Discussion on kinetic and asymptotic diffusion modelsfor gaseous mixtures: Fick vs. Maxwell-Stefan
Bérénice Grec1
in collaboration with L. Boudin and V. Pavan
1MAP5 – Université Paris Descartes, France
Franco-italian meeting on kinetic theory and singular parabolic equationsUniv. Paris Diderot, March 16th, 2017
Diffusion models for mixtures: Fick and Maxwell-Stefan
I Mixture of p ≥ 2 speciesI ni : number of molecules of species iI Ni : flux of species i
I n =
n1...
np
, N =
N1...
Np
I mass conservation: ∂tn +∇ · N = 0
(Generalized) Fick’s law:
N = −F (n)∇x n
Maxwell-Stefan’s equations:
−∇x n = S(n)N
Properties of the matrices F and SI F (n) and S(n) are not invertible (rank p − 1).I Well-posedness of the Cauchy problem [Giovangigli, Bothe, Jüngel & Stelzer]I Use of Moore-Penrose pseudo-inverse: structural similarity
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 2/15
Fick vs. Maxwell-Stefan
Formal analogy of the two systems,but Fick’s and Maxwell-Stefan’s are not obtained in the same way!
Obtention of Fick’s lawI Thermodynamical considerations (entropy minimization)I Thermodynamics of irreversible processes approach
Obtention of Maxwell-Stefan’s equationsI Mechanical considerations on forcesI Extended thermodynamics approach
IdeaUse kinetic theory and study the diffusion asymptotics
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 3/15
Kinetic framework
I Distribution function fi (t, x , v), t ≥ 0, x ∈ Ω, v ∈ R3.
I Number of molecules ni =∫
fidv .
I Mixture of ideal gases without chemical reaction
I Collisions: (mi , v ′), (mj , v ′∗) 7→ (mi , v), (mj , v∗), obtained by the conservationof momentum and kinetic energy
v ′ = 1mi + mj
(miv + mjv∗ + mj |v − v∗|σ)
v ′∗ = 1mi + mj
(miv + mjv∗ −mi |v − v∗|σ)
with mi : mass of species i and σ ∈ S2.
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 4/15
Collision operators
I Collision operator
Qij(f , g)(v) =∫
v∗∈R3
∫σ∈S2
Bij(v , v∗, σ) [f ′g ′∗ − f g∗] dσ dv∗
I Weak form:∫Qij(f , g)ψdv = −1
2
∫∫∫Bij [f ′g ′∗ − f g∗] [ψ′ − ψ] dσ dv∗ dv
I Conservation properties: ∫R3
Qij(f , g)(v) dv = 0∫R3
Qij(f , g)(v)(
mi v12mi |v |2
)dv +
∫R3
Qji (g , f )(v)(
mj v12mj |v |2
)dv = 0
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 5/15
Kinetic model for mixtures
I Diffusive scaling : ε ∼ Kn ∼ Ma
∂t fi + v · ∇x fi =∑
jQij(fi , fj), ∀i
I Study of the equilibrium (H-theorem) [Desvillettes, Monaco, Salvarani]
Mi (v) =( mi2πkT
)3/2exp
(−mi |v |22kT
)I Well-posedness of the Cauchy problem [Briant, Daus]I Asymptotic analysis: introduce the linearized operator L, whose i-th
component is
[Lg ]i =M−1/2i
∑j
ninj(Qij(Mi ,M1/2
j gj) + Qji (M1/2i gi ,Mj)
).
I Compactness of the operator [Boudin, Grec, Pavić, Salvarani]I Spectral gap [Daus, Jüngel, Mouhot, Zamponi]
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 6/15
Kinetic model for mixtures
I Diffusive scaling : ε ∼ Kn ∼ Ma
ε∂t fi + v · ∇x fi = 1ε
∑j
Qij(fi , fj), ∀i
I Study of the equilibrium (H-theorem) [Desvillettes, Monaco, Salvarani]
Mi (v) =( mi2πkT
)3/2exp
(−mi |v |22kT
)I Well-posedness of the Cauchy problem [Briant, Daus]I Asymptotic analysis: introduce the linearized operator L, whose i-th
component is
[Lg ]i =M−1/2i
∑j
ninj(Qij(Mi ,M1/2
j gj) + Qji (M1/2i gi ,Mj)
).
I Compactness of the operator [Boudin, Grec, Pavić, Salvarani]I Spectral gap [Daus, Jüngel, Mouhot, Zamponi]
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 6/15
Fick’s and Maxwell-Stefan’s equations are not obtained in the same wayfrom the Boltzmann equation!
Perturbative method (Fick)I Based on Chapman-Enskog expansion and the use of L−1 [Bardos, Golse,
Levermore]
Moment method (Maxwell-Stefan)I Based on the ansatz (below) and the use of L [Levermore], [Müller, Ruggieri]
Ansatz
f εi (t, x , v) = nεi (t, x)( mi2πkT
)3/2exp
(− mi2kT |v − εu
εi (t, x)|2
)thus
∫v
f εi dv = nεi and∫
vvf εi dv = εnεi uεi .
Maxwell-Stefan’s equations can be written
−∇x ni = 1∑ni
∑j 6=i
njni (ui − uj)Dij
.
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 7/15
ε∂t f εi + v · ∇x f εi = 1ε
∑j
Qij(f εi , f εj ), ∀i
I Mass conservationMoment of order 0:
ε∂
∂t
(∫R3
f εi (v) dv)
+∇x ·(∫
R3v f εi (v) dv
)= 0,
where the collision operator Qij(fi , fj) vanishes by invariance.
∂tnεi +∇x · (nεi uεi ) = 0.
I Momentum equationMoment of order 1:
ε∂
∂t
∫R3
v f εi (v) dv +∫R3
v (v · ∇x f εi (v)) dv = 1ε
∑j 6=i
∫R3
v Qij(f εi , f εj )(v) dv
where the mono-species collision term vanishes by invariance.Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 8/15
Computation of the divergence term
ε∂
∂t
∫R3
v f εi (v) dv +∫R3
v (v · ∇x f εi (v)) dv = 1ε
∑j 6=i
∫R3
v Qij(f εi , f εj )(v) dv
I Use of the Ansatz, translation in v
∂
∂x
(∫R
v2 f εi (v) dv)∝ ∂
∂x nεi∫R
(v + εuiε)2e−mi |v |2/2kT dv
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 9/15
Computation of the divergence term
ε∂
∂t
∫R3
v f εi (v) dv +∫R3
v (v · ∇x f εi (v)) dv = 1ε
∑j 6=i
∫R3
v Qij(f εi , f εj )(v) dv
I Use of the Ansatz, translation in v + parity argument
∂
∂x
(∫R
v2 f εi (v) dv)∝ ∂
∂x nεi∫R
(v2 + ε2(ui
ε)2)
e−mi |v |2/2kT dv
I In terms of macroscopic quantities∂
∂x
(∫R
v2 f εi (v) dv)
= kTmi
∂nεi∂x + ε2
∂
∂x
(nεi (ui
ε)2)
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 9/15
Galilean transformations
I Galilean invariance:
For any translation w ∈ R3, and any rotation Θ ∈ O+(R3):
Bij(v + w , v∗ + w , σ) = Bij(v , v∗, σ) = Bij(Θv ,Θv∗,Θσ), ∀v , v∗, σ
Galilean transformations for the collision rules
Let πij(v , v∗, σ) := v ′ = 1mi + mj
(miv + mjv∗ + mj |v − v∗|σ).
For any translation w ∈ R3, and any rotation Θ ∈ O+(R3):πij(v + w , v∗ + w , σ) = πij(v , v∗, σ) + wπij(Θv ,Θv∗,Θσ) = Θπij(v , v∗, σ)
The same holds for v ′∗.
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 10/15
Computation of the collision term
ε∂
∂t
∫R3
v f εi (v) dv +∫R3
v (v · ∇x f εi (v)) dv = 1ε
∑j 6=i
∫R3
v Qij(f εi , f εj )(v) dv
Let Γεij(εui , εuj) :=∫R3
v Qij(f εi , f εj )(v) dv
∝ nεi nεj∫∫∫v ,v∗,σ
Bij(v , v∗, σ)(
e−mi (v′−εui )2
2kT −mj (v′∗−εuj )2
2kT − e−mi (v−εui )2
2kT −mj (v∗−εuj )2
2kT
)(v ′ − v)
Find Dij such that limε→0
Γεij(εui , εuj)ε
= Dijninj(ui − uj).
I Invariance property: from v ′ = πij(v , v∗, σ) (and the same for v ′∗)
Γεij(εui + w , εuj + w) = Γεij(εui , εuj), Γεij(εΘui , εΘuj) = ΘΓεij(εui , εuj).
I LetHε
ij (u) :=Γεij(εu, 0)
ε, where u := ui − uj .
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 11/15
I Expand
Γεij(εu, 0) ∝ nεi nεj∫∫∫
Bij
(exp
[− mi2kT (v ′2 − 2εv ′ · u + ε2u2)
]exp
[− mj2kT v ′2∗
]− exp
[− mi2kT (v2 − 2εv · u + ε2u2)
]exp
[− mj2kT v∗2
] )(v − v ′)
by the conservation of kinetic energy.I Expansion in ε of the exponentials:
∝ nεi nεj∫∫∫
Bij exp[− mi2kT v2 − mj
2kT v∗2]
(v − v ′)
I Matrix form of the operator Lij(u) = limε→0
Γεijε , involving (v ′ − v)(v ′ − v)ᵀ
I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:
Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,
symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15
I Expand
Γεij(εu, 0) ∝ nεi nεj∫∫∫
Bij
(exp
[− mi2kT (v2 − 2εv ′ · u + ε2u2)
]exp
[− mj2kT v2
∗
]− exp
[− mi2kT (v2 − 2εv · u + ε2u2)
]exp
[− mj2kT v∗2
] )(v − v ′)
by the conservation of kinetic energy.I Expansion in ε of the exponentials:
∝ nεi nεj∫∫∫
Bij exp[− mi2kT v2 − mj
2kT v∗2]
(v − v ′)
I Matrix form of the operator Lij(u) = limε→0
Γεijε , involving (v ′ − v)(v ′ − v)ᵀ
I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:
Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,
symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15
I Expand
Γεij(εu, 0) ∝ nεi nεj∫∫∫
Bij
(exp
[− mi2kT (v2 − 2εv ′ · u + ε2u2)
]exp
[− mj2kT v2
∗
]− exp
[− mi2kT (v2 − 2εv · u + ε2u2)
]exp
[− mj2kT v∗2
] )(v − v ′)
by the conservation of kinetic energy.I Expansion in ε of the exponentials:
∝ nεi nεj∫∫∫
Bij exp[− mi2kT v2 − mj
2kT v∗2]
(1 + ε
mikT v ′ · u − 1− εmi
kT v · u + O(ε2))
(v − v ′)
I Matrix form of the operator Lij(u) = limε→0
Γεijε , involving (v ′ − v)(v ′ − v)ᵀ
I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:
Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,
symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15
I Expand
Γεij(εu, 0) ∝ nεi nεj∫∫∫
Bij
(exp
[− mi2kT (v2 − 2εv ′ · u + ε2u2)
]exp
[− mj2kT v2
∗
]− exp
[− mi2kT (v2 − 2εv · u + ε2u2)
]exp
[− mj2kT v∗2
] )(v − v ′)
by the conservation of kinetic energy.I Expansion in ε of the exponentials:
∝ nεi nεj∫∫∫
Bij exp[− mi2kT v2 − mj
2kT v∗2]
(ε
mikT (v ′ − v) · u + O(ε2)
)(v − v ′)
I Matrix form of the operator Lij(u) = limε→0
Γεijε , involving (v ′ − v)(v ′ − v)ᵀ
I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:
Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,
symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15
I Expand
Γεij(εu, 0) ∝ nεi nεj∫∫∫
Bij
(exp
[− mi2kT (v2 − 2εv ′ · u + ε2u2)
]exp
[− mj2kT v2
∗
]− exp
[− mi2kT (v2 − 2εv · u + ε2u2)
]exp
[− mj2kT v∗2
] )(v − v ′)
by the conservation of kinetic energy.I Expansion in ε of the exponentials:
∝ nεi nεj∫∫∫
Bij exp[− mi2kT v2 − mj
2kT v∗2]
(ε
mikT (v ′ − v) · u + O(ε2)
)(v − v ′)
I Matrix form of the operator Lij(u) = limε→0
Γεijε , involving (v ′ − v)(v ′ − v)ᵀ
I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:
Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,
symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15
I Expand
Γεij(εu, 0) ∝ nεi nεj∫∫∫
Bij
(exp
[− mi2kT (v2 − 2εv ′ · u + ε2u2)
]exp
[− mj2kT v2
∗
]− exp
[− mi2kT (v2 − 2εv · u + ε2u2)
]exp
[− mj2kT v∗2
] )(v − v ′)
by the conservation of kinetic energy.I Expansion in ε of the exponentials:
∝ nεi nεj∫∫∫
Bij exp[− mi2kT v2 − mj
2kT v∗2]
(ε
mikT (v ′ − v) · u + O(ε2)
)(v − v ′)
I Matrix form of the operator Lij(u) = limε→0
Γεijε , involving (v ′ − v)(v ′ − v)ᵀ
I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:
Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,
symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15
Asymptotics
Formal limitni (t, x) = lim
ε→0nεi (t, x), Ni (t, x) = lim
ε→0nεi (t, x)uεi (t, x)
I Mass conservation
∂tni +∇x · Ni = 0
I Momentum equation
ε2mikT
(∂t(nεi uεi ) +∇x · (nεi uεi ⊗ uεi )
)+∇x nεi =
∑j 6=i
nεi nεj uεj − nεj nεi uεi∆ij
−∇x ni =∑j 6=i
njNi − niNj∆ij
.
No closure relation at the limit! The closure relation can be obtained e.g. inthe isothermal setting with equimolar diffusion.
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 13/15
Link with the linearized operator
I We obtained Dij ∝∫∫∫
Bij(v , v∗, σ) exp[− mi
2kT v2 − mj2kT v2
∗]
(v ′ − v)2.
I Link with the linearized Boltzmann operator (dimension 3)
Dij ∝ 〈miC (k)i ,L(mjC (k)
j )〉, for any k = 1, 2, 3,
where C (k)i = (0, · · · , 0, v (k)M1/2
i , 0, · · · , 0)ᵀ.I Matrix form of Maxwell-Stefan’s system: ∇x n = − 1
kT AU, withA ∝ (Dijninj)i,j and U = (u1, · · · , up)ᵀ.
Self-adjointness + non-positivity of L ↔ A symmetric + semi-definite negative
I Diffusion coefficients for Fick’s law
Dij ∝ 〈L−1(C (k)i − αi Γ(k)),C (k)
j 〉, for any k = 1, 2, 3,
where Γ(k) =∑
i miC (k)i and αi Γ(k) the projection of C (k)
i on KerL.
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 14/15
Conclusion and prospects
I Asymptotics of Boltzmann equation for mixtures towards Maxwell-StefanI Explicit form of the diffusion coefficients w.r.t. the cross sectionsI Also true for polyatomic gasesI Same computations with the moment method
I Fick’s law models a gas in the continuous regimeI Maxwell-Stefan’s equations can be invoked when moderate rarefaction occurs
I Maxwell-Stefan’s equations model moderately rarefied mixturesI Apply the formalism of hyperbolic conservation laws with stiff relaxation terms
[Chen, Levermore, Liu]I Obtain an explicit form of Fick’s coefficients from Maxwell-Stefan’s for low
Knudsen numbers (ongoing work with L. Boudin, V. Pavan)I Compare experimental and analytical relaxation times
I Rigorous convergence towards both Fick’s and Maxwell-Stefan’s equations(ongoing works with A. Bondesan, L. Boudin, M. Briant, E. Daus)
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 15/15
Thank you for your attention!
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 15/15
Moment method
I Entropy associated to the Boltzmann equation for mixtures and its associatedconjugate convex function
η(y) =∑
iyi ln(yi )− yi , η∗(z) =
∑i
exp(zi )
I Tensorial moment basis Ni , Ci (v), E (v)
N1 =
10... , Np =
...01
, C1(v) =
m1v0...
, Cp(v) =
...0
mpv
, E (v) = 12
m1v2
...mpv2
I Even for not so small Knudsen numbers, we obtain from the entropy and the
moment basis the form of
fi (v) = ni
( mi2πkT
)3/2exp
(mi (v − ui )2
2πkT
),
and its moments give, with the same computations, the macroscopicequations.
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 15/15
Closure relation
I Moment of order 2
ε∂
∂t
(mi
∫v
|v |2 f εi)
+∇x ·(
mi
∫v
|v |2 v f εi)
= 1ε
∑j 6=i
mi
∫v
|v |2 Qij(f εi , f εj )
I Ansatz ⇒ mi
∫v|v |2f εi dv = 3kTnεi and mi
∫v|v |2v f εi dv = 5kTεnεi uεi
I Sum over i : 3∂t∑
ini + 5∇x ·
∑i
Ni = 0
I Along with mass conservation, it follows ∂t∑
ini = 0 et ∇ ·
∑i
Ni = 0
I If∑
inini is initially uniform in space, then
∑i
ni is constant
Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 15/15