The Homogeneous Landau Equation with Regularised Thermal Noise

Manh Hong Duong; Zhengyan Wu; Zihui He

REVIEW 2 major objections 5 minor 12 references

The Homogeneous Landau Equation with Regularised Thermal Noise

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that a fluctuating Landau equation with regularised thermal noise has probabilistic weak solutions for moderately soft potentials, conserving mass and momentum and dissipating entropy up to an explicit noise-driven constan

desk verdict Main existence theorem for regularised fluctuating Landau equation is solid; the d=2 divergence-free basis construction for the refined entropy theorem is wrong as written, though fixable. read the letter →

arxiv 2607.22329 v1 pith:VFV7QCGB submitted 2026-07-24 math.AP math.PR

Manh Hong Duong , Zihui He , Zhengyan Wu This is my paper · ORCID

classification math.APmath.PR MSC 60H1535Q2035Q84

keywords fluctuatingLandauequationprobabilisticweaksolutionconservativethermalnoisefluctuation–dissipationprinciplemoderatelysoftpotentialsStratonovich-to-ItôcorrectionentropydissipationnonlocalkineticSPDE

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces thermal noise into the homogeneous Landau equation of kinetic gas theory, producing a stochastic PDE whose noise mimics the random fluctuations of a large particle system around its deterministic mean-field limit. Its central claim is an existence theorem: for moderately soft collisions, provided the noise is regularised (a smooth mobility replaces the singular square-root amplitude, only finitely many antisymmetric noise modes act, and the noise intensity is small), at least one probabilistic weak solution exists. The solution keeps the defining structure of the deterministic Landau equation: mass and momentum are conserved almost surely, energy does not increase, and the Boltzmann entropy is dissipated up to an explicit constant depending on the noise coefficient. For a specially chosen class of noise modes satisfying a tangential divergence-free condition, the expected entropy is non-increasing, recovering deterministic dissipativity exactly. This supplies a weak-solution theory for a fluctuating kinetic equation with genuinely nonlocal, pair-coupled noise — groundwork for checking whether such equations correctly reproduce particle-system fluctuations.

What carries the argument

Load-bearing object: the Landau difference gradient ∇̃φ(v,v_*) = Π(v−v_*)⊥(∇_vφ − ∇_v_*φ_*), which turns the Landau operator into a gradient flow of Boltzmann entropy; the thermal noise is placed on the same collision space in divergence form so its covariance matches the particle-system martingale. The singular √(A f f_*) mobility is replaced by a smooth σ(f)σ(f_*) (σ = √r away from vacuum, linear at 0), and the Stratonovich noise is converted to Itô form, producing nonlocal correction terms that couple v, v_*, w. The proof runs on a three-level approximation (Galerkin truncation, coefficient regularisation, artificial diffusion) with L²-then-L¹ compactness, plus an exponential entropy esti

What would settle it

A concrete check: simulate the Galerkin scheme of Section 3 with a divergence-free noise basis supported at scale K and measure the entropy gap E[H(f_t)] + E[∫_0^t D(f_s) ds] − H(f_0) as ε is lowered. The theorem predicts this gap stays bounded by C(‖σ′‖_{L∞}) for all ε below a positive threshold; observing it grow without bound as ε → 0, or finding the threshold collapse to zero as the mode-support scale K grows, would refute the claim.

Watch

Extended reading notes

Core claim

On the paper's own terms: the regularised Itô fluctuating Landau equation, with conservative noise whose covariance matches the fluctuation martingale of the underlying particle system, has at least one probabilistic weak solution for every small noise intensity ε < ε₀, for kernels |v−v_*|^{γ+2}, γ ∈ (−2,0), d ≥ 2, and nonnegative initial data of finite energy and finite Boltzmann entropy. The solution conserves mass and momentum almost surely, obeys the energy inequality, and satisfies the entropy-dissipation inequality up to an additive constant C(‖σ′‖_{L∞}); for a tangential divergence-free noise basis, the expected entropy is non-increasing. The proof uses Galerkin truncation, coefficien

Load-bearing premise

The load-bearing premise is that physically natural noise bases keep the regularity and support constants of the active modes, and the exponential-in-entropy constant entering the small-noise threshold, finite — so that the existence threshold ε₀ stays positive and the Itô-correction terms stay integrable for small noise.

Editorial extensions

If this is right

If correct, the theorem gives a well-defined weak-solution theory for the regularised fluctuating Landau equation with the same conservation laws as the deterministic equation: mass and momentum are conserved almost surely, and the entropy-dissipation inequality holds up to an explicit noise-driven constant.
The paper's own next step comes into reach: the solution theory is the stated foundation for proving that the Gaussian fluctuations and large deviations of the stochastic equation coincide with those of the underlying conservative particle system.
For noise modes satisfying the tangential divergence-free condition, the additive constant disappears: the expected θ-entropy is non-increasing, matching the deterministic Landau dissipation structure exactly.
The exponential entropy-dissipation estimate, derived from the supermartingale structure of the entropy balance, supplies the stochastic substitute for the deterministic entropy bound and is the tool expected to transfer to other singular fluctuating-hydrodynamics SPDEs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the natural next test is the vanishing-noise limit ε → 0 — the theorem does not quantify how fast solutions return to the deterministic Landau flow, so tracing the ε-dependence of all constants would be needed to see the transition.
Beyond the paper: since the regularised mobility σ is linear rather than square-root near vacuum, the genuinely singular noise amplitude is not covered here; extending a renormalised-solution method — which the paper explains fails for the nonlocal Landau operator — is the implied route to removing the regularisation.
Beyond the paper: the divergence-free condition admits a geometric reading — only noise modes acting purely tangentially, without compressing the collision geometry, preserve entropic monotonicity; numerical comparison of entropy production for general versus admissible bases would show how much deterministic dissipativity survives generic thermal noise.
Beyond the paper: the exclusion of the borderline γ = −2, located by the paper in the failure of the uniform L² estimate, suggests an L¹-based compactness argument might extend the existence theorem to that threshold.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the headline: this is the first solution theory for a fluctuating homogeneous Landau equation with conservative velocity-pair noise, and the main existence theorem looks right. The advertised divergence-free basis construction for the refined entropy theorem is, however, wrong in d=2 as written, and the paper needs a repair before that second result can be trusted.

What is new and good: the noise is written in Landau-divergence form, antisymmetric in the pair of velocities, and is motivated by the fluctuation–dissipation principle and the covariance of a Kac-like particle system. The paper proves existence of probabilistic weak solutions for the regularised Itô equation with γ∈(−2,0), conserving mass and momentum, with energy and entropy-dissipation inequalities (the latter up to an explicit additive constant). The Stratonovich-to-Itô correction is handled by replacing the square-root mobility with a regular coefficient, and Remark 5.4 honestly explains why γ=−2 fails in the SPDE setting while deterministic theory covers it. The exponential entropy estimate (Prop. 5.2) is a genuinely stochastic ingredient, and the four-step compactness scheme is carefully laid out. The claims are scoped honestly: covariance matching with the particle system is formal, and fluctuation coincidence is deferred.

Soft spots: Appendix D's d=2 construction is not tangential. T_{l,m}=R∇_{S^1}Y_{l,m}/l gives radial vectors because R e_θ = e_r, so ω·T≠0 and the projected fields G_k vanish identically. Thus Assumption 7.3 is not satisfied non-trivially by that construction in d=2. A valid basis does exist (the constant rotation field e_θ, used with antisymmetric z- and r-factors), so the flaw is repairable, but Theorem 1.2 as written lacks a working d=2 basis. This does not touch Theorem 1.1. A second soft spot is that ε_0 inherits exponential-in-entropy constants from the Desvillettes weighted Fisher-information lemma, so the small-noise threshold could be extremely small, though still positive for fixed f0. Also, the existence theory is for the regularised equation: smooth σ, finite K modes, and the physical limits (σ→√·, K→∞, recovering particle-system fluctuations) are open. The paper says so; it is a scope limitation, not an overclaim. Finally, the proofs are very long and many estimates are delegated to earlier propositions; I saw no circularity, but independent verification of the L2 and L1 passages is warranted.

Who is this for: researchers in stochastic kinetic theory and singular SPDEs, and anyone working on Dean–Kawasaki-type equations with nonlocal mobility. It deserves a serious referee. Recommendation: send it to peer review, but require the authors to fix or restrict Appendix D (e.g., state Theorem 1.2 for d≥3 or give the correct d=2 basis) and to state the dependence of ε_0 on the entropy constants more explicitly.

Referee Report

2 major / 5 minor

Summary. The paper introduces a stochastic perturbation of the homogeneous Landau equation in which the conservative noise is written in nonlocal Landau-divergence form, is antisymmetric in the colliding velocities, and is interpreted in the Stratonovich sense before passing to Itô form. The square-root mobility is replaced by a regular coefficient σ that behaves like √r away from vacuum and linearly near zero, and the noise is projected onto a finite antisymmetric basis. For moderately soft potentials γ∈(−2,0), Theorem 1.1 asserts the existence of probabilistic weak solutions to the regularised Itô equation for sufficiently small noise intensity ε, under regularity and support conditions on the noise basis; the solutions conserve mass and momentum, satisfy an energy inequality, and satisfy the entropy-dissipation inequality (1.18) with an additive constant arising from the Stratonovich–Itô correction. The proof combines Galerkin approximation, artificial diffusion, L²-compactness, an exponential entropy-dissipation estimate, and L¹-compactness via a Desvillettes-type weighted Fisher-information bound. Theorem 1.2 claims a refined entropy inequality for a θ-regularised equation when the noise basis is tangential and divergence-free on the relative-velocity sphere, giving monotonicity of the expected θ-entropy without the additive constant.

Significance. The existence theory for a fluctuating Landau equation with nonlocal, conservative noise is new and the paper is unusually careful about the stochastic singularities: Remark 5.4 honestly explains why γ=−2 cannot be handled in the L² framework; the additive constant in (1.18) is explicitly identified as the cost of the Stratonovich–Itô correction; and the exponential entropy-dissipation estimate (Proposition 5.2) is a genuinely stochastic tool. The formal particle derivation (Appendix A) and the Stratonovich-to-Itô conversion (Appendix B) are worked out in detail and give useful heuristics. The main weakness is that the noise-basis construction behind the second main result is incorrect in d=2 as written, and the small-noise threshold ε₀ is non-explicit and depends on entropy-exponential constants; the latter is acceptable for an existence theorem, while the former needs repair.

major comments (2)

[Appendix D / Theorem 1.2] The d=2 construction of the divergence-free basis is invalid. In (D.3), T_{l,m}=R∇_{S¹}Y_{l,m}/l with R the 90° rotation. On S¹, ∇_{S¹}Y is tangent, so R∇_{S¹}Y is radial; indeed the explicit formula T_{l,1}(ω)=−(1/√π)sin(lθ)ω confirms ω·T_{l,m}≠0. Such radial modes give G_k=√A Π g_k=0, so they do not generate active noise and Assumption 7.3 is not satisfied. Theorem 1.2 is therefore not established for d=2 as stated. Since a valid tangential divergence-free antisymmetric basis exists in d=2 (e.g., the constant rotation field e_θ), the defect is repairable, but the appendix must be rewritten.
[Section 7, Lemma 7.5] The passage α→0 for the θ-regularised equation needs a uniform L¹_t W^{1,1}_v bound on f^α. The proof after (7.16) asserts E∫₀ᵀ ∫_{B_R} |∇θ(f^α)|² ≤ C(f₀) 'directly from [Des15, Theorem 1] applied to θ²', but the required control of the entropy H(θ²(f^α)) and of the dissipation D(θ²(f^α)) is not spelled out. This is a load-bearing step for Theorem 1.2; please expand it with the same detail as Lemma 6.4.

minor comments (5)

[Section 1.3] The cross-reference 'Assumption 2.12' appears twice; the initial-data condition is displayed as (2.12), not as a numbered assumption. Please correct.
[Assumption 2.1, (2.6)] The support condition '∪ Supp G_k ⊂ {r∈R+ | K^{-1} ≤ r ≤ K} × R^d × S^{d-1}' is notationally ambiguous. Please specify the coordinate decomposition r=|v−v_*|, z=(v+v_*)/2, ω=(v−v_*)/|v−v_*|.
[Appendix D] The sentence 'these modes are normalised in L²(S¹;R²)' and the subsequent claim that T_{l,m} is tangential contradict the explicit formula T_{l,1}(ω)=−(1/√π)sin(lθ)ω, which is radial. The text should be revised consistently.
[Lemma 6.3] The proof uses H both for the positive part of entropy and as a placeholder in inequalities; please define H⁺ and H⁻ explicitly and use them consistently to avoid confusion.
[Section 6.2, Step 4] The phrase 'countable dense set of full-measure times' is used to justify the supremum lower semicontinuity; this is standard but could be stated more explicitly for the exponential estimates.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the existence and entropy theorems are derived from stated assumptions, with covariance matching explicitly presented as design rather than as an independent prediction.

full rationale

The main existence result (Theorem 1.1) is a genuine derivation: starting from explicit assumptions on the kernel (2.3), the noise basis (Assumption 2.1), the mobility coefficient (Assumption 2.3), and the initial datum (2.12), the paper constructs Galerkin/regularised equations, proves uniform estimates (Lemmas 3.7, 5.3, 6.4), passes to the limit in L2 and L1 frameworks, and identifies the limiting martingale. No step uses the conclusion that weak solutions exist to impose a hypothesis, and no fitted quantity is relabelled as a prediction. The covariance-matching discussion in Section 1.1 and Appendix A is explicitly a construction: the paper states that the noise term 'is designed so that the continuum SPDE and the Kac-like Landau particle system have the same covariance structure,' and calls the SPDE a 'formal fluctuating hydrodynamic ansatz.' Thus the match with the particle-system bracket (1.11) is a consistency check of the model definition, not a derived prediction that is secretly an input. The self-citations, e.g., the gradient-flow framework [CDDW24] and the authors' fuzzy-Landau papers [DH25a; DH25b; DGH25], are used for motivation and context, not as the load-bearing argument for Theorem 1.1 or Theorem 1.2; even if those citations were weaker than claimed, the existence proof is self-contained from the stated PDE assumptions. Theorem 1.2's refined entropy inequality is also derived under the explicitly imposed tangential divergence-free condition (Assumption 7.3) and a structure-preserving θ-regularisation; the θ-entropy and θ-dissipation are the natural functionals for that regularised equation, and the inequality is proved by the same compactness scheme rather than assumed. The Appendix D construction issue flagged by the skeptic is a correctness/basis-validity concern in d=2, not a circularity, and it does not affect Theorem 1.1. Overall, no claimed result reduces to its own inputs by construction.

Assumptions & free parameters 4 free parameters · 7 assumptions · 1 invented entities

The paper pays for the result with: smallness of epsilon, fixed finite smooth noise modes, a linearly-regularised mobility near vacuum, exclusion of gamma = -2, and, for Theorem 1.2, a divergence-free condition on the noise basis. All are stated explicitly; none is fitted to data. The genuinely new mathematical content is the existence/entropy machinery assembled for this nonlocal SPDE.

free parameters (4)

epsilon (noise intensity) = epsilon < epsilon_0(lambda), never pinned to a specific number
Smallness drives the exponential-martingale entropy estimate (4.47)-(4.51), hence the uniform-in-alpha estimates in Lemma 6.4. The threshold is inherited from the Desvillettes constant C_2^{Des} via epsilon <= epsilon_0(C_2^{Des}). Chosen small by hand, not fitted to data.
r_0 (regularisation threshold in Assumption 2.3) = r_0 in (0,1), fixed
sigma(r) = sqrt(r) for r > r_0 and sigma(r) ~ r near 0; the interpolation bound C_{r0} enters every estimate. Hand-chosen structural constant.
K (number of active noise modes) = fixed integer; all bounds scale with C_K from (2.5)
The noise is a finite-dimensional projection xi^K; the limit K -> infinity is not treated. The W^{2,inf} cap W^{2,1} norm bound C_K is an input to Assumption 2.1, not derived.
Artificial diffusion alpha (approximation-only parameter) = alpha in (0,1), sent to 0 in Section 6
Needed for the L^2 energy estimates at the regularised level; removed in the L^1 limit. Not a parameter of the final equation.

assumptions (7)

standard math Itô formula for semimartingales and its cut-off version (Lemma 4.4)
Used to derive the entropy balance and the Stratonovich-to-Itô correction structure; standard but technically heavy.
standard math Pitt's inequality (Beckner) bounding integrals of |v-v*|^gamma f^2 by the L^2 norm of the gradient
Invoked in Lemma 5.3, inequality (5.23), to obtain n-uniform L^2 estimates; requires gamma > -2 for the absorption argument.
standard math Skorokhod-Jakubowski representation and Aubin-Lions-Simon compactness
Used in each of the three limit passages (Sections 3.3, 5.2, 6.2).
domain assumption Noise-mode regularity and support (Assumption 2.1): G_k = sqrt(A) Pi g_k smooth, antisymmetric, Sigma ||G_k||_{W^{2,inf} cap W^{2,1}} finite, supports in {K^{-1} <= |v-v*| <= K}
Makes the deterministic Itô-correction terms and quadratic variations well defined; Appendix C shows such bases exist, but this excludes natural non-smooth modes.
domain assumption Regularised mobility sigma (Assumption 2.3): sigma = sqrt(r) away from 0, Lipschitz-linear near 0, sigma(0) = 0
The true square-root mobility is singular at the vacuum; the entire existence theory is for the regularised equation, and the limit sigma -> sqrt(.) is open.
domain assumption Tangential divergence-free condition (1.20), Assumption 7.3, for Theorem 1.2
Causes the Itô correction to cancel in the refined theta-entropy balance; Appendix D constructs such bases. A structural restriction on admissible noise.
domain assumption Initial datum f_0 in L^2_1 cap LlogL (Assumption 2.12)
Needed for the mass/energy/entropy framework; standard for Landau existence.

invented entities (1)

Velocity-pair Gaussian noise xi^K = sum_{k<=K} g_k(v,v*) dot B_k with antisymmetric modes g_k and vector fields G_k = sqrt(A) Pi g_k independent evidence
purpose: Models the conservative thermal fluctuation in the collision (v,v*) space; its quadratic variation is designed to match the Kac-like particle system's fluctuation martingale covariance (1.11).
The covariance is pinned by the particle-system covariation computed in Appendix A (eqs. (1.10)-(1.11)), and Appendices C-D give explicit orthonormal-basis constructions, so the object is realised mathematically. No experimental or simulation validation is offered; the matching itself is formal.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Homogeneous Landau Equation with Regularised Thermal Noise." pith.science (2026). https://pith.science/paper/VFV7QCGB

@misc{pith2026260722329,
  author       = {Pith},
  title        = {Pith review of: The Homogeneous Landau Equation with Regularised Thermal Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFV7QCGB}},
  note         = {Machine review of arXiv:2607.22329}
}

abstract

We introduce and analyze a fluctuating homogeneous Landau equation with regularised thermal noise. The model is motivated by the nonlocal gradient flow structure of the deterministic Landau equation, the fluctuation--dissipation principle, and the covariance of the martingale fluctuations of a Kac-like conservative Landau particle system. The noise is written in Landau-divergence form, is antisymmetric in the pair of velocities, and is interpreted in the Stratonovich sense after introducing a velocity correlation. To handle the vacuum singularity of the square-root mobility and the nonlocal Stratonovich-to-It\^o correction, we replace the mobility by a regular coefficient. For moderately soft potentials, we prove the existence of probabilistic weak solutions to the regularised fluctuating homogeneous Landau equation. The proof is based on a three-level approximation scheme combining Galerkin approximations, coefficient regularisations, artificial diffusion, and compactness in both $L^2$ and $L^1$ frameworks. The solutions satisfy mass conservation, an energy inequality, and the entropy dissipation estimate. Finally, for a special class of admissible noise bases satisfying a tangential divergence-free condition, we obtain a refined entropy inequality in which the expected entropy is non-increasing relative to the initial entropy.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 6 linked inside Pith

[1]

71–305.doi:10.1016/S1874-5792(02)80004-0

Amsterdam: North-Holland, 2002, pp. 71–305.doi:10.1016/S1874-5792(02)80004-0. [Vil25] C. Villani.Fisher Information in Kinetic Theory

doi:10.1016/s1874-5792(02)80004-0 2002
[2]

[DH25a] M

arXiv:2512.04713 [math.AP].url:https: //arxiv.org/abs/2512.04713. [DH25a] M. H. Duong and Z. He.On a fuzzy Landau Equation: Part I. A variational approach

arXiv
[3]

Scalar conservation laws with stochastic forcing

arXiv:2504.07666 [math.AP]. [DH25b] M. H. Duong and Z. He.On a fuzzy Landau Equation: Part II. Solvability results. Preprint, arXiv:2507.10288 [math.AP] (2025). 2025.url:https://arxiv.org/ abs/2507.10288. [DV10] A. Debussche and J. Vovelle. “Scalar conservation laws with stochastic forcing”. In:J. Funct. Anal.259.4 (2010), pp. 1014–1042.issn: 0022-1236.do...

arXiv 2025
[4]

From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules

arXiv:2504.18370 [math.PR]. [FG17] N. Fournier and A. Guillin. “From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules”. In:Annales scientifiques de l’ ´Ecole Normale Sup´ erieure. 4th ser. 50.1 (2017), pp. 157–199.doi:10 . 24033/asens.2318. [FG19] B. Fehrman and B. Gess. “Well-posedness of nonlinear diffusion equ...

arXiv 2017
[5]

100 REFERENCES [GHW24] B

arXiv: 2506.14309 [math.AP].url:https://arxiv.org/abs/2506.14309. 100 REFERENCES [GHW24] B. Gess, D. Heydecker, and Z. Wu.Landau–Lifshitz–Navier–Stokes Equations: Large Deviations and Relationship to the Energy Equality

arXiv
[6]

Kac’s program in kinetic theory

Second. Fluid mechanics, Translated from the third Russian edition by J. B. Sykes and W. H. Reid. Pergamon Press, Oxford, 1987, pp. xiv+539.isbn: 0-08-033933-6. [MM13] S. Mischler and C. Mouhot. “Kac’s program in kinetic theory”. In:Inventiones Mathematicae193.1 (2013), pp. 1–147.doi:10.1007/s00222-012-0422-3. [MRZ25] F. M¨ uller, M. von Renesse, and J. Z...

arXiv 1987
[7]

The variational formulation of the Fokker-Planck equation

arXiv: 2511.10194 [math.PR]. [JKO98] R. Jordan, D. Kinderlehrer, and F. Otto. “The variational formulation of the Fokker-Planck equation”. In:SIAM J. Math. Anal.29.1 (1998), pp. 1–17.issn: 0036-1410.doi:10.1137/S0036141096303359.url:https://doi.org/10. 1137/S0036141096303359. [Kaw98] K. Kawasaki. “Microscopic analyses of the dynamical density functional e...

arXiv 1998
[11]

On the Cauchy problem for Landau equations: Sequential stability, global existence

arXiv:2501 . 00925 [math.AP].url:https://arxiv.org/abs/2501.00925. [Vil96] C. Villani. “On the Cauchy problem for Landau equations: Sequential stability, global existence”. In:Adv. Differ. Equ.1.5 (1996), pp. 793–816.issn: 1079-9389. [Vil98] C. Villani. “On the spatially homogeneous Landau equation for Maxwellian molecules”. In:Math. Models Methods Appl. ...

arXiv 1996

Show all 12 references

[12]

arXiv:2208.13142 [math.PR]. (H. Duong)School of Mathematics, University of Birmingham, UK Email address:h.duong@bham.ac.uk 102 REFERENCES (Z. He)F akult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, 33501 Bielefeld, Germany Email address:zihui.he@uni-bielefeld.d...

arXiv
[2005]

Irreversible Processes in Gases. I. The Diagram Technique

[PB57a] I. Prigogine and R. Balescu. “Irreversible Processes in Gases. I. The Diagram Technique”. In:Physica23.1–5 (1957), pp. 28–42.doi:10 . 1016 / S0031 - 8914(57)94080-8. [PB57b] I. Prigogine and R. Balescu. “Irreversible Processes in Gases. II. The Equations of Evolution”....

1957 doi
[2024]

Long-time behavior, invariant measures, and regularizing effects for stochastic scalar conservation laws

arXiv:2311. 02223 [math.PR]. [GS17] B. Gess and P. E. Souganidis. “Long-time behavior, invariant measures, and regularizing effects for stochastic scalar conservation laws”. In:Comm. Pure Appl. Math.70.8 (2017), pp. 1562–1597.issn: 0010-3640,1097-0312.doi:10. 1002/cpa.21646.ur...

2017 arXiv
[2025]

Entropy dissipa- tion and long-range interactions

arXiv:2511.03347 [math.PR].url:https://arxiv.org/abs/2511. 03347. [ADVW00] R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg. “Entropy dissipa- tion and long-range interactions”. In:Arch. Ration. Mech. Anal.152.4 (2000), pp. 327–355.issn: 0003-9527. [A V04] R. Alexand...

2000

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

71–305.doi:10.1016/S1874-5792(02)80004-0

Amsterdam: North-Holland, 2002, pp. 71–305.doi:10.1016/S1874-5792(02)80004-0. [Vil25] C. Villani.Fisher Information in Kinetic Theory

doi:10.1016/s1874-5792(02)80004-0 2002

[2] [2]

[DH25a] M

arXiv:2512.04713 [math.AP].url:https: //arxiv.org/abs/2512.04713. [DH25a] M. H. Duong and Z. He.On a fuzzy Landau Equation: Part I. A variational approach

arXiv

[3] [3]

Scalar conservation laws with stochastic forcing

arXiv:2504.07666 [math.AP]. [DH25b] M. H. Duong and Z. He.On a fuzzy Landau Equation: Part II. Solvability results. Preprint, arXiv:2507.10288 [math.AP] (2025). 2025.url:https://arxiv.org/ abs/2507.10288. [DV10] A. Debussche and J. Vovelle. “Scalar conservation laws with stochastic forcing”. In:J. Funct. Anal.259.4 (2010), pp. 1014–1042.issn: 0022-1236.do...

arXiv 2025

[4] [4]

From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules

arXiv:2504.18370 [math.PR]. [FG17] N. Fournier and A. Guillin. “From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules”. In:Annales scientifiques de l’ ´Ecole Normale Sup´ erieure. 4th ser. 50.1 (2017), pp. 157–199.doi:10 . 24033/asens.2318. [FG19] B. Fehrman and B. Gess. “Well-posedness of nonlinear diffusion equ...

arXiv 2017

[5] [5]

100 REFERENCES [GHW24] B

arXiv: 2506.14309 [math.AP].url:https://arxiv.org/abs/2506.14309. 100 REFERENCES [GHW24] B. Gess, D. Heydecker, and Z. Wu.Landau–Lifshitz–Navier–Stokes Equations: Large Deviations and Relationship to the Energy Equality

arXiv

[6] [6]

Kac’s program in kinetic theory

Second. Fluid mechanics, Translated from the third Russian edition by J. B. Sykes and W. H. Reid. Pergamon Press, Oxford, 1987, pp. xiv+539.isbn: 0-08-033933-6. [MM13] S. Mischler and C. Mouhot. “Kac’s program in kinetic theory”. In:Inventiones Mathematicae193.1 (2013), pp. 1–147.doi:10.1007/s00222-012-0422-3. [MRZ25] F. M¨ uller, M. von Renesse, and J. Z...

arXiv 1987

[7] [7]

The variational formulation of the Fokker-Planck equation

arXiv: 2511.10194 [math.PR]. [JKO98] R. Jordan, D. Kinderlehrer, and F. Otto. “The variational formulation of the Fokker-Planck equation”. In:SIAM J. Math. Anal.29.1 (1998), pp. 1–17.issn: 0036-1410.doi:10.1137/S0036141096303359.url:https://doi.org/10. 1137/S0036141096303359. [Kaw98] K. Kawasaki. “Microscopic analyses of the dynamical density functional e...

arXiv 1998

[8] [11]

On the Cauchy problem for Landau equations: Sequential stability, global existence

arXiv:2501 . 00925 [math.AP].url:https://arxiv.org/abs/2501.00925. [Vil96] C. Villani. “On the Cauchy problem for Landau equations: Sequential stability, global existence”. In:Adv. Differ. Equ.1.5 (1996), pp. 793–816.issn: 1079-9389. [Vil98] C. Villani. “On the spatially homogeneous Landau equation for Maxwellian molecules”. In:Math. Models Methods Appl. ...

arXiv 1996