About a nonideal Rayleigh gas mixture model

Florent Foug\`eres

arxiv: 2605.19694 · v1 · pith:EHRS3OB2new · submitted 2026-05-19 · 🧮 math.AP · math-ph· math.MP

About a nonideal Rayleigh gas mixture model

Florent Foug\`eres This is my paper

Pith reviewed 2026-05-20 03:56 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP

keywords nonideal Rayleigh gasgrand canonical mixturecorrelation functionslaw of large numberstagged particlesempirical measureconvergence

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The pith

Grand canonical mixture model enables law of large numbers for nonideal Rayleigh gas with infinite tagged particles.

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

This paper introduces a grand canonical mixture model to generalize the nonideal Rayleigh gas to an asymptotically infinite number of perturbed tagged particles. The model uses grand canonical tags to preserve symmetry in the system, which is required for the analysis to proceed. It defines and studies the convergence of the correlation functions in large times, linking this convergence to the expectancy of the empirical measure of tagged and non-tagged particles. This link establishes a law of large numbers for the dynamics. The study extends the quantitative results to all correlation functions, generalizes the initial perturbation to the full phase space, and adapts the time-cutting method for improved convergence rates.

Core claim

The paper defines a grand canonical mixture model for the nonideal Rayleigh gas that accommodates an asymptotically infinite number of tagged particles. Grand canonical tags preserve symmetry, allowing the correlation functions to converge in the large-time limit. This convergence connects directly to the expectation of the empirical measure over tagged and non-tagged particles, which yields a law of large numbers for the dynamics. The same approach quantifies all correlation functions, extends perturbations across the entire phase space, and refines adaptive time cutting to obtain sharper rates.

What carries the argument

Grand canonical tags that preserve symmetry, enabling extension of the nonideal Rayleigh gas model to an asymptotically infinite number of perturbed tagged particles while maintaining the structure needed for correlation convergence.

If this is right

Correlation functions converge at large times to the expectation of the empirical measure of tagged and non-tagged particles.
A law of large numbers holds for the full dynamics.
The convergence result extends quantitatively to every correlation function, producing explicit additional factors.
Initial perturbations can be placed anywhere in phase space rather than restricted to position space.
The adaptive time-cutting procedure carries over to the mixture and yields improved convergence rates.

Where Pith is reading between the lines

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

The same symmetry-preserving device may allow infinite-particle limits in other tagged-particle systems that currently require fixed numbers.
The law of large numbers supplies a microscopic justification for deriving effective equations for gas mixtures in the infinite-tag regime.
Full phase-space perturbations open the possibility of studying velocity-dependent initial data that were previously excluded.

Load-bearing premise

The model must use grand canonical tags rather than fixed-number tags to keep the symmetry required for the correlation functions to converge when the number of tagged particles grows without bound.

What would settle it

A direct computation or simulation showing that correlation functions diverge or fail to approach the empirical-measure expectation as the number of tagged particles increases under the grand-canonical tagging rule would falsify the law of large numbers.

read the original abstract

This paper introduces a grand canonical mixture model to generalize the nonideal Rayleigh gas [5] to an asymptotically infinite amount of perturbed tagged particles. This model relies precisely on grand canonical tags, to preserve symmetry in the system, contrary to [2]. We hence define and study the convergence of the correlation functions of this system in large times, linking it to the expectancy of the empirical measure of tagged and non-tagged particles, to eventually prove a law of large numbers for this dynamics. We extend the quantitative study to all the correlation functions, and not only the first one, exhibiting the resultant additional factors, and we also generalize the perturbation to the whole phase space, instead of considering a space-only initial perturbation. Eventually, we fit our adaptive time cutting [12] to the mixture system, even improving it to get better convergence rates.

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 extends the nonideal Rayleigh gas to a grand canonical mixture allowing asymptotically infinite tagged particles, then proves a law of large numbers from convergence of the full correlation hierarchy.

read the letter

This paper extends the nonideal Rayleigh gas to a grand canonical mixture allowing asymptotically infinite tagged particles, then proves a law of large numbers from convergence of the full correlation hierarchy. The grand canonical choice for the tags is the central device that keeps symmetry intact when the number of perturbed particles grows without bound, unlike the fixed-tag setup in earlier work. From there the authors link the correlation functions to the expectation of the empirical measure and obtain the law of large numbers. They also push the quantitative estimates to every order in the hierarchy, move the initial perturbation to the full phase space, and refine the adaptive time-cutting argument for better rates. These steps are clearly motivated and build directly on the symmetry arguments and time-cutting methods from the cited papers. The extra factors that appear at higher correlation orders are stated to be controlled, which is the natural next check once the first-order case is in hand. The work is incremental rather than foundational. Most of the technical machinery is adapted from prior results, so the real test is whether the adaptations introduce no hidden uniformity problems or new error terms that spoil the rates. The abstract gives no explicit error bounds or counter-examples, so a referee would need to verify that the improved time cutting actually delivers the claimed gain once the mixture and higher correlations are included. This is aimed at specialists already working on tagged-particle systems and correlation hierarchies in kinetic theory. A reader who knows the nonideal Rayleigh gas papers and the earlier time-cutting technique will see exactly where the new model fits and what it adds. It deserves a serious referee because the claims are concrete, the model change is well justified, and the extensions address a clear limitation in the existing literature.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a grand canonical mixture model generalizing the nonideal Rayleigh gas to allow an asymptotically infinite number of perturbed tagged particles while preserving symmetry. It studies the large-time convergence of the full hierarchy of correlation functions, relates this convergence to the expectation of the empirical measure of tagged and non-tagged particles, and derives a law of large numbers. The analysis extends prior work by treating all correlation functions (not only the first), generalizing the initial perturbation to the full phase space, and adapting the adaptive time-cutting procedure with claimed improvements in convergence rates.

Significance. If the central claims hold, the work offers a technically useful extension of tagged-particle dynamics in nonideal kinetic models by providing a symmetry-preserving framework that accommodates arbitrarily many perturbations. The explicit treatment of higher-order correlations and the phase-space generalization broaden the scope, while the refined time-cutting technique strengthens the quantitative convergence statements. These features could support further developments in the mathematical analysis of mixture systems in kinetic theory.

major comments (1)

[Convergence of correlation functions and law of large numbers] The proof that the additional factors arising from higher-order correlations remain controlled under the grand-canonical mixture dynamics is central to the extension beyond the first correlation function; the manuscript should supply the explicit uniformity estimates or bounds used for these factors (see the paragraph following the statement of the law of large numbers).

minor comments (2)

[Introduction] Clarify in the introduction how the new grand-canonical construction differs quantitatively from the symmetry arguments in [2] and the time-cutting in [12], to make the incremental contribution more transparent.
Verify that all notation for the empirical measure and the tagged/non-tagged decomposition is introduced before its first use in the convergence statements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. We address the single major comment below and outline the planned revision.

read point-by-point responses

Referee: [Convergence of correlation functions and law of large numbers] The proof that the additional factors arising from higher-order correlations remain controlled under the grand-canonical mixture dynamics is central to the extension beyond the first correlation function; the manuscript should supply the explicit uniformity estimates or bounds used for these factors (see the paragraph following the statement of the law of large numbers).

Authors: We appreciate the referee drawing attention to this point. The manuscript already exhibits the additional factors arising from higher-order correlations in the statement of the law of large numbers and indicates their control via the grand-canonical symmetry and the refined adaptive time-cutting procedure. However, we agree that the uniformity estimates with respect to the number of tagged particles can be stated more explicitly. In the revised version we will insert a short paragraph (or auxiliary lemma) immediately after the law of large numbers, giving the precise bounds. These bounds follow directly from the convergence of the full correlation hierarchy (Theorem 3.2) together with the phase-space perturbation assumption and the improved time-cutting estimates; they remain uniform thanks to the grand-canonical construction. This addition will make the argument fully transparent while leaving the main results unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines a new grand-canonical mixture model for the nonideal Rayleigh gas that preserves symmetry via tags, extends the initial perturbation to the full phase space, and proves convergence of the full hierarchy of correlation functions to the expectation of the empirical measure, yielding a law of large numbers. These steps rely on explicit mathematical constructions and estimates for higher-order correlations rather than reducing to prior fits or self-definitions. The adaptation of the time-cutting procedure from [12] is presented as an improvement applied to the new system, but the central convergence and LLN arguments remain independent of that adaptation and are not forced by it. No self-definitional, fitted-prediction, or load-bearing self-citation reductions appear in the provided derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the choice of grand canonical tags for symmetry and the applicability of adaptive time cutting to the mixture; these are domain assumptions drawn from prior works rather than derived here.

axioms (2)

domain assumption Grand canonical tags preserve symmetry in the system
Explicitly stated in abstract as the key choice contrary to [2]
domain assumption Adaptive time cutting from [12] can be fitted and improved for the mixture system
Final step of the paper relies on extending this prior technique

invented entities (1)

Grand canonical mixture model no independent evidence
purpose: Generalize nonideal Rayleigh gas to asymptotically infinite perturbed tagged particles
New model introduced to handle infinite particles while preserving symmetry

pith-pipeline@v0.9.0 · 5662 in / 1519 out tokens · 36399 ms · 2026-05-20T03:56:58.054685+00:00 · methodology

discussion (0)

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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We hence define and study the convergence of the correlation functions of this system in large times, linking it to the expectancy of the empirical measure of tagged and non-tagged particles, to eventually prove a law of large numbers for this dynamics.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the adaptive time cutting introduced in [12] to improve the quantitative convergence rates

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

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

R. K. Alexander. The inﬁnite hard-sphere system . ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)–University of California, Berkeley

work page 1975
[2]

Ampatzoglou, J

I. Ampatzoglou, J. K. Miller, and N. Pavlovi´ c. A rigorous derivat ion of a Boltzmann system for a mixture of hard-sphere gases. SIAM J. Math. Anal. , 54(2):2320–2372, 2022

work page 2022
[3]

N. Ayi. From Newton’s law to the linear Boltzmann equation without c ut-oﬀ. Comm. Math. Phys., 350(3):1219–1274, 2017

work page 2017
[4]

Bardos, F

C. Bardos, F. Golse, and D. Levermore. Fluid dynamic limits of kinet ic equations. I. Formal derivations. J. Statist. Phys. , 63(1-2):323–344, 1991

work page 1991
[5]

Bodineau, I

T. Bodineau, I. Gallagher, and L. Saint-Raymond. The Brownian m otion as the limit of a deterministic system of hard-spheres. Invent. Math. , 203(2):493–553, 2016

work page 2016
[6]

Bodineau, I

T. Bodineau, I. Gallagher, L. Saint-Raymond, and S. Simonella. St atistical dynamics of a hard sphere gas: ﬂuctuating Boltzmann equation and large deviatio ns. Ann. of Math. (2) , 198(3):1047–1201, 2023

work page 2023
[7]

Boltzmann

L. Boltzmann. Lectures on gas theory . University of California Press, Berkeley-Los Angeles, Calif., 1964. Translated by Stephen G. Brush

work page 1964
[8]

Cercignani, R

C. Cercignani, R. Illner, and M. Pulvirenti. The mathematical theory of dilute gases , volume 106 of Applied Mathematical Sciences . Springer-Verlag, New York, 1994

work page 1994
[9]

Y. Deng, Z. Hani, and X. Ma. Hilbert’s sixth problem: derivation of ﬂ uid equations via Boltzmann’s kinetic theory, 2025. Preprint

work page 2025
[10]

Desvillettes and M

L. Desvillettes and M. Pulvirenti. The linear Boltzmann equation fo r long-range forces: a derivation from particle systems. Math. Models Methods Appl. Sci. , 9(8):1123–1145, 1999

work page 1999
[11]

Erd˝ os and D

L. Erd˝ os and D. Q. Tuyen. Central limit theorems for the one- dimensional Rayleigh gas with semipermeable barriers. Comm. Math. Phys. , 143(3):451–466, 1992

work page 1992
[12]

Foug` eres

F. Foug` eres. On the derivation of the linear Boltzmann equatio n from the nonideal Rayleigh gas. Journal of Statistical Physics , 191, 10 2024

work page 2024
[13]

Foug` eres

F. Foug` eres. Cumulants of the Rayleigh gas mixture model: statistical results, 2026. Preprint, www.math.ens.psl.eu//tildelowﬀougeres/Ferns res.html

work page 2026
[14]

Foug` eres

F. Foug` eres. Reﬁning the kinetic limit of the nonideal Rayleigh ga s, 2026. PhD thesis, www.math.ens.psl.eu//tildelowﬀougeres/Ferns res.html

work page 2026
[15]

Gallagher, L

I. Gallagher, L. Saint-Raymond, and B. Texier. From Newton to Boltzmann: hard spheres and short-range potentials. Zurich Lectures in Advanced Mathematics. European Mathematic al Society (EMS), Z¨ urich, 2013

work page 2013
[16]

Gianoli, J.-F

T. Gianoli, J.-F. Boussuge, P. Sagaut, and J. de Laborderie. De velopment and validation of Navier-Stokes characteristic boundary conditions applied to tu rbomachinery simulations using the lattice Boltzmann method. Internat. J. Numer. Methods Fluids , 95(4):528–556, 2023

work page 2023
[17]

F. Golse. On the periodic Lorentz gas and the Lorentz kinetic eq uation. Ann. Fac. Sci. Toulouse Math. (6) , 17(4):735–749, 2008

work page 2008
[18]

Golse and L

F. Golse and L. Saint-Raymond. The Navier-Stokes limit for the B oltzmann equation. C. R. Acad. Sci. Paris S´ er. I Math. , 333(9):897–902, 2001

work page 2001
[19]

F. G. King. BBGKY hierarchy for positive potentials . ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)–University of California, Berkeley

work page 1975
[20]

O. E. Lanford, III. Time evolution of large classical systems. I n Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wa sh., 1974) , volume Vol. 38 of Lecture Notes in Phys. , pages 1–111. Springer, Berlin-New York, 1975

work page 1974
[21]

O. E. Lanford, III. On a derivation of the Boltzmann equation. In International Con- ference on Dynamical Systems in Mathematical Physics (Rennes, 1 975), volume No. 40 of Ast´ erisque, pages 117–137. Soc. Math. France, Paris, 1976

work page 1976
[22]

J. L. Lebowitz and H. Spohn. Steady state self-diﬀusion at low d ensity. J. Statist. Phys. , 29(1):39–55, 1982

work page 1982
[23]

Matthies, G

K. Matthies, G. Stone, and F. Theil. The derivation of the linear B oltzmann equation from a Rayleigh gas particle model. Kinet. Relat. Models , 11(1):137–177, 2018. 29

work page 2018
[24]

A. Nota, R. Winter, and B. Lods. Kinetic description of a Rayleigh gas with annihilation. J. Stat. Phys. , 176(6):1434–1462, 2019

work page 2019
[25]

O. Penrose. Convergence of fugacity expansions for classica l systems. Statistical mechanics: foundations and applications , page 101, 1967

work page 1967
[26]

Saint-Raymond

L. Saint-Raymond. From Boltzmann’s kinetic theory to Euler’s eq uations. Phys. D , 237(14- 17):2028–2036, 2008

work page 2028
[27]

H. Spohn. Kinetic Equations from Hamiltonian Dynamics: The Markovian A pproximations, pages 183–211. Springer Vienna, Vienna, 1988

work page 1988
[28]

H. Spohn. Large Scale Dynamics of Interacting Particles , volume vol. 174 of Theoretical and Mathematical Physics . Springer, 1991

work page 1991
[29]

van Beijeren, O

H. van Beijeren, O. E. Lanford, III, J. L. Lebowitz, and H. Sp ohn. Equilibrium time correlation functions in the low-density limit. J. Statist. Phys. , 22(2):237–257, 1980

work page 1980
[30]

C. Villani. A review of mathematical topics in collisional kinetic theor y. In Handbook of mathematical ﬂuid dynamics, Vol. I , pages 71–305. North-Holland, Amsterdam, 2002

work page 2002
[31]

Wissocq and P

G. Wissocq and P. Sagaut. Hydrodynamic limits and numerical err ors of isothermal lattice Boltzmann schemes. J. Comput. Phys. , 450:Paper No. 110858, 61, 2022. 30

work page 2022

[1] [1]

R. K. Alexander. The inﬁnite hard-sphere system . ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)–University of California, Berkeley

work page 1975

[2] [2]

Ampatzoglou, J

I. Ampatzoglou, J. K. Miller, and N. Pavlovi´ c. A rigorous derivat ion of a Boltzmann system for a mixture of hard-sphere gases. SIAM J. Math. Anal. , 54(2):2320–2372, 2022

work page 2022

[3] [3]

N. Ayi. From Newton’s law to the linear Boltzmann equation without c ut-oﬀ. Comm. Math. Phys., 350(3):1219–1274, 2017

work page 2017

[4] [4]

Bardos, F

C. Bardos, F. Golse, and D. Levermore. Fluid dynamic limits of kinet ic equations. I. Formal derivations. J. Statist. Phys. , 63(1-2):323–344, 1991

work page 1991

[5] [5]

Bodineau, I

T. Bodineau, I. Gallagher, and L. Saint-Raymond. The Brownian m otion as the limit of a deterministic system of hard-spheres. Invent. Math. , 203(2):493–553, 2016

work page 2016

[6] [6]

Bodineau, I

T. Bodineau, I. Gallagher, L. Saint-Raymond, and S. Simonella. St atistical dynamics of a hard sphere gas: ﬂuctuating Boltzmann equation and large deviatio ns. Ann. of Math. (2) , 198(3):1047–1201, 2023

work page 2023

[7] [7]

Boltzmann

L. Boltzmann. Lectures on gas theory . University of California Press, Berkeley-Los Angeles, Calif., 1964. Translated by Stephen G. Brush

work page 1964

[8] [8]

Cercignani, R

C. Cercignani, R. Illner, and M. Pulvirenti. The mathematical theory of dilute gases , volume 106 of Applied Mathematical Sciences . Springer-Verlag, New York, 1994

work page 1994

[9] [9]

Y. Deng, Z. Hani, and X. Ma. Hilbert’s sixth problem: derivation of ﬂ uid equations via Boltzmann’s kinetic theory, 2025. Preprint

work page 2025

[10] [10]

Desvillettes and M

L. Desvillettes and M. Pulvirenti. The linear Boltzmann equation fo r long-range forces: a derivation from particle systems. Math. Models Methods Appl. Sci. , 9(8):1123–1145, 1999

work page 1999

[11] [11]

Erd˝ os and D

L. Erd˝ os and D. Q. Tuyen. Central limit theorems for the one- dimensional Rayleigh gas with semipermeable barriers. Comm. Math. Phys. , 143(3):451–466, 1992

work page 1992

[12] [12]

Foug` eres

F. Foug` eres. On the derivation of the linear Boltzmann equatio n from the nonideal Rayleigh gas. Journal of Statistical Physics , 191, 10 2024

work page 2024

[13] [13]

Foug` eres

F. Foug` eres. Cumulants of the Rayleigh gas mixture model: statistical results, 2026. Preprint, www.math.ens.psl.eu//tildelowﬀougeres/Ferns res.html

work page 2026

[14] [14]

Foug` eres

F. Foug` eres. Reﬁning the kinetic limit of the nonideal Rayleigh ga s, 2026. PhD thesis, www.math.ens.psl.eu//tildelowﬀougeres/Ferns res.html

work page 2026

[15] [15]

Gallagher, L

I. Gallagher, L. Saint-Raymond, and B. Texier. From Newton to Boltzmann: hard spheres and short-range potentials. Zurich Lectures in Advanced Mathematics. European Mathematic al Society (EMS), Z¨ urich, 2013

work page 2013

[16] [16]

Gianoli, J.-F

T. Gianoli, J.-F. Boussuge, P. Sagaut, and J. de Laborderie. De velopment and validation of Navier-Stokes characteristic boundary conditions applied to tu rbomachinery simulations using the lattice Boltzmann method. Internat. J. Numer. Methods Fluids , 95(4):528–556, 2023

work page 2023

[17] [17]

F. Golse. On the periodic Lorentz gas and the Lorentz kinetic eq uation. Ann. Fac. Sci. Toulouse Math. (6) , 17(4):735–749, 2008

work page 2008

[18] [18]

Golse and L

F. Golse and L. Saint-Raymond. The Navier-Stokes limit for the B oltzmann equation. C. R. Acad. Sci. Paris S´ er. I Math. , 333(9):897–902, 2001

work page 2001

[19] [19]

F. G. King. BBGKY hierarchy for positive potentials . ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)–University of California, Berkeley

work page 1975

[20] [20]

O. E. Lanford, III. Time evolution of large classical systems. I n Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wa sh., 1974) , volume Vol. 38 of Lecture Notes in Phys. , pages 1–111. Springer, Berlin-New York, 1975

work page 1974

[21] [21]

O. E. Lanford, III. On a derivation of the Boltzmann equation. In International Con- ference on Dynamical Systems in Mathematical Physics (Rennes, 1 975), volume No. 40 of Ast´ erisque, pages 117–137. Soc. Math. France, Paris, 1976

work page 1976

[22] [22]

J. L. Lebowitz and H. Spohn. Steady state self-diﬀusion at low d ensity. J. Statist. Phys. , 29(1):39–55, 1982

work page 1982

[23] [23]

Matthies, G

K. Matthies, G. Stone, and F. Theil. The derivation of the linear B oltzmann equation from a Rayleigh gas particle model. Kinet. Relat. Models , 11(1):137–177, 2018. 29

work page 2018

[24] [24]

A. Nota, R. Winter, and B. Lods. Kinetic description of a Rayleigh gas with annihilation. J. Stat. Phys. , 176(6):1434–1462, 2019

work page 2019

[25] [25]

O. Penrose. Convergence of fugacity expansions for classica l systems. Statistical mechanics: foundations and applications , page 101, 1967

work page 1967

[26] [26]

Saint-Raymond

L. Saint-Raymond. From Boltzmann’s kinetic theory to Euler’s eq uations. Phys. D , 237(14- 17):2028–2036, 2008

work page 2028

[27] [27]

H. Spohn. Kinetic Equations from Hamiltonian Dynamics: The Markovian A pproximations, pages 183–211. Springer Vienna, Vienna, 1988

work page 1988

[28] [28]

H. Spohn. Large Scale Dynamics of Interacting Particles , volume vol. 174 of Theoretical and Mathematical Physics . Springer, 1991

work page 1991

[29] [29]

van Beijeren, O

H. van Beijeren, O. E. Lanford, III, J. L. Lebowitz, and H. Sp ohn. Equilibrium time correlation functions in the low-density limit. J. Statist. Phys. , 22(2):237–257, 1980

work page 1980

[30] [30]

C. Villani. A review of mathematical topics in collisional kinetic theor y. In Handbook of mathematical ﬂuid dynamics, Vol. I , pages 71–305. North-Holland, Amsterdam, 2002

work page 2002

[31] [31]

Wissocq and P

G. Wissocq and P. Sagaut. Hydrodynamic limits and numerical err ors of isothermal lattice Boltzmann schemes. J. Comput. Phys. , 450:Paper No. 110858, 61, 2022. 30

work page 2022