩ − k'Ω) in the Balescu–Lenard equation no longer selects specific resonant pairs and the equation becomes ill-posed. The paper's central numerical discovery is that such fully degenerate systems still relax toward the finite-N thermodynamic equilibrium, but on a timescale t_rel ∝ N² t_dyn, instead of the usual t_rel ∝ N t_dyn found for non-degenerate systems such as Plummer or compact-support clusters. The quadratic scaling is read off from the crossing time of a Kolmogorov–Smirnov distance threshold between the instantaneous and equilibrium c

Load-bearing premise

The quadratic scaling is inferred from the crossing time of an adopted threshold D0 = 0.015 in a Kolmogorov–Smirnov distance, measured only for N between 21 and 141 with small-N points excluded as 'polluted by small-N effects'; if these crossing times blend a transient phase with the asymptotic relaxation, the fitted exponent may not be the true large-N law.

Editorial extensions

If this is right

Fully degenerate harmonic 1D clusters relax on N² timescales, an order of magnitude slower in N than the linear law predicted by Balescu–Lenard theory.
Partially degenerate clusters exhibit two scaling regimes, with the N²-to-N crossover shifting to higher N as the fraction of degenerate orbits increases.
Compact but non-degenerate systems relax linearly with N, with a prefactor about ten times larger than that of the infinite-support Plummer system.
The long N² relaxation of harmonic cores implies that density cores in galaxy centers can persist much longer than standard relaxation estimates, affecting how long shallow cores survive and how efficiently substructures sink via dynamical friction.

Reading between the lines

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

The same quadratic law may apply, with a different prefactor, to 3D harmonic spheres, which would extend the paper's 1D conclusion to the globular-cluster core-stalling problem; this is a conjecture beyond the paper's own scope.
The N² law is measured only up to N = 141; a faster approximate integrator could test whether the slope 2 persists at N ~ 1000, or whether the finite-N window is showing a transient dressed by the threshold crossing.
A time-averaged Hamiltonian or renormalized kinetic description (e.g., a point-vortex analogy) might predict the prefactor of the N² law analytically, turning the empirical scaling into a computed one.

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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

Short version: this is a serious, clean numerical study of a regime where standard Balescu–Lenard theory is ill-defined, and it reports a genuinely new scaling law. The claim is plausible but the asymptotic statement outruns the data. The paper deserves refereeing.

What is actually new: for fully degenerate harmonic 1D systems, all orbits share the same frequency, so the BL resonance condition degenerates and the standard 1/N relaxation theory does not apply. The paper measures relaxation time as a function of N with an exact collision-driven integrator (round-off controlled, invariants conserved to ~1e-23) and finds t_rel ∝ N² t_dyn, versus linear for Plummer and Compact. It also shows that partially degenerate Anharmonic systems show N² at low N and N at high N, with the transition moving to larger N as degeneracy increases. That crossover has not been reported before.

What I like: the numerical setup is honest. Ensemble sizes N×N_r ~ 1e5, the threshold choice is tested with D0=0.022 and the same trends appear, and the relaxation is measured against the externally defined finite-N thermodynamic equilibrium from Rybicki, so no constants are fitted to force the scaling. The code is on GitHub. The paper is also explicit that quasilinear expansions fail for flat frequency profiles, and it does not oversell the theoretical interpretation.

Soft spots: the N² result is read off a log-log plot over N=21–141. Small-N points are dropped as “polluted by small-N effects” without a stated criterion, and there is no fit with slope uncertainties. More importantly, the Anharmonic family shows that for every ε>0 the N² branch is a low-N transient that bends to N at larger N. The transition N_trans(ε) grows as ε shrinks. The claim that at ε=0 the N² branch is the true asymptotic law is an extrapolation: it is not measured at large N, and nothing in the paper rules out N_trans remaining finite at ε=0. The cited “thermodynamic blocking” is qualitative and invoked interpretively, not used to predict the exponent. So the headline asymptotic statement is underdetermined; what is solid is the numerical trend over the simulated range and the crossover pattern.

That said, this is not a fatal flaw. The paper frames itself as a first numerical exploration and explicitly lists larger-N inexact integration as future work. For people working on long-range interacting systems or kinetic theory, this is important enough to take seriously and to probe further. I would send it to peer review, and in revision I would ask for a fitting procedure, a stated criterion for dropping small-N points, and ideally larger-N runs for at least one small ε to see whether the crossover persists at ε=0. I would not yet cite the N² law as established beyond the simulated range.

Referee Report

3 major / 6 minor

Summary. The paper studies the very long-term collisional relaxation of one-dimensional self-gravitating systems using an exact, collision-driven N-body integrator. It measures the Kolmogorov–Smirnov distance between the ensemble-averaged cumulative mass profile and the finite-N thermodynamic equilibrium of Rybicki (1971), and defines a relaxation time through an ad hoc threshold D0. The main numerical results are: Plummer and Compact (non-degenerate) systems relax on a timescale t_rel ∝ N t_dyn; Harmonic (fully dynamically degenerate) systems relax as t_rel ∝ N^2 t_dyn; and Anharmonic (partially degenerate) systems show a quadratic branch at low N that transitions to linear scaling at larger N, with the transition pushed to larger N as the degeneracy fraction increases. The paper interprets the slow harmonic relaxation as a manifestation of thermodynamic blocking and discusses astrophysical implications for density cores.

Significance. If the central claim holds, the paper is of genuine interest: it identifies a regime in which the standard inhomogeneous Balescu–Lenard scaling t_rel ∝ N t_dyn fails, and it provides a clean numerical benchmark for future kinetic theories of dynamically degenerate self-gravitating systems. The numerical methodology is a major strength: the collision-driven integrator is exact up to round-off, global invariants are conserved to ~1e-23, and the authors use ensemble averaging and a two-threshold consistency check. The public code and detailed appendices also make the measurements reproducible. However, the asymptotic N^2 scaling for the fully degenerate case is inferred from a narrow N window and from the extrapolation of a trend seen in the Anharmonic series; the current evidence is suggestive but not conclusive.

major comments (3)

[§3.3, Figs. 2–3] The central claim that Harmonic relaxes as t_rel ∝ N^2 is inferred from a narrow window, 21≤N≤141, after discarding small-N points as ‘polluted by small-N effects’ without a stated criterion. The Anharmonic series in Fig. 3 shows that the same N^2 branch is, for every finite ε>0, a pre-asymptotic regime that bends to N at larger N, with the bend moving to larger N as ε decreases. Since no ε=0 run extends beyond N=141, the data do not distinguish between an N_trans(ε) that diverges as ε→0 and a finite N_trans(0)>141; the headline distinction between Harmonic and non-degenerate systems is therefore an extrapolation. A quantitative fit (log t_rel vs log N with slope uncertainty), an explicit estimate of N_trans(ε), or an analytic argument that N_trans diverges is needed to support the asymptotic claim.
[§3.2, Eq. (8)] The relaxation time is defined by an ad hoc threshold D0. The check with D0=0.022 (Appendix F) is reassuring, but both thresholds are crossed in the same finite-N window, so it does not test whether the measured exponent is the asymptotic one. The paper also does not report a regression or slope uncertainties; the lines in Figs. 2–3 are visual guides. Since the inferred exponent is the main quantitative result, the crossing-time definition, the treatment of multiple crossings, and the systematic uncertainty from D0 and the N range should be quantified.
[§3.3, Fig. 2 caption] The statement that small-N measurements are ‘polluted by small-N effects’ is not demonstrated. If the finite-N Rybicki equilibrium (Appendix D) is used exactly, one must specify what the pollution is—for example, the initial KS distance, finite-N corrections to the equilibrium, or remnants of violent relaxation. This matters because excluding those points is part of what determines the fitted exponent.

minor comments (6)

[§2.1] Typo: ‘particule’ should be ‘particle’.
[§3] Typo: ‘mesure’ should be ‘measure’.
[Figs. 2–3] The x-axis begins at 20, while the text states 21≤N≤141. Please make the plotting range and the stated range consistent.
[§3.2] The error-bar definition is unclear: after ensemble averaging there is one DKS(t) curve per N, yet the text says t_rel is the mean of all crossing times with min/max error bars. Clarify whether the scatter is over realizations, over sampling intervals, or over threshold crossings of the averaged curve.
[§3.3 and §4.2] The appeal to Deme & Fouvry’s thermodynamic blocking is qualitative and is not used to predict the N^2 scaling. Please mark this as an interpretative hypothesis or provide a quantitative connection.
[Appendix C.5] Define ε precisely in the main text and figure captions: ε is described as the fraction of non-degenerate orbits, while the constant-density core contains a fraction 1−ε. A reader could confuse the two.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the N^2 scaling is an empirical measurement against an externally defined equilibrium, with no fitted constants or load-bearing self-citations.

full rationale

The paper's central claims are numerical measurements rather than derivations from fitted inputs. The relaxation time is defined by Eq. (8) as the crossing time of the measured KS distance (Eq. 7) below a fixed threshold D0 = 0.015, with the thermodynamical equilibrium M_th taken from Rybicki (1971) (Appendix D), an external result that does not depend on the present authors. No parameter in Figures 2 and 3 is fitted to force either the N or N^2 scaling; the scaling is read off the crossing times over 21 ≤ N ≤ 141, and Appendix F shows the same trends at the alternative threshold D0 = 0.022. The self-citation to Deme & Fouvry (2025) (`thermodynamic blocking') is used only as an interpretive expectation after the measurement, and the paper explicitly defers an a priori prediction of the N-dependence to future work, so it is not load-bearing for the measured scaling. The excluded small-N points and the limited N range are acknowledged statistical and numerical limitations, not definitional reductions. No equation in the paper is equivalent to its own input by construction, and the harmonic N^2 result is not derived by renaming or importing an ansatz from prior work.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

Central scaling is an empirical measurement. The main hand-chosen input is the KS threshold and the ensemble size; the equilibrium benchmark and integrator are adopted from prior work.

free parameters (2)

D0 (relaxation threshold) = 0.015
Ad hoc threshold in Eq. (8) defining t_rel; authors check D0=0.022 in Appendix F, but the choice affects the measured crossing time and is not derived.
N_r×N (ensemble size) = ~1e5
Number of realizations times particles set to ~100,000 to keep the noise floor below threshold; arbitrary but checked.

assumptions (5)

domain assumption Finite-N Rybicki equilibrium is the correct attractor of 1D self-gravitating systems with fixed E_tot and zero momentum.
Appendix D: used to define M_th in Eq. (5)-(6); if wrong, D_KS does not measure relaxation to the true equilibrium.
domain assumption Exact collision-driven integrator with 80-bit double floats reliably tracks dynamics up to 2×10^7 t_dyn.
Appendix A: energy/momentum errors ~1e-23, but round-off still accumulates linearly.
domain assumption Balescu–Lenard equation and its 1/N scaling apply to non-degenerate 1D systems.
Section 3.1: used as baseline to interpret Plummer/Compact linear scaling; not central to harmonic claim.
standard math Eddington inversion yields valid self-consistent equilibria for the chosen densities.
Appendix C: used to generate initial DFs; relies on standard Abel inversion.
domain assumption KS distance threshold crossing gives an operational relaxation time.
Section 3.2: t_rel defined implicitly by D_KS(t_rel)=D0; threshold choice is ad hoc.

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Cite this review

Pith. "Pith review of Very long-term relaxation of harmonic 1D self-gravitating systems." pith.science (2026). https://pith.science/paper/ORBR2A2B

@misc{pith2026260311238,
  author       = {Pith},
  title        = {Pith review of: Very long-term relaxation of harmonic 1D self-gravitating systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ORBR2A2B}},
  note         = {Machine review of arXiv:2603.11238}
}

abstract

One-dimensional self-gravitating systems admit genuine thermodynamical equilibria. For systems with strictly monotonic orbital frequency profile, the Landau and Balescu-Lenard theories predict a relaxation time scaling linearly with the number of particles, $N$, in agreement with simulations. Yet, these theories become ill-posed for degenerate frequency profiles, as is the case in the harmonic potential, where all particles share the exact same mean orbital frequency. Using an exact collision-driven 1D integrator, we investigate numerically the self-consistent relaxation of 1D harmonic self-gravitating systems. We show that harmonic systems relax on a timescale that grows quadratically with $N$. We show that systems that are only partially degenerate display the same quadratic scaling for low $N$, but transition to the linear, non-degenerate behaviour for larger $N$. The larger the fraction of degenerate orbits, the larger the value of $N$ at which this transition of dynamical regime occurs. Finally, we explore the dynamics of fully non-degenerate systems, albeit with finite radial support: we confirm that their relaxation time scales linearly with $N$, though with a substantially larger prefactor than in non-compact systems. Astrophysically, this investigation should offer some new clues on the dynamics of density cores, as in the center of dwarf galaxies.

Figures

Figures reproduced from arXiv: 2603.11238 by the authors.

**Figure 1.** Time evolution of the KS distance (equation 7) for Plummer, Compact, and Harmonic, with N =50 and averaged over Nr =2 000 realizations. The bottom dashed line is the characteristic noise level, 1/ √ Nr×N. The threshold associated with D0 =0.015 is given by the dot-dashed line. It is used to estimate the relaxation time, trel (equation 8). The relaxation times vary greatly between the different potentials, with Ha… view at source ↗

**Figure 2.** Dependence of the relaxation time, trel, with the number of particles, N, for Plummer, Compact and Harmonic ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Same as [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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