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arxiv: 2604.19659 · v1 · submitted 2026-04-21 · 🧮 math.DS · math.AP

Multiscale Kinetic Structures for Living Systems

Diletta Burini , Damian A. Knopoff This is my paper

Pith reviewed 2026-05-10 01:14 UTC · model grok-4.3

classification 🧮 math.DS math.AP
keywords multiscale modelingkinetic theoryactive particlesliving systemsheterogeneityadaptive behaviorcross-scale interactionsnonlinear dynamics
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The pith

Living systems require a multiscale kinetic theory that incorporates a sub-microscopic scale of interacting entities.

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

The paper develops a Multiscale Kinetic Theory of Active Particles that adds a lower scale to the kinetic description of collective dynamics. Classical single-scale models cannot capture the full range of behaviors in living systems because those behaviors emerge from interactions that span sub-microscopic regulatory mechanisms up through collective levels. By treating the activity variable as arising from lower-scale processes while remaining influenced by higher-scale interactions, the framework unifies heterogeneity, adaptive decision-making, nonlinear non-conservative forces, spatial movement, and cross-scale feedbacks. The authors first derive the general mathematical structure and then show how concrete models can be constructed within it. This structure is presented as a way to describe competition and cooperation operating simultaneously at multiple organizational levels.

Core claim

The central claim is that a sub-microscopic scale of interacting entities must be added to the kinetic theory of active particles so that the activity variable becomes an emergent quantity produced by lower-scale regulatory mechanisms and shaped by higher-scale interactions, thereby allowing a single mathematical structure to represent the heterogeneity, adaptation, nonlinear interactions, spatial dynamics, and cross-scale feedbacks characteristic of living systems.

What carries the argument

The Multiscale Kinetic Theory of Active Particles (MS-KTAP), which augments the standard kinetic description by inserting an explicit sub-microscopic scale whose interactions generate the activity variable that drives collective behavior.

If this is right

  • Competition and cooperation can be tracked simultaneously at sub-microscopic, microscopic, and macroscopic levels within one set of equations.
  • The activity variable is no longer an input but is generated by lower-scale regulatory rules that can be altered by higher-scale spatial or social signals.
  • Specific models of biological or social processes follow directly by choosing appropriate interaction kernels at each scale.
  • The same structure can accommodate nonlinear and non-conservative forces without requiring separate ad-hoc terms at each scale.

Where Pith is reading between the lines

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

  • The framework might be tested by deriving reduced single-scale models from the multiscale equations and checking whether the reduction loses predictive accuracy on the same data set.
  • Integration with data-driven methods could allow the lower-scale interaction rules to be learned from observations rather than postulated.
  • The approach could apply to non-living complex systems that also exhibit emergent activity and cross-scale feedbacks, such as certain engineered networks.

Load-bearing premise

That adding a sub-microscopic scale of interacting entities to the kinetic description will overcome the shortcomings of single-scale models for living systems.

What would settle it

Data from a living system, such as cell migration or population dynamics, in which measured adaptive behaviors cannot be reproduced by equations that link a sub-microscopic interaction scale to the emergent activity variable.

Figures

Figures reproduced from arXiv: 2604.19659 by Damian A. Knopoff, Diletta Burini.

Figure 1
Figure 1. Figure 1: On a rationale toward multiscale modeling of behavioural crowds. [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
read the original abstract

This paper develops a conceptual extension of the Kinetic Theory of Active Particles, building upon the framework introduced in [2]. Living systems cannot be adequately described within classical single-scale paradigms, even when refined. To overcome this limitation, we introduce a Multiscale Kinetic Theory of Active Particles (MS-KTAP), in which a sub-microscopic scale of interacting entities is incorporated into the description of collective dynamics. In this framework, the activity variable is interpreted as an emergent quantity arising from lower-scale regulatory mechanisms and influenced by interactions across higher scales. The proposed framework captures key features of living systems, including heterogeneity, adaptive decision-making, nonlinear and non-conservative interactions, spatial dynamics, and cross-scale feedback, within a unified mathematical structure. Competition and cooperation are thus described across multiple levels of organization. The first part of the paper derives the mathematical framework, while the second presents how specific models can be obtained. The paper concludes with perspectives on further developments, including possible integrations with scientific machine learning.

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

1 major / 1 minor

Summary. The manuscript proposes the Multiscale Kinetic Theory of Active Particles (MS-KTAP) as a conceptual extension of the Kinetic Theory of Active Particles from reference [2]. It incorporates a sub-microscopic scale of interacting entities so that the activity variable emerges from lower-scale regulatory mechanisms while being shaped by higher-scale interactions. The framework is presented as unifying heterogeneity, adaptive decision-making, nonlinear and non-conservative interactions, spatial dynamics, and cross-scale feedback within a single mathematical structure, thereby allowing competition and cooperation to be described across multiple levels of organization. The first part derives the general framework; the second part shows how specific models are obtained from it. The paper closes with perspectives on further developments, including possible integrations with scientific machine learning.

Significance. If the derivations are carried through rigorously, the proposal offers a structured way to move beyond single-scale kinetic descriptions by rendering activity emergent rather than prescribed. The explicit construction of models to demonstrate adequacy, rather than external validation, is a constructive feature for a modeling framework. The forward-looking discussion of scientific machine learning integration suggests a route toward data-driven implementations. These elements could influence multiscale modeling in mathematical biology, provided the concrete kinetic equations and emergence mechanism are shown to be free of hidden parameters or circular definitions.

major comments (1)
  1. Abstract: The central claim that incorporating the sub-microscopic scale 'overcome[s] this limitation' of classical single-scale paradigms rests on the derivation promised in the first part of the paper. No equations, interaction operators, or scaling arguments are supplied in the abstract, so it is impossible to verify whether the emergence of the activity variable is achieved without introducing new free parameters or definitional circularity. This is load-bearing for the entire proposal.
minor comments (1)
  1. Abstract: The list of captured features (heterogeneity, adaptive decision-making, etc.) is stated at a high level; a single illustrative equation or operator from the derived framework would help readers assess the unification claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript proposing the Multiscale Kinetic Theory of Active Particles (MS-KTAP). The concern regarding the abstract is noted, and we address it directly below while committing to a revision for improved clarity.

read point-by-point responses

  1. Referee: Abstract: The central claim that incorporating the sub-microscopic scale 'overcome[s] this limitation' of classical single-scale paradigms rests on the derivation promised in the first part of the paper. No equations, interaction operators, or scaling arguments are supplied in the abstract, so it is impossible to verify whether the emergence of the activity variable is achieved without introducing new free parameters or definitional circularity. This is load-bearing for the entire proposal.

    Authors: We agree that the abstract, as a high-level summary, does not include the technical details needed for immediate verification. The full derivation appears in Section 2, where we introduce a sub-microscopic distribution function whose moments define the emergent activity variable through explicit averaging over regulatory mechanisms. The interaction operators are constructed as nonlinear, non-conservative maps with built-in cross-scale feedback terms; these are parameterized solely by the microscopic kernels already present at the sub-scale, without additional free parameters or circular definitions. We will revise the abstract to include a concise reference to this structure (e.g., the multiscale distribution and emergent activity operator) so that the central claim can be assessed at a glance while preserving brevity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a self-contained modeling proposal

full rationale

The paper explicitly constructs MS-KTAP as a conceptual extension of prior KTAP ideas, derives the multiscale structure in its first part, and demonstrates model recovery in the second part. The activity variable is introduced by interpretation as emergent from sub-microscopic scales, but this is a definitional modeling choice rather than a reduction of any derived quantity to its own inputs. No equations or claims are shown to be forced by fitting, self-referential definitions, or load-bearing self-citations that collapse the central result. The framework stands as an independent unification attempt whose adequacy is asserted via explicit construction, not by tautology or external benchmark reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based on the abstract, the main addition is the new multiscale framework; no explicit free parameters are mentioned, and the central claim rests on the domain assumption about the need for multiscale modeling.

axioms (1)
  • domain assumption Living systems cannot be adequately described within classical single-scale paradigms
    This is the starting motivation stated in the abstract.
invented entities (1)
  • Multiscale Kinetic Theory of Active Particles (MS-KTAP) no independent evidence
    purpose: To incorporate sub-microscopic scales and cross-scale feedback into modeling of living systems
    New framework introduced without external validation or specific predictions in the abstract.

pith-pipeline@v0.9.0 · 5461 in / 1358 out tokens · 41549 ms · 2026-05-10T01:14:05.027045+00:00 · methodology

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    Ballerini, N

    M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study,Proceedings National Academy of Sciences of USA,105(4), 1232–1237, (2008)

    work page 2008

  2. [2]

    Bellomo, D

    N. Bellomo, D. Burini, and J. Liao, New trends in kinetic theory toward the complexity of living systems,Mathematical Models and Methods in Applied Sciences,36(2), 341–397, (2026)

    work page 2026

  3. [3]

    Bellomo, D

    N. Bellomo, D. Burini, and N. Outada, Multiscale models of Covid-19 with mutations and variants,Networks and Heterogeneous Media,17(3), 293–310, (2022)

    work page 2022

  4. [4]

    Bellomo and M

    N. Bellomo and M. Egidi, From Herbert A. Simon’s legacy to the evolutionary artificial worldwithheterogeneouscollectivebehaviors,Mathematical Models and Methods in Applied Sciences,34, 145–180, (2024)

    work page 2024

  5. [5]

    Bellomo, R

    N. Bellomo, R. Fabregas, Jie Liao, and P. Terna, Scientific Machine Learning for Collective Systems: physics-brain-guided social-learning to decision-making,Mathematical Models and Methods in Applied Sciences,36(10), (2026)

    work page 2026

  6. [6]

    Bellomo, L

    N. Bellomo, L. Gibelli, and N. Outada, On the interplay between behavioral dynamics and social interactions in human crowds,Kinetic and Related Models,12, 397–409, (2019)

    work page 2019

  7. [7]

    Bellomo, L

    N. Bellomo, L. Gibelli, A. Quaini, and A. Reali, Towards a mathematical theory of behavioral human crowds,Mathematical Models and Methods in Applied Sciences,32, 321–358, (2022)

    work page 2022

  8. [8]

    Bellomo, J

    N. Bellomo, J. Liao, A. Quaini, L. Russo, and C. Siettos, Human behavioral crowds: review, critical analysis, and research perspectives,Mathematical Models and Methods in Applied Sciences,33, 1611-1659, (2023)

    work page 2023

  9. [9]

    Bellouquid, E

    A. Bellouquid, E. De Angelis, and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology,Mathematical Models and Methods in Applied Sciences, 23(5), 949–978, (2013)

    work page 2013

  10. [10]

    Bellouquid and M

    A. Bellouquid and M. Delitala,Modelling Complex Biological Systems - A Kinetic Theory Approach, Birkhäuser-Springer, (2006)

    work page 2006

  11. [11]

    Bengio, Y

    Y. Bengio, Y. LeCun, and G. Hinton, Deep Learning for AI,Communications of the ACM, 64(7), 58–65, (2021). 19

    work page 2021

  12. [12]

    Burini and N

    D. Burini and N. Chouhad, Cross diffusion models in complex frameworks from microscopic to macroscopic,Mathematical Models and Methods in Applied Sciences,33, 1909–1928, (2023)

    work page 1909

  13. [13]

    Burini and D

    D. Burini and D. A. Knopoff, Epidemics and Society - A Multiscale Vision from the Small World to the Globally Interconnected World,Mathematical Models and Methods in Applied Sciences,34(8), 1564–1594 (2024)

    work page 2024

  14. [14]

    Burini and D

    D. Burini and D. A. Knopoff, A multiscale kinetic theory approach to viral-immune com- petition,Mathematical Models and Methods in Applied Sciences,36(6), 1307–1324,(2026)

    work page 2026

  15. [15]

    Burini, D.A

    D. Burini, D.A. Knopoff, and L. Serrano, Multiscale kinetic model for immune reaction in coeliac disease,Mathematics,34(8), (2026)

    work page 2026

  16. [16]

    Conte, Y

    M. Conte, Y. Dzierma, C. Knobe, and C. Surulescu, Mathematical modeling of glioma invasion and therapy approaches via kinetic theory of active particles,Mathematical Models and Methods in Applied Sciences,33(5), 1009–1051, (2023)

    work page 2023

  17. [17]

    Corbin, A

    G. Corbin, A. Klar, C. Surulescu, C. Engwer, M. Wenske, J. Nieto, and J. Soler, Mod- eling glioma invasion with anisotropy- and hypoxia-triggered motility enhancement: From subcellular dynamics to macroscopic PDEs with multiple taxis,Mathematical Models and Methods in Applied Sciences,31(1), 177–222, (2021)

    work page 2021

  18. [18]

    Cucker and S

    F. Cucker and S. Smale, Emergent behavior in flocks,IEEE Transactions Automatic Control,52, 853–862, (2007)

    work page 2007

  19. [19]

    Dolfin and L

    M. Dolfin and L. Leonida, From classical to active particles: Mathematical tools for social dynamics and behavioral economics,Mathematical Models and Methods in Applied Sciences, 36(6), 1325–1353, (2026)

    work page 2026

  20. [20]

    Fabregas, Jie Liao, and N

    R. Fabregas, Jie Liao, and N. Outada, The mathematical theory of behavioural swarms: Towards modelling the collective dynamics of living systems,Mathematical Models and Methods in Applied Sciences,36, (2026). arXiv:2508.12183v1 [nlin.AO] 16 Aug 2025

    work page arXiv 2026

  21. [21]

    Hartwell, J.J

    H.L. Hartwell, J.J. Hopfield, S. Leibler, and A.W. Murray, From molecular to modular cell biology,Nature,402, c47–c52, (1999)

    work page 1999

  22. [22]

    M. A. Herrero, On the role of mathematics in biology,Journal Mathematical Biology,54, 887–889, (2007)

    work page 2007

  23. [23]

    Hilbert, Mathematical problems,Bulletin American Mathematical Society,8(10), 437–479, (1902)

    D. Hilbert, Mathematical problems,Bulletin American Mathematical Society,8(10), 437–479, (1902)

    work page 1902

  24. [24]

    Kahneman,Thinking Fast and Slow, Penguin Books, London, (2012)

    D. Kahneman,Thinking Fast and Slow, Penguin Books, London, (2012)

    work page 2012

  25. [25]

    Kant,Critique of the Power of Judgement, English translation byWerner P.Pluhar, Cambridge University Press, (2002)

    I. Kant,Critique of the Power of Judgement, English translation byWerner P.Pluhar, Cambridge University Press, (2002)

    work page 2002

  26. [26]

    D. C. Krakauer,An Introduction to the Foundation of Complexity Sciences, Santa Fe, NM:SFI Press, (2024). 20

    work page 2024

  27. [27]

    D. C. Krakauer, N. Bertschinger, E. Olbrich, J. C. Flack, and N. Ay, The information theory of individuality,Theory in Biosciences,139, 209–223, (2020)

    work page 2020

  28. [28]

    LeCun, Il manque aux machines le sens commun,La Recherche,Avril/Juin, 20–23, (2024)

    Y. LeCun, Il manque aux machines le sens commun,La Recherche,Avril/Juin, 20–23, (2024)

    work page 2024

  29. [29]

    R. M. May, Uses and abuses of mathematics in biology,Science,303, 338–342, (2004)

    work page 2004

  30. [30]

    Mayr,La biologie de l’évolution, Hermann Editor, Paris, (1981)

    E. Mayr,La biologie de l’évolution, Hermann Editor, Paris, (1981)

    work page 1981

  31. [31]

    The Nobel Assembly at Karolinska Institutet,Discovery of cancer therapy by inhibition of negative immune regulation, Nobel Prize in Physiology or Medicine, (2018)

    work page 2018

  32. [32]

    Parisi, Nobel Lecture: Multiple equilibria,Review Modern Physics,95, paper n.030501, (2023)

    G. Parisi, Nobel Lecture: Multiple equilibria,Review Modern Physics,95, paper n.030501, (2023)

    work page 2023

  33. [33]

    Reed, Why is mathematical biology so hard?Notices of the American Mathematical Society,51, 338–342, (2004)

    R. Reed, Why is mathematical biology so hard?Notices of the American Mathematical Society,51, 338–342, (2004)

    work page 2004

  34. [34]

    Saint-Raymond,Hydrodynamic Limits of the Boltzmann Equation,Lecture Notes in Mathematics, Springer, (1971)

    L. Saint-Raymond,Hydrodynamic Limits of the Boltzmann Equation,Lecture Notes in Mathematics, Springer, (1971)

    work page 1971

  35. [35]

    H. A. Simon, The architecture of complexity,Proceedings of the American Philosophical Society,106(6), 467–482, (1962)

    work page 1962

  36. [36]

    H. A. Simon,The Science of the Artificial, Third Edition, MIT Press, Boston, (2019). 21

    work page 2019