A consistent non-linear Fokker-Planck model for a gas mixture of polyatomic molecules

Marlies Pirner

REVIEW 2 major objections 4 minor 22 references

A consistent non-linear Fokker-Planck model for a gas mixture of polyatomic molecules

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read This paper proposes a coupled Fokker-Planck system for polyatomic gas mixtures and proves it conserves mass, momentum, and total energy, keeps temperatures positive, and satisfies an H-theorem with Maxwellian equilibria.

desk verdict First polyatomic mixture Fokker-Planck model with a sound conservation setup, but the H-theorem proof has a false factorization and an inverted parameter ratio, so the central consistency claim is not established. read the letter →

arxiv 2501.04029 v1 pith:TRYY3MPR submitted 2024-12-31 math.AP

Marlies Pirner This is my paper · ORCID

classification math.AP MSC 35Q8482C4035Q2076P05

keywords multi-fluidmixturekineticmodelFokker-PlanckapproximationpolyatomicmoleculesH-theoremtemperaturerelaxationgas

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

This paper proposes a kinetic Fokker-Planck model for a mixture of two polyatomic gas species, each carrying two temperatures: one for translational motion and one for internal (rotational or vibrational) degrees of freedom. The model couples the two species equations to auxiliary evolution equations for their Maxwellian reference states and to an algebraic energy-matching closure, allowing translational and internal temperatures to relax at different rates. The author proves that this coupled system is consistent: it conserves mass, momentum, and total energy; keeps all temperatures positive under a parameter bound; and satisfies an H-theorem (entropy decay) under Assumption 5.1, with equality only at two Maxwellians sharing a common velocity and common temperature. If the proof is correct, this is the first consistent Fokker-Planck model for polyatomic gas mixtures, giving a tractable kinetic description for applications where internal energy relaxation matters.

What carries the argument

The central object is the coupled system (2), (6), (12): the two Fokker-Planck equations for $f_1$ and $f_2$, the auxiliary evolution equations for the reference Maxwellians $M_k$, and the relaxation equation for the internal temperatures $\Theta_k$. The load-bearing identity is the internal-energy matching condition (5), $\frac{d}{2} n_k \Lambda_k = \frac{d}{2} n_k T^t_k + \frac{l_k}{2} n_k T^r_k - \frac{l_k}{2} n_k \Theta_k$, which makes the momentum and internal-energy exchange computed from (2) and (6) coincide, together with the total temperature definition $T_k = (d\Lambda_k + l_k\Theta_k)/(d+l_k) = (dT^t_k + l_k T^r_k)/(d+l_k)$. The H-theorem proof rewrites every collision operator in divergence form and estimates the entropy production as a sum of non-positive terms; the free parameters $\alpha$, $\delta$, $\gamma$ and the ratio condition $Z^r_2/Z^r_1 = (d+l_1)/(d+l_2)$ are chosen so that the cross-species terms cancel.

What would settle it

Numerically solve the space-homogeneous system (2), (6), (12) for two species with unequal initial translational and internal temperatures and parameter choices outside Assumption 5.1 (for instance $\gamma > m_1(1-\delta)$ or $Z^r_2/Z^r_1 \neq (d+l_1)/(d+l_2)$); if total energy drifts or the entropy (24) increases at any time, the claimed consistency fails. Equivalently, find initial data satisfying the paper's assumptions for which the explicit solution (23) shows $\Theta_k - \Lambda_k$ changing sign, contradicting the sign-preservation argument used in Lemma 3.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a Fokker-Planck description of a polyatomic gas mixture can be made consistent by augmenting the two species equations (2) with separate evolution equations for the reference Maxwellians $M_k$ (equation (6)) and the algebraic closure (5) that defines the partial translational temperature $\Lambda_k$ from the internal-energy balance. With this closure, the author proves conservation of mass, momentum, and total energy (Theorem 1), positivity of all temperatures and internal energies (Theorem 2) under the bound $0 \le \gamma \le m_1(1-\delta)$, and an H-theorem (Theorem 4, Corollary 5) under the parameter restrictions of Assumption 5.1, with equality if and only if $f_1$ and $f_2$ are Maxwellians with equal mean velocity and equal temperatures $T = T^r_1 = T^r_2 = T^t_1 = T^t_2 = \Lambda_1 = \Lambda_2 = \Theta_1 = \Theta_2$. The result extends the known monoatomic multi-species Fokker-Planck theory to polyatomic molecules by letting each species have two temperatures that relax at separate rates.

Load-bearing premise

The load-bearing premise is that the auxiliary evolution equations for the Maxwellians $M_k$ and the algebraic energy-matching condition (5) can be imposed as an independent closure on the Fokker-Planck system (2) without changing the physics; this closure is assumed rather than derived from the original collision operators, and if it fails, or if the parameter and initial-data restrictions of Assumption 5.1 are not met, the conservation, positivity, and entropy results do not follow.

Editorial extensions

If this is right

The model gives a kinetic description of polyatomic gas mixtures in which translational and internal temperatures relax at different rates, while mass, momentum, and total energy are still conserved at every instant.
The H-theorem guarantees that any initial distribution evolves toward the expected local equilibrium: two Maxwellians with equal mean velocity and equal temperatures, so the model reproduces the Maxwellian equilibria of the underlying Boltzmann or Landau-Fokker-Planck equations.
The positivity result (Theorem 2) ensures that the temperatures $\Lambda_k$, $\Theta_k$ and the internal energies $d\Lambda_{kj}+l_k\Theta_{kj}$ stay physically meaningful whenever the parameters obey $0 \le \gamma \le m_1(1-\delta)$.
Because only binary interactions are assumed, the two-species construction extends directly to $N$ species by summing pairwise interaction terms (Remark 4), making the model a building block for multi-component polyatomic mixtures.

Reading between the lines

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

Editorial inference: the same closure strategy—coupling Fokker-Planck equations to auxiliary Maxwellian evolution equations—could be applied to BGK or ES-BGK models for polyatomic mixtures to obtain thermodynamically consistent multi-temperature models.
Editorial inference: the free parameters $\alpha$, $\delta$, $\gamma$, $Z^r_1$, $Z^r_2$ are left unspecified; a testable next step is to calibrate them against macroscopic transport coefficients or direct simulation Monte Carlo data for specific polyatomic gases.
Editorial inference: the entropy inequality proof relies on sign preservation of $\Theta_k - \Lambda_k$ and $T^r_k - T^t_k$; this suggests the sign condition on initial data in Assumption 5.1 may be necessary, not just sufficient, for monotone entropy decay.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Short version: you should know this paper is the first Fokker-Planck model I know of for a polyatomic gas mixture, and the conservation/positivity parts are mostly sound. But the H-theorem proof, which is the centerpiece, has an algebraic error that breaks the argument whenever the species have different internal degrees of freedom. The Z^r ratio in Assumption 5.1 is also inverted, so the entropy is not well-defined as written.

What is genuinely new: the combination of the monoatomic mixture framework with polyatomic internal energy variables and the two-temperature relaxation. The model structure is sensible, the free parameters give flexibility, and the extension to N species is a reasonable remark. The conservation proof (Theorem 1) and positivity (Theorem 2) appear correct; I did not find a problem there.

The soft spots. The stress-test is right: equation (26) claims T21−T12 factors as (α−ε(1−α)(d+l1)/(d+l2))((d+l2)/(d+l1)T2−T1). Expanding the correct expression for T21−T12 gives an extra term (1−(d+l2)/(d+l1))T2. That term vanishes only when l1=l2. For the paper's intended case of different internal degrees, the missing term breaks the sign argument, and the product X(T21−T12) is not definite. The H-theorem is therefore not established. Separately, the assumption Z^r_2/Z^r_1=(d+l1)/(d+l2) is the inverse of what's needed for the single z in the entropy (24); with the printed ratio, the entropy is ill-defined. These are internal algebraic failures, not disputes with any external result. The auxiliary closure (5)-(6) is assumed rather than derived; that is a modeling choice, but it does add to the weight on the H-theorem. Several moment calculations are sketched; that's a presentation issue, minor by comparison.

Bottom line: the model construction is plausible and the paper fills a real gap, but the central consistency claim is not proven as written. The errors look fixable—the Z^r ratio is a typo, and the factorization error might be corrected by adjusting the definition of T12/T21 or the parameter constraints—but that requires real rework, not a cosmetic patch. I'd send it to a serious referee because the topic matters and the framework is worth salvaging, but I would not cite it for the H-theorem until the proof is repaired.

Recommendation: engage with it conditional on major revision; ask the author to fix the algebra and verify the sign argument, or explicitly restrict the H-theorem to the case l1=l2.

Referee Report

2 major / 4 minor

Summary. The manuscript proposes a two-species nonlinear Fokker-Planck model for a polyatomic gas mixture, coupling the distribution functions f1 and f2 to auxiliary Maxwellian relaxation equations for M1 and M2 through an algebraic internal-energy matching condition. It claims consistency of the model: conservation of mass, momentum, and total energy; positivity of temperatures; an H-theorem for a weighted entropy; and a characterization of equilibria as two Maxwellians with equal mean velocity and equal translational and internal temperatures. The conservation and positivity arguments are formal and elementary, while the H-theorem is the main mathematical contribution.

Significance. If the H-theorem were correctly established, the model would address a genuine gap in the kinetic-theory literature on Fokker-Planck models for polyatomic gas mixtures with different internal degrees of freedom. The manuscript is clearly organized, states its modeling assumptions explicitly, and the simple conservation and positivity proofs are transparent. However, the central entropy proof contains a false algebraic identity, and the entropy density is not well-defined under the ratio stated in Assumption 5.1. As it stands, the main consistency claim is not established.

major comments (2)

[Section 5, Eq. (26)] The factorization of T21 - T12 in Eq. (26) is algebraically incorrect. From (16), (17), and the assumptions Lambda12 = Theta12 and Lambda21 = Theta21, writing Yk = d Lambda k + lk Theta k and Tkj = Ykj/(d + lk), one obtains T21 - T12 = (epsilon(1-alpha)(d+l1)/(d+l2) - alpha) T1 + (1 - epsilon(1-alpha) - (1-alpha)(d+l2)/(d+l1)) T2. The claimed right-hand side of (26), (alpha - epsilon(1-alpha)(d+l1)/(d+l2))((d+l2)/(d+l1) T2 - T1), expands to the same coefficient of T1 but to a coefficient of T2 that differs by (d+l2)/(d+l1) - 1. Hence the identity holds only when l1 = l2, precisely the case the paper is designed to go beyond. As a consequence, the sign of X(T21 - T12), with X = (d Lambda1 + l1 Theta1) - (d Lambda2 + l2 Theta2), is not controlled by the parameter restriction on alpha: for l1 != l2 the two linear forms X and T21 - T12 have different zero sets, so their product can take both signs for positive T1, T2. Since this sign estimate is the only mechanism producing the inequality I <= 0 in Theorem 4, the H-theorem and Corollary 5 are not established by the submitted proof. This is a load-bearing error, not a typographical one.
[Section 5, Assumption 5.1 and Eq. (24)] The assumption on the relaxation rates is inconsistent with the definition of the entropy weight. Assumption 5.1 states Z2^r/Z1^r = (d+l1)/(d+l2), but Eq. (24) defines 1/z = (1/Zk^r)(d+lk)/d. For z to be independent of k, one needs Z2^r/Z1^r = (d+l2)/(d+l1). As printed, the entropy (24) carries two possibly different weights for the M1 ln M1 and M2 ln M2 terms, and the subsequent use of a common factor z in the proof of Theorem 4 is not justified. The ratio should be inverted, or the notation defining z must be changed consistently.

minor comments (4)

[Section 5, proof of Theorem 4] In the line preceding Eq. (26), the denominator d - l1 should evidently be d + l1; the printed expression is a typographical error.
[Section 3, Eq. (12) and model summary] In the summary of the model, the last term in the equation for partial_t(nk Theta k) is written as Tkj - Theta_k^r, but the rest of the paper uses Theta_k; this should be corrected.
[Section 6, Eq. (35)] After multiplying (31) by |eta|^2, the equation appears to describe the evolution of Theta_2, not Lambda_2; the left-hand side partial_t(Lambda_2) is inconsistent with the right-hand side terms containing T21 - Theta_2.
[Assumption 5.1] The condition on gamma is typeset ambiguously as "gamma = epsilon 1 + epsilon m1(1 - delta)"; the intended closed form should be written explicitly, since the subsequent proof of Theorem 4 refers to this condition to eliminate velocity terms.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity; conservation and entropy are enforced by openly chosen free parameters, so the 'consistency' theorems are verifications rather than predictions.

self definitional [Section 3, Theorem 1, equations (16)-(17)]
"Assume that Λ 12 and Θ 12 satisfy (16)... Then we have conservation of total energy ... provided that dΛ 21 + l2Θ 21 = [expression (17)]."

The combination dΛ21+l2Θ21 is not fixed by an independent constitutive relation; equation (17) is exactly the value that makes the sum of the two energy-exchange terms vanish. The proof substitutes (14), (15), (16), and (17) and concludes conservation, so the theorem's conclusion is the defining equation. This is transparent model calibration rather than a hidden circularity, and the paper explicitly says free parameters are 'used to fix exchange terms of momentum and energy.'

full rationale

The paper is self-contained in the sense relevant to circularity: it does not fit parameters to data and relabel them as predictions, and it does not import any uniqueness theorem from the author's prior work to forbid alternatives. The model is explicitly a construction with free parameters, and the statements labelled theorems are conditional verifications that those parameters can be chosen to deliver conservation and an entropy inequality. The one strict self-definitional element is equation (17): dΛ21+l2Θ21 is defined as the cancellation condition for the two energy-exchange terms, so Theorem 1's conservation result restates that defining choice. This is transparent in the text and does not affect the independent content of the H-theorem, which uses additional explicit assumptions (Λ12=Θ12, parameter restrictions, and the Z^r ratio condition) and nontrivial algebra. Citations [20] and [21] are by the same author, but they motivate the construction and supply an elementary momentum-conservation lemma with stated parameter conditions; they are not a black-box uniqueness argument and are not load-bearing for the polyatomic H-theorem. A separate note on correctness rather than circularity: the factorization in (26) appears algebraically fragile when l1≠l2, but that is a correctness risk outside this pass. Overall circularity score 2.

Assumptions & free parameters 6 free parameters · 6 assumptions · 3 invented entities

The model is a constructed system with several free parameters and auxiliary Maxwellian distributions; the consistency theorems hold only under the stated closure and Assumption 5.1. The entropy functional and auxiliary entities are mathematically introduced for the proofs, without independent physical evidence.

free parameters (6)

epsilon
Ratio c12 = epsilon c21 of interspecies friction constants, introduced in (4); a free knob of the model.
delta
Weight in u12 = delta u1 + (1 - delta) u2, equation (14); constrained by positivity (20) and Assumption 5.1.
alpha
Weight in the energy exchange condition (16); restricted by Assumption 5.1 for the H-theorem.
gamma
Coefficient of |u1-u2|^2 in (16), controlling velocity-dependent energy exchange; bounded by (20) and Assumption 5.1.
Z^r_1, Z^r_2
Internal temperature relaxation rates in equation (12); their ratio is fixed in Assumption 5.1.
c11, c22, c12, c21
Friction constants for intra- and inter-species collisions in (2); assumed positive and time-independent in Assumption 5.1.

assumptions (6)

standard math The distribution functions f1, f2 are positive and decay sufficiently fast at infinity so that integrations by parts are valid.
Used throughout the moment and entropy computations in Sections 3-5.
domain assumption Only binary interactions occur and chemical reactions are absent.
Stated in the introduction and used to justify the sum of two interaction operators per species.
domain assumption The internal energy variable eta is continuous, and the mean internal energy is zero (bar-eta_k = 0).
Introduced after (1); needed for the Gaussian form of the Maxwellians.
ad hoc to paper The closure (5) holds: d/2 n_k Lambda_k = d/2 n_k T^t_k + l_k/2 n_k T^r_k - l_k/2 n_k Theta_k, and the Maxwellians M_k evolve by the auxiliary equations (6).
This is the defining closure of the model, imposed rather than derived; the consistency and H-theorem apply to this closed system.
ad hoc to paper Assumption 5.1 holds: parameter restrictions on gamma, delta, alpha, the initial sign condition for Theta_k(0) - Lambda_k(0), and the ratio Z^r_2/Z^r_1.
Required for the H-theorem in Theorem 4 and Corollary 5.
ad hoc to paper The weighted entropy (24) with z M ln M terms is the appropriate Lyapunov functional.
The proof of the entropy inequality uses this specific functional; no physical derivation of the weight z is given.

invented entities (3)

Auxiliary interspecies Maxwellians M12 and M21
purpose: Define the Fokker-Planck interaction operators between species in (2)
Their parameters n_kj, u_kj, Lambda_kj, Theta_kj are determined by conservation and symmetry conditions, not by independent physics.
Auxiliary single-species Maxwellians M_k, M_tilde_k, M_tilde_kj
purpose: Define the relaxation targets for the internal energy equations (6) and (8)-(9)
Mathematical constructs used to separate translational and internal temperature relaxation; no independent experimental handle.
Weighted entropy H = integral(f1 ln f1 + z M1 ln M1) + integral(f2 ln f2 + z M2 ln M2)
purpose: Provide a Lyapunov functional for the H-theorem
The weight z and the inclusion of M ln M terms are chosen so that the entropy inequality can be proven.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A consistent non-linear Fokker-Planck model for a gas mixture of polyatomic molecules." pith.science (2026). https://pith.science/paper/TRYY3MPR

@misc{pith2026250104029,
  author       = {Pith},
  title        = {Pith review of: A consistent non-linear Fokker-Planck model for a gas mixture of polyatomic molecules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRYY3MPR}},
  note         = {Machine review of arXiv:2501.04029}
}

read the original abstract

We consider a multi component gas mixture with translational and internal energy degrees of freedom without chemical reactions assuming that the number of particles of each species remains constant. We will illustrate the derived model in the case of two species, but the model can be generalized to multiple species. The two species are allowed to have different degrees of freedom in internal energy and are modeled by a system of kinetic Fokker-Planck equations featuring two interaction terms to account for momentum and energy transfer between the species. We prove consistency of our model: conservation properties, positivity of the temperatures, H-theorem and we characterize the equilibrium as two Maxwell distributions where all temperatures coincide.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Agrawal, S

S. Agrawal, S. Singh, and S. Ansumali. Fokker–planck model for b inary mixtures. Journal of Fluid Mechanics , 899:25, 2020

work page 2020
[2]

Andries and B

P. Andries and B. Perthame. The es-bgk model equation with cor rect prandtl number. In AIP Conference Proceedings, volume 30, pages 30–36, 2001

work page 2001
[3]

Bernard, A

F. Bernard, A. Iollo, and G. Puppo. Polyatomic models for rareﬁe d ﬂows. 2017. submitted

work page 2017
[4]

Cercignani

C. Cercignani. The Boltzmann equation and its applications . Springer, 1975

work page 1975
[5]

Cercignani

C. Cercignani. Rareﬁed Gas Dynamics, From Basic Concepts to Actual Calcula - tions. Cambridge University Press, 2000

work page 2000
[6]

Chapman and T

S. Chapman and T. Cowling. The Mathematical Theory of Non–Uniform Gases . Cambridge University Press, Cambridge, third edition edition, 1970

work page 1970
[7]

Crouseilles and F

N. Crouseilles and F. Filbet. Numerical approximation of collisional p lasmas by high order methods. Journal of Computational Physics , 201(2):546–572, 2004

work page 2004
[8]

Filbet and C

F. Filbet and C. Negulescu. Fokker–planck multi-species equation s in the adiabatic asymptotics. Journal of Computational Physics , 471:111642, 2022

work page 2022

Show all 22 references

[9]

Gorji and P

H. Gorji and P. Jenny. A kinetic model for gas mixtures based on a fokker-planck equation. In Journal of Physics: Conference series , volume 362, page 012042, 2012

2012
[10]

Habbershaw, C

E. Habbershaw, C. Hauck, J. Hu, and J. Haack. A nonlinear, co nservative, entropic fokker-planck model for multi-species collisions. arXiv, 2404.11775, 2024

2024 arXiv
[11]

C. Hepp, M. Grabe, and K. Hannemann. A kinetic fokker-planck approach to model hard-sphere gas mixtures. Physics of Fluids , 32(2):1–18, 2020

2020
[12]

M. Herda. On massless electron limit for a multispecies kinetic syst em with external magnetic ﬁeld. Journal of Diﬀerential Equations , 260:7861–7891, 2016

2016
[13]

J. Hu, A. J¨ ungel, and N. Zamponi. Global weak solutions for a no n-local multi- species fokker-planck-landau system. arXiv, 2305.17447, 2024

2024 arXiv
[14]

J. Kelly. Semiclassical Statistical Mechanics . 2002. Lecture notes. 17

2002
[15]

Klingenberg, M

C. Klingenberg, M. Pirner, and G. Puppo. A consistent kinetic mo del for a two- component mixture with an application to plasma. Kinetic and Related Models , 10:445–465, 2017

2017
[16]

Mathiaud and L

J. Mathiaud and L. Mieussens. A fokker-planck model of the bo ltzmann equation with correct prandtl number for polyatomic gases. Journal of Statistical Physics , 168:1031–1055, 2017

2017
[17]

Mathiaud and L

J. Mathiaud and L. Mieussens. Bgk and fokker-planck models of the boltzmann equation for gases with discrete levels of vibrational energy. Journal of Statistical Physics, 178(5):1076–1095, 2020

2020
[18]

T. Morse. Kinetic model for gases with internal degrees of fre edom. The Physics of Fluids , 7:159–169, 1964

1964
[19]

Nagel, L

A. Nagel, L. Basov, and M. Grabe. Modeling of polyatomic gas ﬂow s within a ki- netic fokker-planck approach using a direct modeling method. In Aerospace Europe Conference 2023 – 10th EUCASS – 9th CEAS , 2023

2023
[20]

M. Pirner. A bgk model for gas mixtures of polyatomic molecules a llowing for slow and fast relaxation of the temperatures. Journal of Statistical Physics , 173(6):1660– 1687, 2018

2018
[21]

M. Pirner. A consistent kinetic fokker-planck model for gas mix tures. Journal of Statistical Physics, 2024. accepted for publication

2024
[22]

Wu, T.-C

H. Wu, T.-C. Lin, and C. Liu. Diﬀusion limit of kinetic equations for mu l- tiple species charged particles. Archive for Rational Mechanics and Analysis , 215:419–441, 2015. 18

2015

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Agrawal, S

S. Agrawal, S. Singh, and S. Ansumali. Fokker–planck model for b inary mixtures. Journal of Fluid Mechanics , 899:25, 2020

work page 2020

[2] [2]

Andries and B

P. Andries and B. Perthame. The es-bgk model equation with cor rect prandtl number. In AIP Conference Proceedings, volume 30, pages 30–36, 2001

work page 2001

[3] [3]

Bernard, A

F. Bernard, A. Iollo, and G. Puppo. Polyatomic models for rareﬁe d ﬂows. 2017. submitted

work page 2017

[4] [4]

Cercignani

C. Cercignani. The Boltzmann equation and its applications . Springer, 1975

work page 1975

[5] [5]

Cercignani

C. Cercignani. Rareﬁed Gas Dynamics, From Basic Concepts to Actual Calcula - tions. Cambridge University Press, 2000

work page 2000

[6] [6]

Chapman and T

S. Chapman and T. Cowling. The Mathematical Theory of Non–Uniform Gases . Cambridge University Press, Cambridge, third edition edition, 1970

work page 1970

[7] [7]

Crouseilles and F

N. Crouseilles and F. Filbet. Numerical approximation of collisional p lasmas by high order methods. Journal of Computational Physics , 201(2):546–572, 2004

work page 2004

[8] [8]

Filbet and C

F. Filbet and C. Negulescu. Fokker–planck multi-species equation s in the adiabatic asymptotics. Journal of Computational Physics , 471:111642, 2022

work page 2022