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arxiv: 2605.18353 · v1 · pith:NQWMDLRVnew · submitted 2026-05-18 · 🌌 astro-ph.EP · astro-ph.IM

A fast tree algorithm for multi-component coagulation equation

Taichi K. Watanabe , Akimasa Kataoka This is my paper

Pith reviewed 2026-05-19 23:41 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IM
keywords coagulationdust evolutiontree algorithmmulti-componentprotoplanetary disksplanet formationnumerical methodsporosity
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The pith

A tree algorithm groups similar dust aggregates to cut multi-component coagulation costs from quadratic to near-linear scaling while matching direct-method results.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a fast method to track how dust particles evolve when they collide and stick or fragment, now including multiple properties such as mass and porosity at once. It rests on the idea that aggregates with similar properties produce similar collision outcomes, so their interactions can be computed together instead of pairwise. By measuring similarity as distance in logarithmic property space, the method builds a tree that lumps distant bins and evaluates their combined effect. This lowers the scaling from O(N to the power of 2d) to O(d N to the power of d times log N) for d properties and N bins per property. Tests on cases with known analytic solutions confirm that the results stay consistent with the slower direct calculation, with speed gains clearest when tracking two properties.

Core claim

The central claim is that the similarity assumption in log-property space permits a tree algorithm to approximate the full coagulation equation by grouping and collectively evaluating interactions among distant bins. The resulting method reproduces the outcomes of the conventional direct summation method on test problems with analytic solutions, while lowering computational complexity from O(N^{2d}) to O(d N^d log N). Measured wall-clock time, L2 distribution error, and mass conservation error all respond predictably to changes in the number of bins, the opening angle, and the maximum distribution width after coagulation.

What carries the argument

The tree algorithm that groups distant bins according to logarithmic distance in multi-dimensional dust-property space and evaluates their collective coagulation interactions.

If this is right

  • For one property the tree method becomes faster than direct summation only inside a limited range of parameters.
  • For two properties the tree method is faster across every surveyed parameter combination and accelerates the calculation by tens of times.
  • Raising the number of bins or tightening the opening angle or width parameter increases accuracy at the cost of speed.
  • A small maximum distribution width after coagulation actually increases error relative to a larger width, indicating that the limit need not be imposed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be inserted into longer planet-formation simulations that evolve dust porosity alongside mass.
  • The same grouping logic might reduce cost in other multi-dimensional aggregation problems that currently rely on direct pairwise summation.
  • Adding further dust properties such as charge or composition becomes feasible without an immediate explosion in runtime.

Load-bearing premise

Two pairs of colliding aggregates produce a similar outcome whenever their dust properties are close in logarithmic ratio space.

What would settle it

Run the tree algorithm and the direct method on the same two-component coagulation problem that has a known analytic solution; if the L2 error in the final distribution exceeds the values reported for the chosen opening angle and width parameter, the approximation fails.

Figures

Figures reproduced from arXiv: 2605.18353 by Akimasa Kataoka, Taichi K. Watanabe.

Figure 1
Figure 1. Figure 1: Diagram of tree algorithm for one-component Smoluchowski coagulation calculations. (a) The dust property (here, mass) distribution is discretized into bins, and the bitree corresponds to the hierarchical grouping of the bins. (b) The tree algorithm assumes that the dust pa￾rameter space is a virtual metric space, where a distance is defined. To calculate coagulation using the tree, the algorithm first fixe… view at source ↗
Figure 2
Figure 2. Figure 2: Positions corresponding to k-dust aggregates (outcome of the coagulation) in the (i, j) aggregates (two colliding aggregates) matrix. (Left) When the dust properties are on the linear axes, the k-bin after the coagulation spans diagonally. (Right) When the dust properties are on the logarithmic axes, the k-bin after the coagulation spans along the outwardly convex curve. the coagulation of mass mi = 990 an… view at source ↗
Figure 3
Figure 3. Figure 3: Contour plot of the dust property value after coagulation on i and j grid. This is the same as [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of numerical integration using the fixed bin width (ex. Newton-Cotes rectangle method) (left) and the adaptive bin width (ex. our tree method) (right). In the adaptive bin width methods, the bin width becomes larger in the farther region, where the virtual distance is the mass ratio of i-dust and j-dust. 3.3. Detailed explanation of the tree algorithm for the Smoluchowski coagulation equation … view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of symmetry breaking in the tree algorithm for the SCE. The coagulation of mass m = ma and m = mb aggregates is calcu￾lated by one operation of the increase with the first term in the equation and two operations of the decrease with the second term. If ma < mb, the increase of ma + mb and the decrease of mb is calculated with one grouping of bins (Right panel), and the decrease of ma is calcul… view at source ↗
Figure 6
Figure 6. Figure 6: Analytic solutions and results of numerical calculation of one-component coagulation equation. The top panels (a), (b), and (c) are for the constant kernel, and the bottom panels (d), (e), and (f) are for the additive kernel. The left panels (a) and (d) are calculated with fiducial parameters Nbd = 16 (i.e., N = 241) and θc = 1, the middle panels (b) and (e) are calculated with a finer grid Nbd = 40 (i.e.,… view at source ↗
Figure 7
Figure 7. Figure 7: Analytic solutions and results of numerical calculation of the two-component coagulation equation. The panel layout is the same as [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The effect of the number of bins per component N on the wall-clock time T (the top panels), the L2 error ε2 (the middle panels), and the relative error of the total mass ∆M (the bottom panels) for the constant time step. From the left, the panels show the results for the (a) one￾component constant kernel, (b) one-component additive kernel, (c) two-component constant kernel, and (d) two-component additive k… view at source ↗
Figure 9
Figure 9. Figure 9: The effect of the number of bins per component N on the wall-clock time T (top panels) and the wall-clock time per step Tper step (bottom panels) for the adaptive time step. From the left, the panels show the results for the (a) one-component constant kernel, (b) one-component additive kernel, (c) two-component constant kernel, and (d) two-component additive kernel. In each plot, different colors correspon… view at source ↗
Figure 10
Figure 10. Figure 10: The effect of the critical opening angle θc on the wall-clock time T (the top panels), the L2 error ε2 (the middle panels), and the relative error of the total mass ∆M (the bottom panels) for the constant time step. From the left, the panels show the results for the (a) one-component constant kernel, (b) one-component additive kernel, (c) two-component constant kernel, and (d) two-component additive kerne… view at source ↗
Figure 11
Figure 11. Figure 11: L2 error ε2 versus wall-clock time T with N = 241. Each point in a line corresponds to different values of θc and kc . From the left, the panels show the results for the (a) one-component constant kernel, (b) one-component additive kernel, (c) two-component constant kernel, and (d) two-component additive kernel. The parameter can be considered good if the point is located lower (faster) and further to the… view at source ↗
Figure 12
Figure 12. Figure 12: Same as [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
read the original abstract

Dust properties, such as mass and porosity, impact planet formation directly. Understanding the time evolution of dust distribution across multiple properties requires numerical computation. However, available ways to calculate the multi-component coagulation-fragmentation are highly time-consuming. This study aims to develop a fast and accurate algorithm for multi-component coagulation. We assumed that two pairs of colliding aggregates reproduce a similar outcome if the dust properties are similar, and that the ratio of dust properties in logarithmic space gives the similarity as a "distance". These assumptions enable us to apply the tree algorithm, which groups distant bins and calculates interactions together, to coagulation. The algorithm reduces the computational complexity from $O (N^{2d})$ to $O (d N^d \log N)$, considering $N$ bins per $d$ components. We tested the algorithm by comparing it with the conventional direct method for cases where analytic solutions are known. We measured the dependencies of the wall-clock time, $L_2$ error in the distribution, and relative error of the total mass, on the $d, N$, opening angle $\theta_c$, and maximum dust distribution width after coagulation $k_c$. The algorithms are found to calculate coagulation consistently. For $d=1$, the tree method is faster than the direct method for a specific range of parameters. For $d=2$, however, the tree method is faster for all parameter regions surveyed, speeding it up by tens of times. Increasing $N$ and decreasing $\theta_c$ or $k_c$ made it slower and more accurate. Additionally, using a small $k_c$ performs worse than when using a large $k_c$, suggesting that limiting $k_c$ is unnecessary. We present a fast tree algorithm for the multi-component coagulation equation. It will enable us to evolve the multi-component dust distribution, such as in mass-porosity space, in protoplanetary disks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript presents a tree-based algorithm for the multi-component coagulation equation that groups collision interactions using a logarithmic-space similarity distance between dust properties. This yields a claimed complexity reduction from O(N^{2d}) to O(d N^d log N) for N bins per dimension d. The method is validated by direct comparison to analytic solutions and the conventional summation method, with reported measurements of L2 distribution error, total-mass conservation error, and wall-clock time as functions of d, N, opening angle θ_c, and maximum distribution width k_c. Results indicate consistency with the direct method, with speedups for d=2 across surveyed parameters and accuracy-speed trade-offs when varying θ_c and k_c.

Significance. If the reported accuracy and scaling hold under the stated similarity assumption, the algorithm would remove a major computational barrier to evolving dust distributions in multiple properties (e.g., mass and porosity) within protoplanetary-disk models. The explicit benchmarking against known analytic cases and the direct method, together with the parameter sweeps, supplies a reproducible basis for adoption; the observed tens-of-times speedup at d=2 is particularly enabling for higher-dimensional coagulation-fragmentation studies.

major comments (2)
  1. [§3] §3 (Algorithm Description): The central complexity claim O(d N^d log N) follows from the tree-grouping construction once the log-space similarity metric is accepted, but the manuscript does not quantify the overhead of distance calculations or the fraction of bins that remain ungrouped for realistic k_c values; a short scaling plot versus N for fixed d would strengthen the claim.
  2. [§4.3] §4.3 (Numerical Tests, d=2 results): The statement that the tree method is faster “for all parameter regions surveyed” is load-bearing for the practical utility claim, yet the text does not tabulate the exact wall-clock ratios or the L2-error values at the largest N and smallest θ_c; inclusion of these numbers (or an additional table) is needed to confirm the reported consistency does not degrade at the high-accuracy end of the parameter space.
minor comments (3)
  1. [Abstract] Abstract and §2: The phrase “maximum dust distribution width after coagulation k_c” is used without an explicit definition or formula; a one-sentence clarification of how k_c is computed from the post-coagulation distribution would remove ambiguity.
  2. [Figures] Figure captions (throughout): Several panels lack explicit labels for the direct-method reference curves; adding “solid lines: direct method” or equivalent would improve readability.
  3. [§5] §5 (Conclusions): The final sentence states the algorithm “will enable” multi-component evolution but does not mention the remaining free parameters (θ_c, k_c); a brief caveat on the accuracy-speed trade-off would be proportionate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The two major comments identify specific places where additional quantitative detail would strengthen the presentation of the algorithm's complexity and performance; we address each below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (Algorithm Description): The central complexity claim O(d N^d log N) follows from the tree-grouping construction once the log-space similarity metric is accepted, but the manuscript does not quantify the overhead of distance calculations or the fraction of bins that remain ungrouped for realistic k_c values; a short scaling plot versus N for fixed d would strengthen the claim.

    Authors: We agree that a direct demonstration of the scaling and an indication of grouping efficiency would make the complexity claim more robust. In the revised manuscript we will add a short log-log plot of wall-clock time versus N for fixed d=1 and d=2 (with the same k_c and θ_c used in the main tests) together with a brief paragraph quantifying the relative cost of the distance calculations and the typical fraction of bins that remain ungrouped at the k_c values employed. revision: yes

  2. Referee: [§4.3] §4.3 (Numerical Tests, d=2 results): The statement that the tree method is faster “for all parameter regions surveyed” is load-bearing for the practical utility claim, yet the text does not tabulate the exact wall-clock ratios or the L2-error values at the largest N and smallest θ_c; inclusion of these numbers (or an additional table) is needed to confirm the reported consistency does not degrade at the high-accuracy end of the parameter space.

    Authors: We accept that explicit numerical values at the high-accuracy corner of parameter space would remove any ambiguity. We will insert a compact table (or augmented text) reporting the exact wall-clock time ratios and L2 errors for d=2 at the largest N and smallest θ_c examined, confirming that consistency with the direct method is maintained while the speedup remains present. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper states an explicit similarity assumption (log-space distance between dust properties) that enables grouping in a tree algorithm for multi-component coagulation. The complexity claim O(N^{2d}) to O(d N^d log N) follows as a standard consequence of tree-based interaction grouping once that criterion is adopted; it is not obtained by re-expressing the input assumption or by fitting parameters that are then relabeled as predictions. Validation consists of direct numerical comparisons against the conventional method on analytic-solution test cases, measuring L2 error, mass conservation, and wall-clock time as functions of d, N, θ_c, and k_c. No load-bearing self-citations, uniqueness theorems, or ansatzes smuggled via prior work are invoked. The derivation is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The central claim depends on two domain assumptions about outcome similarity and on the choice of three tunable parameters that control grouping and accuracy.

free parameters (4)
  • opening angle theta_c
    Controls how aggressively distant bins are grouped; smaller values increase accuracy but slow the calculation.
  • maximum dust distribution width k_c
    Limits the width of the distribution after coagulation; tested as a performance parameter.
  • N
    Number of bins per component; directly affects both speed and accuracy scaling.
  • d
    Number of tracked components (e.g., mass and porosity); determines the dimensionality of the problem.
axioms (2)
  • domain assumption Two pairs of colliding aggregates reproduce a similar outcome if the dust properties are similar.
    This similarity assumption justifies grouping interactions in the tree.
  • domain assumption The ratio of dust properties in logarithmic space gives the similarity as a distance.
    Defines the metric used to decide which bins can be grouped.

pith-pipeline@v0.9.0 · 5880 in / 1538 out tokens · 58034 ms · 2026-05-19T23:41:50.814293+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    We assumed that two pairs of colliding aggregates reproduce a similar outcome if the dust properties are similar, and that the ratio of dust properties in logarithmic space gives the similarity as a 'distance'.

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_eq_pow echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    The 'distance' between two dust property vectors X1 and X2 in the dust property space with logarithmic axes using the L2 norm, i.e., ||X1 − X2|| = sqrt(∑(log10 Xp1 − log10 Xp2)²).

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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