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Solvable model of the collective motion of heterogeneous particles interacting on a sphere

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Takuma Tanaka

Adaptation and Self-Organizing Systems · Submitted 14 Jan 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim I show that the motion of the centre of mass of an infinite number of heterogeneous particles whose natural-frequency matrices are obtained from a class of multivariate Lorentz distribution is given by an M-dimensional ordinary differential equation in closed form.

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arXiv ID1401.3109 (version 1)
Pith IDLQTGPJ66, Pith's permanent record of this paper and its review
CitationTakuma Tanaka. "Solvable model of the collective motion of heterogeneous particles interacting on a sphere." arXiv:1401.3109 (2014). https://pith.science/paper/LQTGPJ66
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,451 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.80 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

I propose a model of mutually interacting particles on an M-dimensional unit sphere. I derive the dynamics of the particles by extending the dynamics of the Kuramoto-Sakaguchi model. The dynamics include a natural-frequency matrix, which determines the motion of a particle with no external force, and an external force vector. The position (state variable) of a particle at a given time is obtained by the projection transformation of the initial position of the particle. The same projection transformation gives the position of the particles with the same natural-frequency matrix. I show that the motion of the centre of mass of an infinite number of heterogeneous particles whose natural-frequency matrices are obtained from a class of multivariate Lorentz distribution is given by an M-dimensional ordinary differential equation in closed form. This result is an extension of the Ott-Antonsen theory.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 I propose a model of mutually interacting particles on an M-dimensional unit sphere. contribution 0.92 0.91 0.04 asserted 0.97
s1 I derive the dynamics of the particles by extending the dynamics of the Kuramoto-Sakaguchi model. method 0.94 0.94 0.04 asserted 1.00
s2 The dynamics include a natural-frequency matrix, which determines the motion of a particle with no external force, and an external force vector. method 0.93 0.74 0.04 asserted 1.00
s3 The position (state variable) of a particle at a given time is obtained by the projection transformation of the initial position of the particle. result 0.75 0.85 0.06 asserted 1.00
s4 The same projection transformation gives the position of the particles with the same natural-frequency matrix. result 0.80 0.83 0.06 asserted 1.00
s5 I show that the motion of the centre of mass of an infinite number of heterogeneous particles whose natural-frequency matrices are obtained from a class of multivariate Lorentz distribution is given by an M-dimensional ordinary differential equation in closed form. result 0.99 0.94 0.82 asserted 1.00
s6 This result is an extension of the Ott-Antonsen theory. interpretation 0.85 0.51 0.12 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

I propose a model of mutually interacting particles on an M-dimensional unit sphere. I derive the dynamics of the particles by extending the dynamics of the Kuramoto-Sakaguchi model. The dynamics include a natural-frequency matrix, which determines the motion of a particle with no external force, and an external force vector. The position (state variable) of a particle at a given time is obtained by the projection transformation of the initial position of the particle. The same projection transformation gives the position of the particles with the same natural-frequency matrix. I show that the motion of the centre of mass of an infinite number of heterogeneous particles whose natural-frequency matrices are obtained from a class of multivariate Lorentz distribution is given by an M-dimensional ordinary differential equation in closed form. This result is an extension of the Ott-Antonsen theory.

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