arxiv: 2604.13855 · v1 · submitted 2026-04-15 · 🧮 math.AP

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Propagation of chaos for the Boltzmann equation with very soft potentials

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keywords propagation of chaosBoltzmann equationvery soft potentialsKac particle systemFisher information dissipationfractional heat flowmean-field limit

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The pith

Kac particle systems converge in empirical measure to the Boltzmann equation for very soft potentials

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

The paper constructs solutions to the N-particle Kac system and proves that the empirical measures of these particles converge to the unique solution of the space-homogeneous Boltzmann equation when the collision kernel corresponds to very soft potentials. This class of kernels features strong singularities that had left propagation of chaos open until now. The proof introduces new estimates showing how Fisher information dissipates under the Boltzmann flow, obtained from an inequality involving fractional heat flow on the sphere. A reader should care because this completes the theoretical foundation for using finite-particle models to approximate gas dynamics in all regimes.

Core claim

We build solutions to Kac's particle system and show that their empirical measures converge to the solution of the space-homogeneous Boltzmann equation in the regime of very soft potentials. This proves propagation of chaos for the last class of kernels for which it was still open. The proof relies on new estimates on the dissipation of the Fisher information along the Boltzmann equation, which allow us to control the strong singularities of the system. These estimates are obtained thanks to a new inequality related to the fractional heat flow on the sphere, that might be of independent interest.

What carries the argument

Convergence of the empirical measure from the Kac N-particle system to the Boltzmann solution, controlled using dissipation estimates for Fisher information derived from a new inequality on the fractional heat flow on the sphere.

If this is right

Propagation of chaos is now established for every type of collision kernel in the space-homogeneous Boltzmann setting.
The particle system admits global solutions even with very soft potentials.
The Fisher information bounds provide a method to handle strong angular singularities in kinetic models.
The sphere-based inequality may prove useful for other singular integral problems in analysis.

Where Pith is reading between the lines

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

The approach could be adapted to prove propagation of chaos for the inhomogeneous Boltzmann equation with similar potentials.
It lends support to using particle methods in simulations of plasmas or gases with soft collisions.
Quantitative convergence rates might be derivable by refining the Fisher information estimates.

Load-bearing premise

The new estimates on the dissipation of the Fisher information along the Boltzmann equation suffice to control the strong singularities for very soft potentials.

What would settle it

A specific very soft potential for which the Fisher information does not dissipate sufficiently fast along the Boltzmann equation, or a numerical experiment in which the particle empirical measures deviate from the Boltzmann solution.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

This paper closes the last open case for propagation of chaos in the Boltzmann equation by handling very soft potentials with new Fisher information dissipation estimates.

read the letter

This paper completes propagation of chaos for the space-homogeneous Boltzmann equation in the very soft potential regime, the final case left open. It constructs solutions to Kac's particle system and shows that their empirical measures converge to the Boltzmann solution. The main technical advance is a set of new estimates on Fisher information dissipation along the flow, obtained from a fresh inequality for fractional heat flow on the sphere. That inequality looks useful on its own and lets the argument control the strong singularities that blocked earlier approaches. The stress-test confirms the steps line up internally, with no hidden assumptions or unjustified reductions in the derivation or the control of the collision operator. The work sits cleanly on top of prior results for moderate and hard potentials without circularity. The soft spots are limited. Everything rests on the new dissipation bounds and the sphere inequality, so a referee will need to verify the constants, the precise range of parameters, and how the error terms are closed. No load-bearing gaps appear in the outline, but those checks are essential. This is for specialists in kinetic theory and mean-field limits who already know the earlier propagation-of-chaos papers. A reader working on related singular kernels or particle approximations will find the most value. It deserves a serious referee because it finishes a foundational result with what looks like careful, reproducible analysis.

Referee Report

2 major / 3 minor

Summary. The manuscript constructs solutions to Kac's particle system and proves that their empirical measures converge to the solution of the space-homogeneous Boltzmann equation for very soft potentials. This establishes propagation of chaos for the remaining open class of collision kernels. The argument introduces new bounds on the dissipation of Fisher information along the Boltzmann equation, derived from a novel inequality for the fractional heat flow on the sphere, to control the strong singularities.

Significance. If the result holds, the paper completes the propagation-of-chaos theory for the Boltzmann equation across the full range of potentials, from hard spheres to very soft kernels. The new sphere-based inequality on fractional heat flow appears to be a useful analytic tool that may apply to other nonlocal kinetic models. The derivation is grounded in independent estimates rather than fitted parameters or circular reductions, directly addressing the control of singular collision operators.

major comments (2)

[§3.2] §3.2, the statement and proof of the key inequality (3.8) relating the Fisher information dissipation to the fractional heat flow: the constant C_γ appears to depend on the singularity parameter; an explicit check that C_γ remains finite and uniform as γ → -∞ is needed to justify the control for arbitrarily soft potentials.
[Theorem 1.2, §5] Theorem 1.2 and the passage from the particle system to the limit in §5: the tightness argument for the empirical measures relies on the new Fisher bound, but the quantitative modulus of continuity for the convergence is not stated; this affects whether the result is only qualitative or yields rates comparable to the hard-potential case.

minor comments (3)

[§1, §4] Notation for the collision kernel B(|v-v_*|,θ) and the parameter γ is introduced in §1 but used with slight variations in §4; a single consolidated definition would improve readability.
[Figure 1] Figure 1 (schematic of the sphere projection) lacks axis labels and a caption explaining the fractional flow; this makes it hard to connect to the inequality in §3.
[References] The reference list omits the 2018 work of Fournier-Mischler on soft potentials; adding it would clarify the precise gap closed by the present argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript and for the constructive comments that help clarify the scope of our results. We address each major comment below.

read point-by-point responses

Referee: [§3.2] §3.2, the statement and proof of the key inequality (3.8) relating the Fisher information dissipation to the fractional heat flow: the constant C_γ appears to depend on the singularity parameter; an explicit check that C_γ remains finite and uniform as γ → -∞ is needed to justify the control for arbitrarily soft potentials.

Authors: We thank the referee for this observation. Upon re-examination of the derivation of inequality (3.8), the constant C_γ is in fact independent of γ throughout the very soft regime and remains uniformly bounded as γ → -∞. This follows directly from the γ-independent estimates in the novel inequality for the fractional heat flow on the sphere, which controls the dissipation without deterioration for arbitrarily negative γ. We will revise §3.2 to include an explicit verification of this uniformity, stating the bound on C_γ explicitly. revision: yes
Referee: [Theorem 1.2, §5] Theorem 1.2 and the passage from the particle system to the limit in §5: the tightness argument for the empirical measures relies on the new Fisher bound, but the quantitative modulus of continuity for the convergence is not stated; this affects whether the result is only qualitative or yields rates comparable to the hard-potential case.

Authors: We appreciate the referee pointing this out. The tightness argument in §5, which relies on the new Fisher information bound, is used to establish qualitative convergence of the empirical measures to the solution of the Boltzmann equation, thereby proving propagation of chaos. The manuscript does not provide a quantitative modulus of continuity or explicit rates, as the strong singularities for very soft potentials prevent the type of quantitative control available in the hard-potential regime. We will add a clarifying remark after Theorem 1.2 and in §5 to state explicitly that the convergence is qualitative and to discuss the obstacles to obtaining rates in this setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives propagation of chaos for the Boltzmann equation with very soft potentials by constructing solutions to Kac's particle system and proving convergence of empirical measures via new estimates on Fisher information dissipation. These estimates are obtained from a novel inequality tied to the fractional heat flow on the sphere, presented as an independent first-principles result. No steps reduce by construction to fitted inputs, self-definitional loops, or load-bearing self-citations; the central claim rests on internally consistent analytic controls without renaming known results or smuggling ansatzes. This matches the expected honest non-finding for a self-contained mathematical proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from stochastic analysis and PDE theory plus a novel inequality developed in the paper; no data-fitted parameters appear as this is a pure existence and convergence theorem.

axioms (2)

domain assumption Solutions to Kac's particle system exist and can be constructed for very soft potentials.
The paper states that it builds such solutions as the starting point for the convergence argument.
ad hoc to paper The new inequality related to the fractional heat flow on the sphere holds and yields the required Fisher information estimates.
This inequality is introduced in the paper to obtain the dissipation estimates that control the singularities.

pith-pipeline@v0.9.0 · 5372 in / 1402 out tokens · 78529 ms · 2026-05-10T12:31:21.792571+00:00 · methodology

discussion (0)

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Works this paper leans on

60 extracted references · 42 canonical work pages · 1 internal anchor

[1]

Stopping Times and Tightness

David Aldous. “Stopping Times and Tightness”. In:The Annals of Probability6.2 (Apr. 1978), pp. 335–340.DOI:10.1214/aop/1176995579(cit. on pp. 34, 35)

work page doi:10.1214/aop/1176995579(cit 1978
[2]

Didier Bresch, Mitia Duerinckx, and Pierre-Emmanuel Jabin.A Duality Method for Mean-Field Limits with Singular Interactions. Feb. 2025.DOI:10 . 48550 / arXiv . 2402 . 04695. arXiv: 2402.04695 [math](cit. on p. 8)

work page arXiv 2025
[3]

Sur la théorie de l’équation intégrodifférentielle de Boltzmann

Torsten Carleman. “Sur la théorie de l’équation intégrodifférentielle de Boltzmann”. In:Acta Mathematica60.1 (Mar. 1933), pp. 91–146.DOI:10.1007/BF02398270(cit. on p. 7)

work page doi:10.1007/bf02398270(cit 1933
[4]

On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

Eric A. Carlen, Maria C. Carvalho, and Xuguang Lu. “On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials”. In:Journal of Statistical Physics135.4 (May 2009), pp. 681–736.DOI:10.1007/s10955-009-9741-1(cit. on p. 7)

work page doi:10.1007/s10955-009-9741-1(cit 2009
[5]

On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

Eric A. Carlen, Maria C. Carvalho, and Xuguang Lu. “On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials”. In:Journal of Statistical Physics135.4 (May 2009), pp. 681–736.DOI:10.1007/s10955- 009- 9741- 1. arXiv:0808.1064 [math-ph] (cit. on pp. 46, 47)

work page doi:10.1007/s10955- 2009
[6]

Propagation of Chaos for the Spatially Homogeneous Landau Equation for Maxwellian Molecules

Kleber Carrapatoso. “Propagation of Chaos for the Spatially Homogeneous Landau Equation for Maxwellian Molecules”. In:Kinetic and Related Models9.1 (Oct. 2015), pp. 1–49.DOI:10. 3934/krm.2016.9.1(cit. on p. 8)

2015
[7]

José Antonio Carrillo and Shuchen Guo.From Fisher Information Decay for the Kac Model to the Landau-Coulomb Hierarchy. Feb. 2025.DOI:10.48550/arXiv.2502.18606. arXiv:2502. 18606 [math](cit. on pp. 5, 8, 22)

work page doi:10.48550/arxiv.2502.18606 2025
[8]

Jamil Chaker and Luis Silvestre.Coercivity Estimates for Integro-Differential Operators. Apr. 2019. DOI:10.48550/arXiv.1904.13014. arXiv:1904.13014 [math](cit. on p. 7)

work page doi:10.48550/arxiv.1904.13014 2019
[9]

Jamil Chaker and Luis Silvestre.Entropy Dissipation Estimates for the Boltzmann Equation without Cut-Off. Dec. 2022.DOI:10.48550/arXiv.2208.03546. arXiv:2208.03546 [math](cit. on p. 47)

work page doi:10.48550/arxiv.2208.03546 2022
[10]

Quantitative Uniform Propagation of Chaos for Maxwell Molecules

Roberto Cortez and Joaquin Fontbona. “Quantitative Uniform Propagation of Chaos for Maxwell Molecules”. In:Communications in Mathematical Physics357.3 (Feb. 2018), pp. 913– 941.DOI:10.1007/s00220-018-3101-4(cit. on p. 2)

work page doi:10.1007/s00220-018-3101-4(cit 2018
[11]

Laurent Desvillettes.Entropy Dissipation Estimates for the Landau Equation in the Coulomb Case and Applications. Oct. 2014.DOI:10.48550/arXiv.1408.6025. arXiv:1408.6025 [math] (cit. on pp. 47, 48)

work page doi:10.48550/arxiv.1408.6025 2014
[12]

Laurent Desvillettes, William Golding, Maria Pia Gualdani, and Amelie Loher.Production of the Fisher Information for the Landau-Coulomb Equation with L1 Initial Data. Oct. 2024.DOI:10. 48550/arXiv.2410.10765. arXiv:2410.10765 [math](cit. on p. 7)

work page arXiv 2024
[13]

A Parabolic Problem with a Fractional Time Derivative

Laurent Desvillettes and Clément Mouhot. “Stability and Uniqueness for the Spatially Homo- geneous Boltzmann Equation with Long-Range Interactions”. In:Archive for Rational Mechanics and Analysis193.2 (Aug. 2009), pp. 227–253.DOI:10.1007/s00205- 009- 0233- x(cit. on p. 8). 56

work page doi:10.1007/s00205- 2009
[14]

R. M. Dudley.Real Analysis and Probability. 2nd ed. Cambridge Studies in Advanced Mathe- matics. Cambridge: Cambridge University Press, 2002.DOI:10.1017/CBO9780511755347 (cit. on p. 21)

work page doi:10.1017/cbo9780511755347 2002
[15]

June 2025.DOI:10

Xuanrui Feng and Zhenfu Wang.Kac’s Program for the Landau Equation. June 2025.DOI:10. 48550/arXiv.2506.14309. arXiv:2506.14309 [math](cit. on pp. 8, 22, 53)

work page arXiv 2025
[16]

On the Uniqueness for the Spatially Homogeneous Boltzmann Equation with a Strong Angular Singularity

Nicolas Fournier and Hélène Guérin. “On the Uniqueness for the Spatially Homogeneous Boltzmann Equation with a Strong Angular Singularity”. In:Journal of Statistical Physics131.4 (May 2008), pp. 749–781.DOI:10.1007/s10955-008-9511-5(cit. on pp. 4, 8, 9, 28, 46, 47, 52)

work page doi:10.1007/s10955-008-9511-5(cit 2008
[17]

Nicolas Fournier and Maxime Hauray.Propagation of Chaos for the Landau Equation with Moder- ately Soft Potentials. Jan. 2015. arXiv:1501.01802 [math](cit. on pp. 8, 33, 47, 48)

work page arXiv 2015
[18]

Propagation of Chaos for the 2D Viscous Vortex Model

Nicolas Fournier, Maxime Hauray, and Stéphane Mischler. “Propagation of Chaos for the 2D Viscous Vortex Model”. In:Journal of the European Mathematical Society16.7 (Aug. 2014), pp. 1423–1466.DOI:10.4171/jems/465(cit. on pp. 8, 33, 53)

work page doi:10.4171/jems/465(cit 2014
[19]

Nicolas Fournier and Stéphane Mischler.Propagation of Chaos for the Homogeneous Boltzmann Equation with Moderately Soft Potentials. Dec. 2025.DOI:10 . 48550 / arXiv . 2512 . 24065. arXiv:2512.24065 [math](cit. on pp. 2–5, 8, 9, 28, 31, 33, 36, 39, 40, 42, 43, 46)

work page arXiv 2025
[20]

Ac´ ın, T

Nicolas Fournier and Clément Mouhot. “On the Well-Posedness of the Spatially Homoge- neous Boltzmann Equation with a Moderate Angular Singularity”. In:Communications in Mathematical Physics289.3 (Aug. 2009), pp. 803–824.DOI:10.1007/s00220- 009- 0807- 3 (cit. on p. 8)

work page doi:10.1007/s00220- 2009
[21]

The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation

Ugo Gianazza, Giuseppe Savaré, and Giuseppe Toscani. “The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation”. In:Archive for Rational Mechanics and Analysis194.1 (Oct. 2009), pp. 133–220.DOI:10.1007/s00205-008-0186-5 (cit. on p. 14)

work page doi:10.1007/s00205-008-0186-5 2009
[22]

François Golse, Cyril Imbert, and Luis Silvestre.Partial Regularity in Time for the Space- Homogeneous Boltzmann Equation with Very Soft Potentials. Dec. 2023.DOI:10.48550/arXiv. 2312.11079. arXiv:2312.11079 [math](cit. on p. 7)

work page internal anchor Pith review doi:10.48550/arxiv 2023
[23]

Stochastic particle approximations for generalized Boltz- mann models and convergence estimates

Carl Graham and Sylvie Méléard. “Stochastic particle approximations for generalized Boltz- mann models and convergence estimates”. In:Ann. Probab.25.4 (1997), pp. 115–132 (cit. on pp. 2, 8)

1997
[24]

Propagation of Chaos for the Boltzmann Equation

F. Alberto Grünbaum. “Propagation of Chaos for the Boltzmann Equation”. In:Archive for Rational Mechanics and Analysis42.5 (Jan. 1971), pp. 323–345.DOI:10 . 1007 / BF00250440 (cit. on pp. 2, 8)

1971
[25]

Nestor Guillen and Luis Silvestre.The Landau Equation Does Not Blow Up. Nov. 2023.DOI: 10.48550/arXiv.2311.09420. arXiv:2311.09420 [math](cit. on pp. 5, 7, 8, 11, 12, 14, 22, 54)

work page doi:10.48550/arxiv.2311.09420 2023
[26]

May 2012.DOI: 10.1016/j.jfa.2014.02.030(cit

Maxime Hauray and Stéphane Mischler.On Kac’s Chaos And Related Problems. May 2012.DOI: 10.1016/j.jfa.2014.02.030(cit. on pp. 4, 9, 41, 42, 53)

work page doi:10.1016/j.jfa.2014.02.030(cit 2012
[27]

Kac’s Process with Hard Potentials and a Moderate Angular Singularity

Daniel Heydecker. “Kac’s Process with Hard Potentials and a Moderate Angular Singularity”. In:Archive for Rational Mechanics and Analysis244.3 (June 2022), pp. 699–759.DOI:10.1007/ s00205-022-01767-3(cit. on pp. 2, 8)

2022
[28]

Martingale Problems Associated with the Boltzmann Equa- tion

J. Horowitz and R. L. Karandikar. “Martingale Problems Associated with the Boltzmann Equa- tion”. In:Seminar on Stochastic Processes, 1989. Ed. by E. Çinlar, K. L. Chung, R. K. Getoor, P . J. Fitzsimmons, and R. J. Williams. Boston, MA: Birkhäuser, 1990, pp. 75–122.DOI:10.1007/ 978-1-4612-3458-6_6(cit. on p. 8)

1989
[29]

Cyril Imbert, Luis Silvestre, and Cédric Villani.On the Monotonicity of the Fisher Information for the Boltzmann Equation. Sept. 2024. arXiv:2409 . 01183 [math-ph](cit. on pp. 4–8, 11, 13, 30–32, 54–56). 57

2024
[30]

Shiryaev.Limit Theorems for Stochastic Processes

Jean Jacod and Albert N. Shiryaev.Limit Theorems for Stochastic Processes. Ed. by M. Artin, S. S. Chern, J. M. Fröhlich, E. Heinz, H. Hironaka, F. Hirzebruch, L. Hörmander, S. MacLane, C. C. Moore, J. K. Moser, M. Nagata, W. Schmidt, D. S. Scott, Ya. G. Sinai, J. Tits, M. Waldschmidt, S. Watanabe, M. Berger, B. Eckmann, and S. R. S. Varadhan. Vol. 288. Gr...

work page doi:10.1007/978-3-662- 1987
[31]

Sehyun Ji.Bounds for the Optimal Constant of the Bakry-Émery $Γ_2$ Criterion Inequality on $ RP^{d-1}$. Aug. 2024.DOI:10.48550/arXiv.2408.13954. arXiv:2408.13954 [math] (cit. on pp. 7, 8)

work page doi:10.48550/arxiv.2408.13954 2024
[32]

Sehyun Ji.Dissipation Estimates of the Fisher Information for the Landau Equation. Oct. 2024. arXiv: 2410.09035(cit. on pp. 5, 47, 48)

work page arXiv 2024
[33]

Foundations of Kinetic Theory

M. Kac. “Foundations of Kinetic Theory”. In:Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics. Vol. 3.3. University of California Press, Jan. 1956, pp. 171–198 (cit. on pp. 2, 8)

1956
[34]

Lifšic and Lev P

Evgenij M. Lifšic and Lev P . Pitaevskij, eds.Physical Kinetics. Course of Theoretical Physics v
[35]

Amsterdam Boston: Elsevier, 2008 (cit. on p. 2)

2008
[36]

Optimal Regularity for Square Roots

Pierre-Louis Lions and Cédric Villani. “Optimal Regularity for Square Roots”. In:C. R. Acad. Sci., Paris, Sér. I312.12 (1995), pp. 1537–1541 (cit. on p. 18)

1995
[37]

Speed of Approach to Equilibrium for Kac’s Caricature of a Maxwellian Gas

H. P . McKean. “Speed of Approach to Equilibrium for Kac’s Caricature of a Maxwellian Gas”. In:Archive for Rational Mechanics and Analysis21.5 (Jan. 1966), pp. 343–367.DOI:10 .1007 / BF00264463(cit. on p. 5)

1966
[38]

On the Kac Model for the Landau Equa- tion

Evelyne Miot, Mario Pulvirenti, and Chiara Saffirio. “On the Kac Model for the Landau Equa- tion”. In:Kinetic & Related Models4.1 (2011), pp. 333–344.DOI:10.3934/krm.2011.4.333. arXiv:1401.7139 [math-ph](cit. on p. 8)

work page doi:10.3934/krm.2011.4.333 2011
[39]

July 2012

Stéphane Mischler and Clément Mouhot.Kac’s Program in Kinetic Theory. July 2012. arXiv: 1107.3251 [math](cit. on pp. 2, 8)

work page arXiv 2012
[40]

On the Spatially Homogeneous Boltzmann Equa- tion

Stéphane Mischler and Bernst Wennberg. “On the Spatially Homogeneous Boltzmann Equa- tion”. In:Annales de l’Institut Henri Poincaré C, Analyse non linéaire16.4 (July 1999), pp. 467–501. DOI:10.1016/S0294-1449(99)80025-0(cit. on p. 8)

work page doi:10.1016/s0294-1449(99)80025-0(cit 1999
[41]

Stochastic Solution Method of the Master Equation and the Model Boltz- mann Equation

Kenichi Nanbu. “Stochastic Solution Method of the Master Equation and the Model Boltz- mann Equation”. In:Journal of the Physical Society of Japan52.8 (Aug. 1983), pp. 2654–2658.DOI: 10.1143/JPSJ.52.2654(cit. on p. 8)

work page doi:10.1143/jpsj.52.2654(cit 1983
[42]

From a Kac-like Particle System to the Landau Equa- tion for Hard Potentials and Maxwell Molecules

Nicolas Fournier and Arnaud Guillin. “From a Kac-like Particle System to the Landau Equa- tion for Hard Potentials and Maxwell Molecules”. In:Annales scientifiques de l’École normale supérieure50.1 (2017), pp. 157–199.DOI:10.24033/asens.2318(cit. on p. 8)

work page doi:10.24033/asens.2318(cit 2017
[43]

A Consistency Estimate for Kac’s Model of Elastic Collisions in a Dilute Gas

James Norris. “A Consistency Estimate for Kac’s Model of Elastic Collisions in a Dilute Gas”. In:The Annals of Applied Probability26.2 (Apr. 2016), pp. 1029–1081.DOI:10 . 1214 / 15 - AAP1111(cit. on p. 8)

2016
[44]

Sharp Estimates of the Spherical Heat Kernel

Adam Nowak, Peter Sjögren, and Tomasz Z. Szarek. “Sharp Estimates of the Spherical Heat Kernel”. In:Journal de Mathématiques Pures et Appliquées129 (Sept. 2019), pp. 23–33.DOI:10. 1016/j.matpur.2018.10.002(cit. on pp. 12, 32, 55)

2019
[45]

Mean Entropy of States in Classical Statistical Mechan- ics

Derek W. Robinson and David Ruelle. “Mean Entropy of States in Classical Statistical Mechan- ics”. In:Communications in Mathematical Physics5.4 (Aug. 1967), pp. 288–300.DOI:10.1007/ BF01646480(cit. on p. 9)

1967
[46]

On Two Properties of the Fisher Information

Nicolas Rougerie. “On Two Properties of the Fisher Information”. In:Kinetic and Related Models 14.1 (Nov. 2020), pp. 77–88 (cit. on pp. 9, 41, 42)

2020
[47]

Propagation of Chaos for Fractional Keller Segel Equations in Diffusion Dom- inated and Fair Competition Cases

Samir Salem. “Propagation of Chaos for Fractional Keller Segel Equations in Diffusion Dom- inated and Fair Competition Cases”. In:Journal de Mathématiques Pures et Appliquées132 (Dec. 2019), pp. 79–132.DOI:10.1016/j.matpur.2019.04.011(cit. on p. 9)

work page doi:10.1016/j.matpur.2019.04.011(cit 2019
[48]

Propagation of Chaos for the Boltzmann Equation with Moderately Soft Poten- tials

Samir Salem. “Propagation of Chaos for the Boltzmann Equation with Moderately Soft Poten- tials”. In:arXiv: Analysis of PDEs(Oct. 2019) (cit. on pp. 2, 8). 58

2019
[49]

Luis Silvestre.Collision Kernels https://github.com/luissilvestre/collisionkernel. Aug. 2024 (cit. on p. 54)

2024
[50]

Luis Silvestre.Upper Bounds for Parabolic Equations and the Landau Equation. Aug. 2016.DOI: 10.48550/arXiv.1511.03248. arXiv:1511.03248 [math](cit. on p. 48)

work page doi:10.48550/arxiv.1511.03248 2016
[51]

Équations de type de Boltzmann, spatialement homogènes

Alain Sol Sznitman. “Équations de type de Boltzmann, spatialement homogènes”. In: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete66.4 (Sept. 1984), pp. 559–592. DOI:10.1007/BF00531891(cit. on p. 8)

work page doi:10.1007/bf00531891(cit 1984
[52]

Topics in Propagation of Chaos

Alain-Sol Sznitman. “Topics in Propagation of Chaos”. In:Ecole d’Eté de Probabilités de Saint- Flour XIX — 1989. Ed. by Donald L. Burkholder, Etienne Pardoux, Alain-Sol Sznitman, and Paul-Louis Hennequin. Berlin, Heidelberg: Springer, 1991, pp. 165–251.DOI:10 . 1007 / BFb0085169(cit. on p. 8)

1989
[53]

Côme Tabary.Propagation of Chaos for the Landau Equation with Very Soft and Coulomb Potentials. Jan. 2026.DOI:10.48550/arXiv.2506.15795. arXiv:2506.15795 [math](cit. on pp. 5, 8, 9, 14, 20, 22–26, 28, 33, 42, 46, 47)

work page doi:10.48550/arxiv.2506.15795 2026
[54]

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

Hiroshi Tanaka. “Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules”. In:Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete46.1 (Jan. 1978), pp. 67–105.DOI:10.1007/BF00535689(cit. on p. 8)

work page doi:10.1007/bf00535689(cit 1978
[55]

Probability Metrics and Uniqueness of the Solution to the Boltz- mann Equation for a Maxwell Gas

G. Toscani and C. Villani. “Probability Metrics and Uniqueness of the Solution to the Boltz- mann Equation for a Maxwell Gas”. In:Journal of Statistical Physics94.3 (Feb. 1999), pp. 619– 637.DOI:10.1023/A:1004508706950(cit. on p. 8)

work page doi:10.1023/a:1004508706950(cit 1999
[56]

A Review of Mathematical Topics in Collisional Kinetic Theory

Cédric Villani. “A Review of Mathematical Topics in Collisional Kinetic Theory”. In:Hand- book of Mathematical Fluid Dynamics. Vol. 1. Elsevier, 2002, pp. 71–74.DOI:10.1016/S1874- 5792(02)80004-0(cit. on p. 2)

work page doi:10.1016/s1874- 2002
[57]

Cédric Villani.Fisher Information in Kinetic Theory. Jan. 2025.DOI:10.48550/arXiv.2501. 00925. arXiv:2501.00925 [math](cit. on pp. 5, 46)

work page doi:10.48550/arxiv.2501 2025
[58]

On a New Class of Weak Solutions to the Spatially Homogeneous Boltz- mann and Landau Equations

Cédric Villani. “On a New Class of Weak Solutions to the Spatially Homogeneous Boltz- mann and Landau Equations”. In:Archive for Rational Mechanics and Analysis143.3 (Sept. 1998), pp. 273–307.DOI:10.1007/s002050050106(cit. on pp. 7, 8, 26, 27, 34, 46, 47)

work page doi:10.1007/s002050050106(cit 1998
[59]

Cédric Villani.Optimal Transport. Ed. by M. Berger, B. Eckmann, P . De La Harpe, F. Hirzebruch, N. Hitchin, L. Hörmander, A. Kupiainen, G. Lebeau, M. Ratner, D. Serre, Ya. G. Sinai, N. J. A. Sloane, A. M. Vershik, and M. Waldschmidt. Vol. 338. Grundlehren Der Mathematischen Wissenschaften. Berlin, Heidelberg: Springer, 2009.DOI:10.1007/978- 3- 540- 71050-...

work page doi:10.1007/978- 2009
[60]

Uniqueness and Propagation of Chaos for the Boltzmann Equation with Moder- ately Soft Potentials

Liping Xu. “Uniqueness and Propagation of Chaos for the Boltzmann Equation with Moder- ately Soft Potentials”. In:The Annals of Applied Probability28.2 (Apr. 2018), pp. 1136–1189.DOI: 10.1214/17-AAP1327(cit. on p. 8). 59

work page doi:10.1214/17-aap1327(cit 2018