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DIFFUSION APPROXIMATION AND HOMOGENIZATION OF THE SEMICONDUCTOR BOLTZMANN EQUATION NAOUFEL BEN ABDALLAH∗ AND MOHAMED LAZHAR TAYEB† Abstract. The paper deals with the diffusion approximation of the Boltzmann equation for semiconductors in the presence of spatially oscillating electrostatic potential. When the oscillation period is of the same order of magnitude as the mean free path, the asymptotics leads to the Drift-Diffusion equation with a homogenized electrostatic potential and a diffusion matrix involving the small scale information. The convergence is proven rigorously for Boltzmann statistics, while it is incomplete for Fermi-Dirac statistics. Key words. Semiconductor Boltzmann equation, Drift-Diffusion equation, diffusion approximation, homogenization, two-scale convergence, Relative entropy. AMS subject classifications. 35B27, 35B25, 82B21, 54C70.
1. Introduction. The Drift-Diffusion equation is a standard model of particle transport in many applications like neutron transport [17, 18], plasmas [7, 22], semiconductors [26, 34, 40, 41, 42], gas discharges [39], etc. The Drift-Diffusion model, very suited for numerical simulations, is designed to describe the macroscopic behaviour of the device. It consists of a mass balance equation relying the particle and current densities: (1)
∂t ρ − ∇x .(D∇x ρ + µρF ) = 0
where ρ is the particle concentration, D is a diffusion coefficient, µ the mobility, and F the electrostatic field. This model can be obtained from a diffusion approximation of Boltzmann like equations: (2)
∂t f ε +
1 ε ε Q(f ε ) T f = ε ε2
where f ε = f ε (t, x, v) is the distribution function of particles, T ε is a first order operator which includes free steaming and acceleration terms , while Q is the collision operator which relaxes the distribution function to the local equilibrium. The small parameter ε stands for the scaled mean R free path. The collision operator is usually mass conserving ( Q(f )dv = 0) which implies the continuity equation (3)
∂t ρε + ∇x .j ε = 0
where the particle density and current density are given by Z Z 1 ε ε ε (4) ρ (t, x) = f (t, x, v)dv, j (t, x) = vf ε (t, x, v)dv. ε Such asymptotics have been widely studied in radiative transport [2, 5, 6, 17, 28] and in semiconductors [8, 11, 12, 29, 37]. Other diffusion models like the Energy Transport Model or the SHE "Spherical Harmonics Expansion" model have also been obtained by the same methodology (for different collision operators) [9, 10, 32]. In all these references, the inputs of the Boltzmann equation vary on the macroscopic scale and (like the collision cross sections or the force fields) the only small parameter involved in the asymptotics is the scaled mean free path which is related to the distances over which the local equilibrium [Q(f ) = 0] is reached. When the inputs of the kinetic equation vary at the microscopic scale, the diffusion equation obtained in the limit contains in a homogenized way the small scale information of the Boltzmann problem. This has been proven for neutron transport problems in [2, 43] by using the notion of two-scale limit. This notion has been introduced in [35] and further developed in [1] and is an alternative approach to the energy ∗ Mathématiques pour l’Industrie et la Physique, UMR, CNRS 5640, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France. ([email protected]). † Laboratoire d’Ingénierie Mathématique, Ecole Polytechnique de Tunisie, rue El Khawarezmi, 2078, La Marsa, Tunisie. ([email protected])
1
method of Tartar [46]. This situation holds in particular for superlattices which are obtained by growing periodically successive slices of two materials resulting in a periodic electrostatic potential [15, 20, 21, 50]. We shall here consider that transport is classical and that thickness of the potential barriers is of the order of magnitude of the collision mean free path. In a forthcoming paper the second author analyzes the self-consistent affects obtained by adding to the potential, the Hartree part deduced from the resolution of the Poisson equation [49]. In this work, we consider the diffusion approximation of the Boltzmann equation of semiconductors, when both the scattering cross section and the electrostatic potential have variation at the mean free path scale ε. More precisely, we consider the asymptotics (Bαε ) : 1 Qα, σε (f ε ) , ∂t f ε + TΦε f ε = ε ε2 with (5)
TΦε (f ε ) = v.∇x − ∇x Φε .∇v ,
(x, v) ∈ Rd × Rd
and the collision operator is given by Z (6) Qα, σε (f ) = σε (x, v, v ′ )(m f (v ′ )(1 − αf ) − m(v ′ )f (1 − αf (v ′ )))dv ′ , Rd
M being the normalized maxwellian 2
e−|v| /2 m(v) = √ 2π d
(7)
and Φε (t, x), σε (x, v, v ′ ) are respectively the electrostatic potential and the collision cross-section satisfying respectively assumptions (A2) and (A3), detailed later. The parameter α is equal to 0 (Boltzmann statistics) or 1 (Fermi-Dirac statistics). We shall assume that (8)
x Φε (t, x) = Φ(t, x, ) ε
where Φ(t, x, y) is a given regular potential which is periodic with respect to y. For simplicity we assume that the period is the unit cube [0, 1]d . The initial value of the distribution function is (9)
f ε (0, x, v) = f0ε (x, v)
where f0ε is a given function which might depend on ε. 2. Notations and main result. We shall denote by Y the unit cube of Rd . C# (Y ) and ) denote respectively continuous and infinitely differentiables functions defined on Y and extended on Rd by Y −periodicity. For p ≥ 1, Lp# (Y ) will denote the space of functions of Lploc (Rd ) and Y −periodic. For an open subset Ω ⊂ Rd , Lp (Ω; C# (Y )) is the space of functions of Lp (Ω) ∞ ∞ with value in C# (Y ). The following spaces D′ (Ω; C# (Y )), D(Ω; C# (Y )), . . . are defined in the same manner. For Y −periodic functions on y, the notations (f )ε , (∇y f )ε ,. . . will be used instead of f (t, x, xε , v), ∇y f (t, x, xε , v), . . . We define the maxwellians MΦ and MΦε by ∞ C# (Y
(1)
MΦ (t, x, y, v) = m(v) exp(Φ0 (t, x) − Φ(t, x, y)), x MΦε (t, x, v) = MΦ t, x, , v ε
where Φ0 is the homogenized potential: (2)
Φ0 (t, x) = −log
Z
Y
2
e
−Φ(t,x,y)
dy .
The collision cross-section σ0 := σ0 (x, y, v, v ′ ) is the two-scale limit of σε (see Section 5.1) and Qα, σi is the collision operator associated with σi (for i ≥ 0): Z (3) Qα, σi (f ) = σi (x, y, v, v ′ )(m f (v ′ )(1 − αf ) − m(v ′ )f (1 − αf (v ′ )))dv ′ . Rd
Assumptions.
Throughout this study we shall use the following assumptions:
(A1) The initial data f0ε is uniformly bounded in L1 (R2d ) and ∃ c0 > 0,
0 ≤ f0ε ≤ c0 exp(−
v2 − Φε (t = 0)). 2
+ 2, ∞ + ∞ 2d (A2) The potential Φ belongs to L∞ (R2d )) and ∂t Φ ∈ L∞ loc (R ; W loc (R ; L (R )). (A3) The cross-section σε is uniformly bounded, satisfying the micro-reversibility principle:
0 < σ ≤ σε (x, v, v ′ ) = σε (x, v ′ , v) ≤ σ ¯
(4)
and converging in two-scale strong (see section 5, or [1, 31] ). Our main result is the following Theorem 2.1. Assume that α = 0 and assumptions (A1), (A2) and (A3) are satisfied. Let f˜0 be the two-scale limit of f0ε . Then, any weak solution f ε of the linear Boltzmann equation (B0ε ) + 1 ∞ 2d associated with (5)−(9), converges in L∞ loc (R ; L ∩L (R )) weak star to ρ m where the function ρ is the solution of the Drift-Diffusion equation ∂t ρ + ∇x .J(ρ) = 0 J(ρ) = −D(t, x)(∇x ρ + ρ∇x Φ0 ) (5) ZZ f˜0 (y, v)dydv, ρ(t = 0) = ρ0 = Y ×Rd
where Φ0 is defined by (2) and D is the positive definite matrix given by (11). For the sake of clarity , we postpone to Section 6 the discussion of the Fermi-Dirac statistics (α = 1) for which our result is still formal.
The outline of the paper is as follows. Section 3, is devoted to the formal analysis of the asymptotics in the linear case, using two-scale expansions. This analysis leads to an auxiliary cell problem, studied in section 4. In Section 5 , the convergence result (Theorem 2.1) is proven in the linear case by the two-scale limit technique. Finally Section 6 is devoted to a formal analysis for the Fermi-Dirac model in which we present the formal derivation of the fluid system and study the corresponding cell problem. 3. Formal expansion for the Boltzmnn statistics. In this section, we shall present the formal analysis allowing to describe the limit model, in the case of Boltzmann statistics (Q0, σε ). To this aim, we assume that f ε and σε verify when ε goes to zero (1) (2)
x x x f ε (t, x, v) = f0 t, x, , v + ε f1 t, x, , v + ε2 f2 t, x, , v + . . . , ε ε ε x x x σε (x, v, v ′ ) = σ0 x, , v, v ′ + ε σ1 x, , v, v ′ + ε2 σ2 x, , v, v ′ + . . . ε ε ε
where the coefficients fi and σi are Y −periodic with respect to y.
1 Substituting (1) and (2) into (B0ε ), replacing ∇x (fi )ε by (∇x fi )ε + (∇y fi )ε and letting the ε coefficients of the powers of ε equal to zero, we get a hierarchy of equations: (3)
ε−2 :
L0, σ0 f0 = 0,
3
(4)
(5)
ε−1 :
L0, σ0 f1 = −v.∇x f0 + ∇x Φ(t, x, y).∇v f0 + Q0, σ1 (f0 ),
ε0 :
L0, σ0 f2 = −∂t f0 − v.∇x f1 + ∇x Φ(t, x, y).∇v f1 + Q0, σ1 (f1 ) +Q0, σ2 (f0 ),
where (6)
L0, σ0 = v.∇y − ∇y Φ.∇v − Q0, σ0
and (7)
Q0, σ0 (f ) =
Z
Rd
σ0 (x, y, v, v ′ )(m f (v ′ ) − m(v ′ )f )dv ′ .
To obtain the evolution equation satisfied by f0 , we need to study the spectral properties of L0, σ0 . This operator, acting on the Hilbert space L2MΦ = f ∈ L2 (dydv/MΦ ) / f : Y-periodic with respect to y , is unbounded with domain:
D(L0, σ0 ) = {f ∈ L2MΦ / v.∇y f − ∇y Φ.∇v f ∈ L2MΦ }. It satisfies the following Proposition 3.1. The operator L0, σ0 is maximal monotone on L2MΦ and satisfies 1. N (L0, σ0 ) = R MΦ , ZZ g(y, v)dydv = 0 . 2. R(L0, σ0 ) = g ∈ L2MΦ / Y ×Rd
3. For all g ∈ R(L0, σ0 ), thereZexists f ∈ D(L0, σ0 ) such that L0, σ0 f = g. This solution is Z unique under the condition f (y, v)dydv = 0. Y ×Rd
This proposition will be proven in section 4. Let us continue the derivation of the fluid system. According to (3), the two-scale limit of f ε has the form: (8)
f0 (t, x, y, v) = ρ(t, x)MΦ (t, x, y).
As a consequence, f0 ∈ N (Q0, σ1 ) and a simple computation of the right hand side of (4) gives: (9)
f1 (t, x, y, v) = −(∇x ρ + ρ∇x Φ0 )(t, x).L−1 0, σ0 (vMΦ )(t, x, y, v).
denoting by χ the unique solution in [R(L0, σ0 ) ∩ D(L0, σ0 )]d of (10)
L0, σ0 χ = vMΦ
and D the diffusion matrix: (11)
D(t, x) =
ZZ
Y ×Rd
v ⊗ χ(t, x, y, v)dydv
Integrating (5) with respect to dydv, the solvability condition gives the Drift-Diffusion equation associated to the normalized potential Φ0 (given in (2)): (12)
∂t ρ + ∇x .J(ρ) = 0
where (13)
J(ρ)(t, x) = −D(t, x)[∇x ρ + ρ∇x Φ0 ](t, x). 4
4. The cell Problem. We are concerned in this section with the proof of Proposition 3.1. The monotonicity of L0, σ0 is a consequence of the entropy inequality ZZ
Y ×Rd
L0, σ0 (f )
f dydv = − MΦ
Z
eΦ−Φ0
Y
Z
Q0, σ0 (f ) Rd
f dv dy ≥ 0. m
This relation implies that L0, σ0 and its adjoint L∗0, σ0 are monotone and then L0, σ0 is maximal since it is closed and has a dense domain in L2MΦ (see for instance [16] page 113). To prove 2), we remark that the two-scale limit σ0 of σε satisfies (4) and then ZZ f dydv ≥ Ckf − ρ mk2L2 − Q0, σ0 (f ) MΦ M d Φ Y ×R R where ρ(.) = Rd f (., v)dv. As a consequence, any function belonging to the null-space N (L0, σ0 ) has the form f = ρ(y)m(v). Inserting this expression in L0, σ0 (f ) = 0, we obtain ∇y ρ + ρ∇y Φ = 0. Therefore, N (L0, σ0 ) is spanned MΦ . RR To prove 3), we shall show that, for ε tending to zero and g satisfying Y ×Rd g(y, v)dydv = 0, the solution of εf ε + L0, σ0 (f ε ) = g
f ε : Y-periodic with respect to y.
is uniformly bounded. To this aim, we proceed by contradiction and assume that (f ε ) is unbounded in L2MΦ . Denoting F ε = f ε /kf ε kL2M
Φ
and Gε = g/kf ε kL2M , Φ
the sequence (F ε )ε is bounded in L2MΦ and (1)
εF ε + L0, σ0 (F ε ) = Gε .
Besides, F ε converges weakly to a function F ∈ N (L0, σ0 ). An integration of the above equation, RR the condition Y ×Rd g dydv = 0 leads to F ≡ 0. Multiplying (1) by F ε /MΦ . Integrating with respect to dydv, using Jensen’s inequality and taking advantage of the identity (v.∇y − ∇y Φ.∇v )MΦ (t, x, y) = 0, one can deduce (2)
F ε − m(v)ρF ε → 0 in L2MΦ
where (3)
ρF ε (y) =
Z
F ε (y, v)dv.
Rd
Multiplying (1) by v and integrating with respect to v, we obtain, thanks to identity (2) ∇y ρF ε = (S ε − ρF ε ∇y Φ)/d where S ε belongs to a compact subset of [H −1 (Y )]d . Consequently, F ε converges (up to an extraction of a subsequence) strongly in L2MΦ . This is not possible because kF ε kL2M = 1 and its weak Φ limit is equal to zero. 5
5. Rigorous convergence Proof for the Boltzmann statistics . The proof of Theorem 2.1 is done in two steps. In subsection 5.3, we begin by establishing useful Lp −estimates on the charge and current densities. Lemma 5.5 allows to deal with weak and two-scale convergence of f ε , ρε and j ε . In Subsection 5.4) we shall characterize the limit , as ε goes to zero, of ρε and j ε . Before developing this proof let us briefly review (in Subsections 5.1 and 5.2) some definitions and useful properties of the two-scale convergence notion. We refer to [1, 31, 35] for further details. 5.1. Two-scale convergence. Definition 5.1. Let Ω be an open set of Rn (n ≥ 1) and (uε ) a bounded sequence of L2 (Ω). The sequence (uε ) is said to be two-scale convergent, as ε goes to zero, to a function u ∈ L2 (Ω × Y ) if : Z Z Z x (1) dx = u(x, y)ψ(x, y)dxdy lim uε (x)ψ x, ε→0 Ω ε Ω Y ∞ for any smooth function ψ which is Y −periodic in y (ψ ∈ D(Ω; C# (Y ))).
The previous definition is justified by the following compactness theorem Theorem 5.2. [1]. Let (uε ) be a bounded sequence of L2 (Ω). Then, there exists a subsequence of uε which two-scale converges to a function u ∈ L2 (Ω × Y ) in the sense of the previous definition. Let us give some examples of two scale limits x converges in two-scale to u. 1. Let u ∈ L2 (Ω; C# (Y )). Then, the sequence uε (x) = u x, ε 2. RAny sequence (uε ) that converges in two-scale to a limit u, converges L2 − weakly to u(., y)dy Y 3. Any sequence (uε ) that converges strongly in L2 (Ω) to a function u, two-scale converges to the same limit u. 4. Any sequence (uε ) that admits the expansion uε (x) = u0 (x, x/ε) + ε u1 (x, x/ε) + ε2 u2 (x, x/ε) + . . . where the functions ui are smooth and Y −periodic in y, two-scale converges to the first term u0 . 5.2. Strong two scale limit. Definition 5.3. A sequence (uε )ε of L2 (Ω) is said to be two-scale strong convergent to a function u ∈ L2 (Ω × Y ) if: for any two-scale convergent sequence vε to a limit v and any function ψ ∈ D(Ω; C# (Y )), we have Z Z Z x lim uε (x)vε (x)ψ x, dx = u(x, y)v(x, y)ψ(x, y)dxdy. ε→0 Ω ε Ω Y
The following theorem which is very important in the analysis of nonlinear problems, gives a sufficient condition (in terms of energy conservation) for strong two-scale convergence. Theorem 5.4. Let (uε )ε be sequence of L2 (Ω) that two-scale convergent to a function u ∈ 2 L (Ω × Y ). Assume that lim kuε kL2 (Ω) = kukL2 (Ω×Y ) .
ε→0
Let (vε ) ∈ L2 (Ω) a two-scale convergent sequence to a limit v ∈ L2 (Ω × Y ). Then, we have Z uε vε ⇀ u(., y)v(., y)dy in D′ (Ω). Y
2
Moreover, if u ∈ L (Ω; C# (Y ), then
.
lim uε (.) − u ., 2 = 0. ε→0 ε Lloc (Ω)
Remark 1. Previous definitions of two-scale convergence can be extended to Lp spaces for p ∈]1, +∞] and also to any dual of separable Banach spaces. 6
5.3. A priori estimates. Lemma 5.5. Assume that (A1), (A2) and (A3) are satisfied. Let f ε be a weak solution of the linear Boltzmann equation (B0ε ) associated with (5)−(9). Then, f ε satisfies + 1 ∞ 2d 1. Lp −estimates: f ε is uniformly bounded in L∞ loc (R ; L ∩ L (R )), (2)
kf ε (t)kL1 (R2d ) = kf0ε kL1 (R2d ) ,
(3)
0 ≤ f ε ≤ α(t) exp −|v|2 /2 − Φε (t) ,
where
α(t) = α0 + exp
Z
0
t
k∂t Φ(s)kL∞ (Rd ) ds .
2. Distance to equilibrium: Z
(4)
0
TZ
R2d
(f ε − ρε m)2 dtdxdv ≤ CT ε2 . me−Φε
3. The current density j ε is bounded in L2loc (R+ ; L2 (Rd )).
Proof of Lemma 5.5. Inequality (2) is immediate. Inequality (3) is deduced from the weak maximum principle. Indeed, the maxwellian
2
˜ ε (t, x, v) = α(t) exp − |v| − Φε (t, x, x ) M 2 ε
solves the following scaled Boltzmann equation ˜ ε (t, x, v) = (k∂t Φε (t)kL∞ − ∂t Φε (t, x))M ˜ ε (t, x, v) TΦε M ˜ ε − f ε is then a weak solution of the system satisfying where TΦε is defined in (5). The function M
˜ ε − f ε ) ≥ 0, TΦε (M
(M ˜ ε − f ε )(t = 0) ≥ 0.
˜ ε − f ε . To establish the estimate (4), we multiply (B0ε ) The maximum principle implies the positivity of M ε −Φε by f /(m(v)e ) and integrate on all variables, we obtain d dt
Z
R2d
f ε2 dxdv m(v)e−Φε
+
1 ε2
Z
R2d
(f ε − ρε m)2 dxdv ≤ α′ (t) m(v)e−Φε
Z
R2d
f ε2 dxdv m(v)e−Φε
A Gronwall lemma yields, thanks to assumption (A3), to (3) and the boundedness of the current density in L2loc (R+ ; L2 (Rd )). Corollary 5.6. The charge and current densities, given by (4), satisfy the following Green formula (5)
Z
0
+∞
Z n
ρε (∂t ψ)ε + j ε .(∇x ψ)ε +
Rd
1 ε j .(∇y ψ)ε ε
d ∞ for all ψ ∈ D(R+ t × Rx ; C# (Y )).
o
+
Z
R2d
f0ε ψ 0, x,
x dxdv = 0 ε
5.4. Passing to the limit: duality method. Lemma 5.7. The charge density ρε and the current density j ε satisfy ρε ⇀ ρ˜ := ρ(t, x) exp(Φ0 (t, x) − Φ(t, x, y)), j ε ⇀ ˜j :=
Z
Rd
two-scale
(v ⊗ χ(t, x, y, v)) .(∇x ρ + ρ∇x Φ0 )(t, x),
two-scale
where Φ0 and χ are defined in (2) and (10) respectively and ρ is the weak limit of ρε . Proof of Lemma
5.7. Let us denote g ε = (f ε − ρε m)/ε. Integrating (B0ε ) with respect to vdv, we obtain Z Z 2 ε ε (6) ε ∂t j + ε∇x . (v ⊗ v)g dv + d∇x ρε = −dρε ∇x Φε + vQ0, σε (g ε )dv Rd
Rd
7
∞ Multiplying this equation by a test function εψ(t, x, x/ε) where ψ ∈ [D(R∗+ × Rdx ; C# (Y ))]d and integrating with respect to dtdx yields to Z Z 1 (7) ερε (∇x .ψ)ε + (∇y .ψ)ε − ∇x Φε .(ψ)ε dtdx = §ε , d ε R+ Rd
where ε
§ :=
−ε
Z
Z
−ε
Z
Z
(8)
R+ Rd
R+
Z 2 ε ε (∂t ψ)ε j +
ε
Rd
(v ⊗ v)g dv (ε∇x ψ + ∇y ψ)ε dtdx
v Q0, σε (g ε )(ψ)ε dtdxdv.
R2d
By analyzing (2), (3) and (4), one can show that §ε converges to zero. Moreover, Φε is regular enough to get ε∇x Φε −→ ∇y Φ
two-scale strong.
Therefore, by passing to the limit in (7), the two-scale limit ρ˜ of ρε satisfies Z +∞Z Z ∞ (Y ))]d . ρ˜ [∇y .ψ − ∇y Φ.ψ] dtdxdy = 0, ∀ ψ ∈ [D(]0, +∞[×Rdx; C# Rd
0
Y
An integration by parts gives: d d ρ˜ ∈ L∞ (R+ loc × Rx × Ry )
and ∇y ρ˜ = −ρ˜∇y Φ,
1,∞ d wich in turn implies ρ˜ ∈ L∞ (R+ loc × R ; W# (Y )) and
ρ˜(t, x, y) = ρ(t, x) exp(Φ0 (t, x) − Φ(t, x, y)) where ρ (the average of ρ˜) is the weak limit of ρε . Let now ψ ε be a test function belonging to D(R+ × R2d ). The weak formulation associated to (B0ε ) associated with (5) − (9) reads Z 1 ε ε ε ε ε f ε∂t ψ + v∇x ψ − (∇x Φ)ε .∇v ψ − (∇y Φ)ε .∇v ψ dtdxdv ε R+ ×R2d (9) Z Z 1 + Q0,σε (f ε )ψ ε dtdxdv + ε f0ε (x, v)ψ ε (0, x, v)dxdv = 0. ε R+ ×R2d 2d R Choosing ψ ε (t, x, v) :=
εϕ(t, x, xε , v)
x
m(v)e−Φ(t,x, ε )
∞ with ϕ ∈ D(R+ × Rdx ; C# (Y ) × L∞ (Rdv )) and ϕ/m(v) ∈ L∞ , and using the property ε
[v.∇x − ∇x Φε ∇v ](m(v)e−Φ
(t,x)
) = 0,
the above formulation becomes Z Z fε (ε∂ ϕ + εϕ∂ Φ + v.∇ ϕ − ∇ Φ.∇ ϕ) + f0ε ψ ε (0) t t x x v ε ε −Φ R+ ×R2d m(v)e R2d (10)
−
Z
R+ ×R2d
1 = ε
Z
gε (v.∇y ϕ − ∇y Φ.∇v ϕ − Q0, σε (ϕ))ε m(v)e−Φε
R+ ×Rd
Z ∇y .
Rd
vϕdv
ε
ρε e Φ ε
8
where the notation (.)ε means that we take expression between brackets at point (t, x, x/ε, v) and we used f ε = ρε m(v) + ε g ε . Let us denote D# and E# Z d ∞ ∞ d D# = D(R+ × R ; C (Y ) × L (R )), E = ϕ ∈ D / ∇ . vϕdv = 0 # # y t x # m v Rd
and pass to the limit only for ϕ ∈ E# in order to overcome the difficulty due to the right hand side of (10). We will show later that this choice allows to describe the limit equation. From Lemma 5.7, Estimate (4) and thanks to two-scale strong convergence of Φε and σε , one can deduce ε
eΦ f ε m(v)
⇀
2d If we denote g˜ ∈ L2 (R+ × Y, loc × R
dtdxdydv m(v) )
ε
g ε eΦ m(v)
ρ(t, x)eΦ0 (t,x) ,
⇀
two-scale.
the two-scale limit of g ε , then
g˜ eΦ g˜ eΦ0 (t,x) := , m(v) MΦ
two-scale
Q0, σ0 (˜ g eΦ ) , m(v)
two-scale.
and ε
Q0, σε (g ε eΦ ) m(v) We deduce from (10) " Z
ρe
R+ ×Y ×Rd
Φ0
⇀
# (L∗0, σ0 ϕ)˜ g eΦ0 (v∇x ϕ − ∇x Φ.∇v ϕ) − dtdydv = 0, MΦ
∀ϕ ∈ E# .
which means that v∇x (ρeΦ0 ) +
L0, σ0 (˜ g eΦ0 ) ⊥ ∈ E# . MΦ
The following lemma proven by Poupaud and Goudon, characterizes the orthogonal of E# . ′ ⊥ ′ Lemma 5.8. [31]. Let T ∈ D# . Then, T belongs to E# if and only if there exists S ∈ D# such that ∇v S = 0 and T = v.∇y S. ′ According to this lemma, there exists S ∈ D# satisfying ∇v S = 0 and
v∇x (ρeΦ0 ) +
L0, σ0 (˜ g eΦ0 ) = v.∇y S. MΦ
Multiplying this equation by MΦ , combined with (10) imply L0, σ0 (˜ geΦ0 + ∇x (ρeΦ0 )χ − SMΦ ) = 0 and then g˜eΦ0 = MΦ S − (∇x ρ + ρ∇x Φ0 )eΦ0 (t,x) χ(t, x, y, v). Since j ε (t, x) =
Z
vg ε (t, x, v)dv,
Rd
the two-scale limit of j ε is given by the formula: Z ˜j(t, x, y) = − (v ⊗ χ(t, x, y, v))dv .(∇x ρ + ρ∇x Φ0 )(t, x). Rd
9
Integrating this equation on the unit cell, we obtain j ε ⇀ J(ρ) := −D(t, x). [∇x ρ(t, x) + ρ(t, x)∇x Φ0 (t, x)] where D(t, x) is defined by (11). Enf of the Proof of Theorem 2.1: Conclusion To complete the proof of Theorem 2.1, we pass to the limit in (5) with test function ϕ(t, x, v) not depending on y. This leads to the weak formulation of the Drift-Diffusion equation (5). Corollary 5.9. The diffusion matrix D(t, x) is positive definite. Proof. Let define by Ms the symmetric part of the matrix D. For all i, j ∈ {1, 2, . . . , n} Msi,j =
1 2
Z
Y ×Rd
(L0, σ0 (χi )χj + χi L0, σ0 (χj ))
dydv MΦ
where L0, σ0 (χi ) = vi MΦ (y, v) and L0, σ0 is defined by (6). Let η ∈ Rd and U = show, thanks to entropy inequality, that s
(M η, η) = where ρU =
R
Rd
Z
Y ×Rd
L0, σ0 (U )U
d X
χi ηi . It is simple to
i=1
dydv ≥ CkU − ρU mk2L2 MΦ MΦ
U dv. Therefore, if (Ms η, η) = 0 then U belongs to N (L0, σ0 ) and for all v ∈ Rd L0, σ0 (U ) = (η, v)MΦ = 0
which implies that η = 0 and M is definite positive.
6. Formal analysis of the Fermi-Dirac statistics. This section is devoted to the formal derivation of the fluid model corresponding to the Fermi-Dirac model (α = 1). We are concerned with the limit (ε → 0) of the solution f ε of ε ∂t f ε + 1 TΦε (f ε ) = Q1, σε2(f ) , ε
(1)
(x, v) ∈ R2d
ε
f ε (x, v, 0) = f0ε (x, v)
where Φε and Q1, σε are given by (8) and Q1, σε (f ) =
Z
Rd
σε (x, v, v ′ )(m f (v ′ )(1 − f ) − m(v ′ )f (1 − f (v ′ )))dv ′ .
We assume instead of (A1): (A’1)
0 ≤ f0ε ≤ 1 and f0ε is uniformly bounded in L1 (R2d ).
Φ and σε satisfy assumptions (A2) and (A3) respectively. It is well known that under the natural assumption (A’1), the system (1) has a weak solution satisfying kf ε (t)kL1 (R2d ) = kf0ε kL1 (R2d ) and 0 ≤ f ε ≤ 1. Thus, one can deal with the two-scale convergence for f ε .
6.1. Fluid system. Assume that f ε and σε admit (1) and (2) as an asymptotic expansion. One can show that Q1, σε (f ε ) = Q1, σ0 (f0 ) + ε (Q1, σ1 (f0 ) + DQ1, σ0 (f0 )f1 ) (2)
+ε2 DQ1, σ0 (f0 )f2 + Q0, σ2 (f0 ) + DQ1, σ1 (f0 )f1 +
Z
Rd
+...
σ0 f1 f1 (v ′ )(m(v ′ ) − m)dv ′
where DQ1, σi is the linearized of Q1, σi : DQ1, σi (f )(g) = Q0, σi (g) +
Z
Rd
σi (f (v ′ )g + f g(v ′ ))(m(v ′ ) − m)dv ′ . 10
Inserting the asymptotic expansion (1) in (1), thanks to (2), an identification of coefficients with same power on ε leads to (3)
L1, σ0 f0 = 0,
(4)
(v.∇y − ∇y Φ.∇v − DQ1, σ0 (f0 ))f1 = −v.∇x f0 + ∇x Φ.∇v f0 + Q1, σ1 (f0 ), (v.∇y − ∇y Φ.∇v − DQ1, σ0 (f0 ))f2 = −∂t f0 − v.∇x f1 + ∇x Φ.∇v f1 + Q0, σ2 (f0 )
(5)
+DQ1, σ1 (f0 )f1 +
Z
Rd
σ0 f1 f1 (v ′ )(m(v ′ ) − m)dv ′
where ε
2
L1, σ0 = v.∇y − ∇y Φ.∇v − Q1, σ0 .
The solution f is bounded in L . Assume that f ε has (1) as an asymptotic expansion. Then, it converges in two-scale to the leading term f0 . Letting ε goes to zero, we wish to describe the equation satisfied by the average of f0 . This is the homogenized fluid model. To obtain this model, we shall prove the existence and uniqueness of the solution of (3) and then study the cell problem associated with the linearized operator. Namely, a necessary solvability condition of (4) and (5) provides Drift-Diffusion model. First equation. We consider the function (6)
H(f ) =
f . (1 − f )MΦ
Then L1, σ0 satisfies Lemma 6.1. 1. Entropy inequality:
Z
Y ×Rd
(7)
1 = 2
Z
L1, σ0 (f )H(f )dydv = −
Y ×R2d
Z
Q1, σ0 (f )H(f )dydv
Y ×Rd
σ0 (1 − f )(1 − f (v ′ ))MΦ m(v ′ ) H(f (v ′ )) − H(f )
2. The null-space N (L1, σ0 ) is (8)
N (L1, σ0 ) = {F (µ(t, x) + Φ0 (t, x) − Φ(t, x, y)),
2
dydvdv ′ ≥ 0.
¯ µ ∈ R}
where F (η) is the Fermi-Dirac function associated with the chemical potential η: F (η) =
1 1 + exp(|v|2 /2 − η)
¯ ∀η ∈ R.
Proof. Multiplying (3) by H(f ) and integrating gives
Z
L1, σ0 (f )H(f )dydv = −
Y ×Rd
Z
Q1, σ0 (f )H(f )dydv
Y ×Rd
since (v.∇y − ∇y Φ.∇v )(MΦ ) = 0. Using micro-reversibility property of the cross-section σ0 , we have
Z
2
L1, σ0 (f )H(f ) =
Y ×Rd
Z
Y ×R2d
σ0 (1 − f )(1 − f (v ′ ))MΦ m(v ′ )[H(f (v ′ )) − H(f )]2 dydvdv ′ ≥ 0.
This implies if L1, σ0 (f ) = 0 then H(f ) doesn’t depend on v. Therefore, ∀ y ∈ Y,
f (y) =
h MΦ (y) ∈ N (Q1, σ0 ) (1 + h MΦ (y))
where h is independent on v. Rewrite (3), gives 0 = (v.∇y − ∇y Φ.∇v )f = v.∇y h 11
MΦ (1 + h MΦ )2
which implies that h does not depend on y. Hence f= where µ(t, x) = log
h(t, x)MΦ = [1 + exp(|v|2 /2 − µ(t, x) − Φ0 + Φ(t, x, y)]−1 . (1 + h(t, x)MΦ )
h(t,x) (2π)d/2
.
As a consequence of this lemma, the solution f0 of (3) is a Fermi-Dirac distribution associated with the chemical potential µ(t, x) + Φ0 (t, x) − Φ(t, x, y). ¯ Let us now define for all µ(t, x) ∈ R
F (µ(t, x), y, v) =
(9)
1 , 1 + e|v|2 /2−µ(t,x)+Φ(t,x,y)
L(µ, σ) = v.∇y − ∇y Φ.∇v − DQ1, σ (F (µ)),
In all the sequel, F (µ) = F (t, x, y, v) will denotes the Fermi-Dirac function associated with µ(t, x) − Φ(t, x, y). The following proposition will provide an equation on the averaged of f0 = F (µ + Φ0 ). ¯ and σ a cross section satisfying (4). Then, L(µ, σ) is maximal Proposition 6.2. Let µ(t, x) ∈ R monotone on LF (1−F )(µ) identified to L2# (Y ; L2 (Rd )) with the inner product (10)
hf, giF (1−F )(µ) =
Z
Y ×Rd
fg dydv. F (µ)(1 − F (µ))
1. The null-space of L(µ) is equal to
N (L(µ, σ)) = R F (µ)(1 − F (µ))
if µ ∈ R
N (L(−∞, σ)) = R MΦ
(11)
2. R(L(µ, σ)) =
N (L(+∞, σ)) = R MΦ−1 .
g ∈ LF (1−F )(µ) ;
Z
g(y, v)dydv = 0
Y ×Rd
tion if and only if f and g ∈ R(L(µ, σ)).
and L(µ, σ)(f ) = g has only one solu-
As a consequence, there exists one solution h(µ) ∈ [R(L(µ, σ))]d of (12)
L(µ, σ)(h(µ, σ)) = vF (µ)(1 − F (µ)).
Moreover, (13)
Π(µ, σ) =
Z
Y ×Rd
v ⊗ h(µ, σ0 , y, v)dydv
is positive definite for all µ ∈ R and Π(±∞, σ) = 0. This proposition will be proved (partially) in the last paragraph. Let us continue the formal derivation and deduce from (4) and (5) an equation on ρ=
Z
F (µ + Φ0 )dydv.
Y ×Rd
To do this, we replace in (4) f0 by F (µ + Φ0 ). We obtain L(µ + Φ0 , σ0 )f1
=
−∇x (µ + Φ0 ).vF (µ + Φ0 )(1 − F (µ + Φ0 )).
In view of (12), there exists f1 ∈ [R(L(µ + Φ0 , σ0 )) ∩ D(L(µ + Φ0 ))]d and satisfying f1 = −∇x (µ + Φ0 ).h(µ + Φ0 , σ0 ). Therefore, equation (5) becomes L(µ + Φ0 , σ0 )(f2 ) = −∂t F (µ + Φ0 ) + ∇x . [v ⊗ h(µ + Φ0 , σ0 )∇x (µ + Φ0 )] Integrating this equation with respect to dydv leads to ∂t ρ − ∇x [Π(µ + Φ0 , σ0 )∇x (µ + Φ0 )] = 0 12
and if f0 is the two-scale limit of f0ε , then ρ is a solution of the following drift-diffusion system
∂t ρ(t, x) + ∇x .J(µ + Φ0 ) = 0 J(µ + Φ ) = −Π(µ + Φ , σ )(t, x).∇ (µ + Φ )(t, x) 0 0 0 x 0 Z f0 (x, y, v)dydv. ρ(0, x) =
(14)
Y ×Rd
6.2. Spectral analysis of L(µ). This subsection is devoted to the proof of proposition 6.2. The outline is exactly the same as in the Boltzmann statistics. We shall only explain how to prove the monotony of the linearized operator L(µ). We begin by writing DQ1,σ (F (µ)) in the following form DQ1,σ (F (µ))(r)
=
K(µ)(r) − λ(µ)r
where λ(µ) and K(µ) are given by λ(µ) =
Z
σ[F ′ (µ)m + (1 − F (µ)′ )m′ ]dv ′
Rd
and
Z
K(µ)(r) =
Rd
σ[F (µ)m′ + (1 − F (µ))m]r ′ dv ′ .
Using property of σ, one can deduce
(1 − F ′ )m′ (1 − F )m ′ m′ m )r − ( + )r dv ′ −DQ1,σ (F (µ))(r) = σF (µ)F (µ) ( + ′ ′ F F F F FF′ Rd
Z
′
Let denote, for µ ∈ R, G(µ) =
(1 − F ′ )m′ m + , F FF′
a simple computation gives √ (1 − F (µ))m eµ˜ = e−µ˜ , G(µ) = and e−µ˜ = e−µ+Φ /( 2π)d F (µ) F (µ)(1 − F (µ)) and −DQ1,σ (F (µ)) satisfies (for all µ ∈ R) (15)
−2
Z
Rd
DQ1,σ (F (µ))(r)r = e−µ˜ F (1 − F )(µ)
Using the property (16)
ZZ
R2d
σF (µ)F (µ)′ G(µ, v)r − G(µ, v ′ )r ′
2
dvdv ′ .
(v.∇y − ∇y Φ.∇v ) [F (µ)(1 − F (µ))] = 0,
we deduce easily the monotony of L(µ) : (17)
Z
Y ×Rd
L(µ)(r) r dydv =− F (µ)(1 − F (µ))
Z
DQ1,σ (F (µ))(r) r
Y ×Rd
dydv ≥ 0. F (µ)(1 − F (µ))
and by the same argument as proposition 3.1, L(µ) is maximal monotone. Moreover, if L(µ)(r) = 0, then r = k(t, x, y)F (µ)(1 − F (µ)) ∈ DQ1,σ (F (µ)). Writing L(µ)(r), we obtain L(µ)(kF (µ)(1 − F (µ))) = ∇y k.vF (µ)(1 − F (µ)) = 0. ¯ N (L(µ)) = R F (µ)(1 − F (µ)). Therefore, for µ ∈ R, The proof of the solvability condition of L(µ)f = g: (18)
Z
g(y, v)dydv = 0. Y ×Rd
is exactly the same as in the first case. It is based on the entropy inequality which allows to deduce that under this condition all solution of (19) εf ε + L(µ)f ε = g.
is bounded in LF (1−F )(µ) (as ε goes to zero). The positivity of the diffusion matrix is a consequence of the monotony of L(µ). We refer to [29] for further details. 13
6.3. Concluding remarks. Two Drift-Diffusion models are derived from Semiconductors Boltzmann equations, corresponding to a given electrostatic potential varying on the scale of the mean free path of particles. When the collision operator is linear, the proof of convergence is rigorous. In the nonlinear case, the Fermi-Dirac statistics, only the formal derivation and Existence and uniqueness of solutions of the cell problem are studied. We think that these results can be improved but some serious difficulties must be overcome . It seems that rigorous proof for the nonlinear case needs a compactness argument like two-scale averaging lemma. The difficulty here arises from the fact that the double scale limit of f ε has a double scale dependence which is not the case for periodic homogenization for nonlinear elliptic partial differential equations [13, 47] etc. Indeed, in the elliptic case, the limit varies only on the macroscopic scale and ellipticity estimates often provides uniform estimates on the gradient of the solution. In a forthcoming paper however, the second author tackles this problem by taking advantage of the explicit knowledge of the double scale dependence of the limit solution. We illustrate this difficulty by explaining why the Hilbert expansion method fails in proving the convergence. In order to avoid technicality (like linearization and boundary layers) we consider the linear case in the whole space setting. In analogy to [12, 37, 48], we write the Boltzmann equation for the remainder (20)
r ε (t, x, v) := f ε (t, x, v) − f0 (t, x,
x x x , v) − εf1 (t, x, , v) − ε2 f2 (t, x, , v) ε ε ε
and try to prove the convergence of r ε in L1 . In this case, f0 and f1 are given by (8) and (9) respectively and f2 is a solution of (5), which can be written under the following form −L0, σ0 f2
=
(∂t ρ)MΦ + ∇x .[(v ⊗ χ).D−1 J(ρ)] + [∇x Φ ⊗ D−1 J(ρ)].∇v χ +Q0,σ1 (f1 ) + ρ∂t MΦ .
where χ, D and J(ρ) are given by (10) (11) and (12) respectively. This implies f2 = χ1 + χ2 + χ3 + ρ(t, x)χ4 where χ1 , χ2 and χ3 are respectively the solutions (on component wise) of −L0, σ0 χ1 = (∂t ρ)MΦ + ∇x .[(v ⊗ χ)D−1 J(ρ)], −L0, σ0 χ2 = [∇x Φ(t, x, y) ⊗ D−1 J(ρ)].∇v χ, −L0, σ0 χ3 = Q0,σ1 (f1 ) and χ4 ∈ D(L0, σ0 ) ∩ R(L0, σ0 ) the solution of −L0, σ0 χ4 = ∂t (MΦ ). The remainder r ε defined by (20) is then a solution of the following scaled Boltzmann equation : (21)
TΦε (r ε ) = S ε = −ε{(∂t f1 )ε + ε(∂t f2 )ε + v.(∇x f2 )ε − (∇x Φ.∇v f2 )ε }
where TΦε is given by (5). The L1 −estimate associated to this equation is kr ε (t)kL1 (Ω) ≤ kr ε (t = 0)kL1 (Ω) +
Z
t 0
kS ε (s)kL1 (Ω) ds.
According to this estimate, the convergence of r ε towards zero "needs" an approximation of the right hand side of (21) in L1 norm. In view of expressions of f1 and f2 , this property requires Sobolev regularities on the solution χ of
(
L0, σ0 (χ) := v.∇y χ − ∇y Φ(t, x, y).∇v χ − Q(χ) = vMΦ , χ:
Y-periodic on y.
Due to the coupling between variables y and v in the transport part of L0, σ0 , it seems to be difficult to have more than L2MΦ −regularity on χ. Acknowledgments. The authors acknowledge support from the European IHP network No. RNT2 2001 349 entitled "HYperbolic and Kinetic Equations : Asymptotics, Numerics, Analysis" and from the CNRSDGRST bilateral project No. 13904 entitled "Modèles classiques et quantiques de transport électronique : analyse asymptotique et simulations". 14
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16
1. Introduction. The Drift-Diffusion equation is a standard model of particle transport in many applications like neutron transport [17, 18], plasmas [7, 22], semiconductors [26, 34, 40, 41, 42], gas discharges [39], etc. The Drift-Diffusion model, very suited for numerical simulations, is designed to describe the macroscopic behaviour of the device. It consists of a mass balance equation relying the particle and current densities: (1)
∂t ρ − ∇x .(D∇x ρ + µρF ) = 0
where ρ is the particle concentration, D is a diffusion coefficient, µ the mobility, and F the electrostatic field. This model can be obtained from a diffusion approximation of Boltzmann like equations: (2)
∂t f ε +
1 ε ε Q(f ε ) T f = ε ε2
where f ε = f ε (t, x, v) is the distribution function of particles, T ε is a first order operator which includes free steaming and acceleration terms , while Q is the collision operator which relaxes the distribution function to the local equilibrium. The small parameter ε stands for the scaled mean R free path. The collision operator is usually mass conserving ( Q(f )dv = 0) which implies the continuity equation (3)
∂t ρε + ∇x .j ε = 0
where the particle density and current density are given by Z Z 1 ε ε ε (4) ρ (t, x) = f (t, x, v)dv, j (t, x) = vf ε (t, x, v)dv. ε Such asymptotics have been widely studied in radiative transport [2, 5, 6, 17, 28] and in semiconductors [8, 11, 12, 29, 37]. Other diffusion models like the Energy Transport Model or the SHE "Spherical Harmonics Expansion" model have also been obtained by the same methodology (for different collision operators) [9, 10, 32]. In all these references, the inputs of the Boltzmann equation vary on the macroscopic scale and (like the collision cross sections or the force fields) the only small parameter involved in the asymptotics is the scaled mean free path which is related to the distances over which the local equilibrium [Q(f ) = 0] is reached. When the inputs of the kinetic equation vary at the microscopic scale, the diffusion equation obtained in the limit contains in a homogenized way the small scale information of the Boltzmann problem. This has been proven for neutron transport problems in [2, 43] by using the notion of two-scale limit. This notion has been introduced in [35] and further developed in [1] and is an alternative approach to the energy ∗ Mathématiques pour l’Industrie et la Physique, UMR, CNRS 5640, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France. ([email protected]). † Laboratoire d’Ingénierie Mathématique, Ecole Polytechnique de Tunisie, rue El Khawarezmi, 2078, La Marsa, Tunisie. ([email protected])
1
method of Tartar [46]. This situation holds in particular for superlattices which are obtained by growing periodically successive slices of two materials resulting in a periodic electrostatic potential [15, 20, 21, 50]. We shall here consider that transport is classical and that thickness of the potential barriers is of the order of magnitude of the collision mean free path. In a forthcoming paper the second author analyzes the self-consistent affects obtained by adding to the potential, the Hartree part deduced from the resolution of the Poisson equation [49]. In this work, we consider the diffusion approximation of the Boltzmann equation of semiconductors, when both the scattering cross section and the electrostatic potential have variation at the mean free path scale ε. More precisely, we consider the asymptotics (Bαε ) : 1 Qα, σε (f ε ) , ∂t f ε + TΦε f ε = ε ε2 with (5)
TΦε (f ε ) = v.∇x − ∇x Φε .∇v ,
(x, v) ∈ Rd × Rd
and the collision operator is given by Z (6) Qα, σε (f ) = σε (x, v, v ′ )(m f (v ′ )(1 − αf ) − m(v ′ )f (1 − αf (v ′ )))dv ′ , Rd
M being the normalized maxwellian 2
e−|v| /2 m(v) = √ 2π d
(7)
and Φε (t, x), σε (x, v, v ′ ) are respectively the electrostatic potential and the collision cross-section satisfying respectively assumptions (A2) and (A3), detailed later. The parameter α is equal to 0 (Boltzmann statistics) or 1 (Fermi-Dirac statistics). We shall assume that (8)
x Φε (t, x) = Φ(t, x, ) ε
where Φ(t, x, y) is a given regular potential which is periodic with respect to y. For simplicity we assume that the period is the unit cube [0, 1]d . The initial value of the distribution function is (9)
f ε (0, x, v) = f0ε (x, v)
where f0ε is a given function which might depend on ε. 2. Notations and main result. We shall denote by Y the unit cube of Rd . C# (Y ) and ) denote respectively continuous and infinitely differentiables functions defined on Y and extended on Rd by Y −periodicity. For p ≥ 1, Lp# (Y ) will denote the space of functions of Lploc (Rd ) and Y −periodic. For an open subset Ω ⊂ Rd , Lp (Ω; C# (Y )) is the space of functions of Lp (Ω) ∞ ∞ with value in C# (Y ). The following spaces D′ (Ω; C# (Y )), D(Ω; C# (Y )), . . . are defined in the same manner. For Y −periodic functions on y, the notations (f )ε , (∇y f )ε ,. . . will be used instead of f (t, x, xε , v), ∇y f (t, x, xε , v), . . . We define the maxwellians MΦ and MΦε by ∞ C# (Y
(1)
MΦ (t, x, y, v) = m(v) exp(Φ0 (t, x) − Φ(t, x, y)), x MΦε (t, x, v) = MΦ t, x, , v ε
where Φ0 is the homogenized potential: (2)
Φ0 (t, x) = −log
Z
Y
2
e
−Φ(t,x,y)
dy .
The collision cross-section σ0 := σ0 (x, y, v, v ′ ) is the two-scale limit of σε (see Section 5.1) and Qα, σi is the collision operator associated with σi (for i ≥ 0): Z (3) Qα, σi (f ) = σi (x, y, v, v ′ )(m f (v ′ )(1 − αf ) − m(v ′ )f (1 − αf (v ′ )))dv ′ . Rd
Assumptions.
Throughout this study we shall use the following assumptions:
(A1) The initial data f0ε is uniformly bounded in L1 (R2d ) and ∃ c0 > 0,
0 ≤ f0ε ≤ c0 exp(−
v2 − Φε (t = 0)). 2
+ 2, ∞ + ∞ 2d (A2) The potential Φ belongs to L∞ (R2d )) and ∂t Φ ∈ L∞ loc (R ; W loc (R ; L (R )). (A3) The cross-section σε is uniformly bounded, satisfying the micro-reversibility principle:
0 < σ ≤ σε (x, v, v ′ ) = σε (x, v ′ , v) ≤ σ ¯
(4)
and converging in two-scale strong (see section 5, or [1, 31] ). Our main result is the following Theorem 2.1. Assume that α = 0 and assumptions (A1), (A2) and (A3) are satisfied. Let f˜0 be the two-scale limit of f0ε . Then, any weak solution f ε of the linear Boltzmann equation (B0ε ) + 1 ∞ 2d associated with (5)−(9), converges in L∞ loc (R ; L ∩L (R )) weak star to ρ m where the function ρ is the solution of the Drift-Diffusion equation ∂t ρ + ∇x .J(ρ) = 0 J(ρ) = −D(t, x)(∇x ρ + ρ∇x Φ0 ) (5) ZZ f˜0 (y, v)dydv, ρ(t = 0) = ρ0 = Y ×Rd
where Φ0 is defined by (2) and D is the positive definite matrix given by (11). For the sake of clarity , we postpone to Section 6 the discussion of the Fermi-Dirac statistics (α = 1) for which our result is still formal.
The outline of the paper is as follows. Section 3, is devoted to the formal analysis of the asymptotics in the linear case, using two-scale expansions. This analysis leads to an auxiliary cell problem, studied in section 4. In Section 5 , the convergence result (Theorem 2.1) is proven in the linear case by the two-scale limit technique. Finally Section 6 is devoted to a formal analysis for the Fermi-Dirac model in which we present the formal derivation of the fluid system and study the corresponding cell problem. 3. Formal expansion for the Boltzmnn statistics. In this section, we shall present the formal analysis allowing to describe the limit model, in the case of Boltzmann statistics (Q0, σε ). To this aim, we assume that f ε and σε verify when ε goes to zero (1) (2)
x x x f ε (t, x, v) = f0 t, x, , v + ε f1 t, x, , v + ε2 f2 t, x, , v + . . . , ε ε ε x x x σε (x, v, v ′ ) = σ0 x, , v, v ′ + ε σ1 x, , v, v ′ + ε2 σ2 x, , v, v ′ + . . . ε ε ε
where the coefficients fi and σi are Y −periodic with respect to y.
1 Substituting (1) and (2) into (B0ε ), replacing ∇x (fi )ε by (∇x fi )ε + (∇y fi )ε and letting the ε coefficients of the powers of ε equal to zero, we get a hierarchy of equations: (3)
ε−2 :
L0, σ0 f0 = 0,
3
(4)
(5)
ε−1 :
L0, σ0 f1 = −v.∇x f0 + ∇x Φ(t, x, y).∇v f0 + Q0, σ1 (f0 ),
ε0 :
L0, σ0 f2 = −∂t f0 − v.∇x f1 + ∇x Φ(t, x, y).∇v f1 + Q0, σ1 (f1 ) +Q0, σ2 (f0 ),
where (6)
L0, σ0 = v.∇y − ∇y Φ.∇v − Q0, σ0
and (7)
Q0, σ0 (f ) =
Z
Rd
σ0 (x, y, v, v ′ )(m f (v ′ ) − m(v ′ )f )dv ′ .
To obtain the evolution equation satisfied by f0 , we need to study the spectral properties of L0, σ0 . This operator, acting on the Hilbert space L2MΦ = f ∈ L2 (dydv/MΦ ) / f : Y-periodic with respect to y , is unbounded with domain:
D(L0, σ0 ) = {f ∈ L2MΦ / v.∇y f − ∇y Φ.∇v f ∈ L2MΦ }. It satisfies the following Proposition 3.1. The operator L0, σ0 is maximal monotone on L2MΦ and satisfies 1. N (L0, σ0 ) = R MΦ , ZZ g(y, v)dydv = 0 . 2. R(L0, σ0 ) = g ∈ L2MΦ / Y ×Rd
3. For all g ∈ R(L0, σ0 ), thereZexists f ∈ D(L0, σ0 ) such that L0, σ0 f = g. This solution is Z unique under the condition f (y, v)dydv = 0. Y ×Rd
This proposition will be proven in section 4. Let us continue the derivation of the fluid system. According to (3), the two-scale limit of f ε has the form: (8)
f0 (t, x, y, v) = ρ(t, x)MΦ (t, x, y).
As a consequence, f0 ∈ N (Q0, σ1 ) and a simple computation of the right hand side of (4) gives: (9)
f1 (t, x, y, v) = −(∇x ρ + ρ∇x Φ0 )(t, x).L−1 0, σ0 (vMΦ )(t, x, y, v).
denoting by χ the unique solution in [R(L0, σ0 ) ∩ D(L0, σ0 )]d of (10)
L0, σ0 χ = vMΦ
and D the diffusion matrix: (11)
D(t, x) =
ZZ
Y ×Rd
v ⊗ χ(t, x, y, v)dydv
Integrating (5) with respect to dydv, the solvability condition gives the Drift-Diffusion equation associated to the normalized potential Φ0 (given in (2)): (12)
∂t ρ + ∇x .J(ρ) = 0
where (13)
J(ρ)(t, x) = −D(t, x)[∇x ρ + ρ∇x Φ0 ](t, x). 4
4. The cell Problem. We are concerned in this section with the proof of Proposition 3.1. The monotonicity of L0, σ0 is a consequence of the entropy inequality ZZ
Y ×Rd
L0, σ0 (f )
f dydv = − MΦ
Z
eΦ−Φ0
Y
Z
Q0, σ0 (f ) Rd
f dv dy ≥ 0. m
This relation implies that L0, σ0 and its adjoint L∗0, σ0 are monotone and then L0, σ0 is maximal since it is closed and has a dense domain in L2MΦ (see for instance [16] page 113). To prove 2), we remark that the two-scale limit σ0 of σε satisfies (4) and then ZZ f dydv ≥ Ckf − ρ mk2L2 − Q0, σ0 (f ) MΦ M d Φ Y ×R R where ρ(.) = Rd f (., v)dv. As a consequence, any function belonging to the null-space N (L0, σ0 ) has the form f = ρ(y)m(v). Inserting this expression in L0, σ0 (f ) = 0, we obtain ∇y ρ + ρ∇y Φ = 0. Therefore, N (L0, σ0 ) is spanned MΦ . RR To prove 3), we shall show that, for ε tending to zero and g satisfying Y ×Rd g(y, v)dydv = 0, the solution of εf ε + L0, σ0 (f ε ) = g
f ε : Y-periodic with respect to y.
is uniformly bounded. To this aim, we proceed by contradiction and assume that (f ε ) is unbounded in L2MΦ . Denoting F ε = f ε /kf ε kL2M
Φ
and Gε = g/kf ε kL2M , Φ
the sequence (F ε )ε is bounded in L2MΦ and (1)
εF ε + L0, σ0 (F ε ) = Gε .
Besides, F ε converges weakly to a function F ∈ N (L0, σ0 ). An integration of the above equation, RR the condition Y ×Rd g dydv = 0 leads to F ≡ 0. Multiplying (1) by F ε /MΦ . Integrating with respect to dydv, using Jensen’s inequality and taking advantage of the identity (v.∇y − ∇y Φ.∇v )MΦ (t, x, y) = 0, one can deduce (2)
F ε − m(v)ρF ε → 0 in L2MΦ
where (3)
ρF ε (y) =
Z
F ε (y, v)dv.
Rd
Multiplying (1) by v and integrating with respect to v, we obtain, thanks to identity (2) ∇y ρF ε = (S ε − ρF ε ∇y Φ)/d where S ε belongs to a compact subset of [H −1 (Y )]d . Consequently, F ε converges (up to an extraction of a subsequence) strongly in L2MΦ . This is not possible because kF ε kL2M = 1 and its weak Φ limit is equal to zero. 5
5. Rigorous convergence Proof for the Boltzmann statistics . The proof of Theorem 2.1 is done in two steps. In subsection 5.3, we begin by establishing useful Lp −estimates on the charge and current densities. Lemma 5.5 allows to deal with weak and two-scale convergence of f ε , ρε and j ε . In Subsection 5.4) we shall characterize the limit , as ε goes to zero, of ρε and j ε . Before developing this proof let us briefly review (in Subsections 5.1 and 5.2) some definitions and useful properties of the two-scale convergence notion. We refer to [1, 31, 35] for further details. 5.1. Two-scale convergence. Definition 5.1. Let Ω be an open set of Rn (n ≥ 1) and (uε ) a bounded sequence of L2 (Ω). The sequence (uε ) is said to be two-scale convergent, as ε goes to zero, to a function u ∈ L2 (Ω × Y ) if : Z Z Z x (1) dx = u(x, y)ψ(x, y)dxdy lim uε (x)ψ x, ε→0 Ω ε Ω Y ∞ for any smooth function ψ which is Y −periodic in y (ψ ∈ D(Ω; C# (Y ))).
The previous definition is justified by the following compactness theorem Theorem 5.2. [1]. Let (uε ) be a bounded sequence of L2 (Ω). Then, there exists a subsequence of uε which two-scale converges to a function u ∈ L2 (Ω × Y ) in the sense of the previous definition. Let us give some examples of two scale limits x converges in two-scale to u. 1. Let u ∈ L2 (Ω; C# (Y )). Then, the sequence uε (x) = u x, ε 2. RAny sequence (uε ) that converges in two-scale to a limit u, converges L2 − weakly to u(., y)dy Y 3. Any sequence (uε ) that converges strongly in L2 (Ω) to a function u, two-scale converges to the same limit u. 4. Any sequence (uε ) that admits the expansion uε (x) = u0 (x, x/ε) + ε u1 (x, x/ε) + ε2 u2 (x, x/ε) + . . . where the functions ui are smooth and Y −periodic in y, two-scale converges to the first term u0 . 5.2. Strong two scale limit. Definition 5.3. A sequence (uε )ε of L2 (Ω) is said to be two-scale strong convergent to a function u ∈ L2 (Ω × Y ) if: for any two-scale convergent sequence vε to a limit v and any function ψ ∈ D(Ω; C# (Y )), we have Z Z Z x lim uε (x)vε (x)ψ x, dx = u(x, y)v(x, y)ψ(x, y)dxdy. ε→0 Ω ε Ω Y
The following theorem which is very important in the analysis of nonlinear problems, gives a sufficient condition (in terms of energy conservation) for strong two-scale convergence. Theorem 5.4. Let (uε )ε be sequence of L2 (Ω) that two-scale convergent to a function u ∈ 2 L (Ω × Y ). Assume that lim kuε kL2 (Ω) = kukL2 (Ω×Y ) .
ε→0
Let (vε ) ∈ L2 (Ω) a two-scale convergent sequence to a limit v ∈ L2 (Ω × Y ). Then, we have Z uε vε ⇀ u(., y)v(., y)dy in D′ (Ω). Y
2
Moreover, if u ∈ L (Ω; C# (Y ), then
.
lim uε (.) − u ., 2 = 0. ε→0 ε Lloc (Ω)
Remark 1. Previous definitions of two-scale convergence can be extended to Lp spaces for p ∈]1, +∞] and also to any dual of separable Banach spaces. 6
5.3. A priori estimates. Lemma 5.5. Assume that (A1), (A2) and (A3) are satisfied. Let f ε be a weak solution of the linear Boltzmann equation (B0ε ) associated with (5)−(9). Then, f ε satisfies + 1 ∞ 2d 1. Lp −estimates: f ε is uniformly bounded in L∞ loc (R ; L ∩ L (R )), (2)
kf ε (t)kL1 (R2d ) = kf0ε kL1 (R2d ) ,
(3)
0 ≤ f ε ≤ α(t) exp −|v|2 /2 − Φε (t) ,
where
α(t) = α0 + exp
Z
0
t
k∂t Φ(s)kL∞ (Rd ) ds .
2. Distance to equilibrium: Z
(4)
0
TZ
R2d
(f ε − ρε m)2 dtdxdv ≤ CT ε2 . me−Φε
3. The current density j ε is bounded in L2loc (R+ ; L2 (Rd )).
Proof of Lemma 5.5. Inequality (2) is immediate. Inequality (3) is deduced from the weak maximum principle. Indeed, the maxwellian
2
˜ ε (t, x, v) = α(t) exp − |v| − Φε (t, x, x ) M 2 ε
solves the following scaled Boltzmann equation ˜ ε (t, x, v) = (k∂t Φε (t)kL∞ − ∂t Φε (t, x))M ˜ ε (t, x, v) TΦε M ˜ ε − f ε is then a weak solution of the system satisfying where TΦε is defined in (5). The function M
˜ ε − f ε ) ≥ 0, TΦε (M
(M ˜ ε − f ε )(t = 0) ≥ 0.
˜ ε − f ε . To establish the estimate (4), we multiply (B0ε ) The maximum principle implies the positivity of M ε −Φε by f /(m(v)e ) and integrate on all variables, we obtain d dt
Z
R2d
f ε2 dxdv m(v)e−Φε
+
1 ε2
Z
R2d
(f ε − ρε m)2 dxdv ≤ α′ (t) m(v)e−Φε
Z
R2d
f ε2 dxdv m(v)e−Φε
A Gronwall lemma yields, thanks to assumption (A3), to (3) and the boundedness of the current density in L2loc (R+ ; L2 (Rd )). Corollary 5.6. The charge and current densities, given by (4), satisfy the following Green formula (5)
Z
0
+∞
Z n
ρε (∂t ψ)ε + j ε .(∇x ψ)ε +
Rd
1 ε j .(∇y ψ)ε ε
d ∞ for all ψ ∈ D(R+ t × Rx ; C# (Y )).
o
+
Z
R2d
f0ε ψ 0, x,
x dxdv = 0 ε
5.4. Passing to the limit: duality method. Lemma 5.7. The charge density ρε and the current density j ε satisfy ρε ⇀ ρ˜ := ρ(t, x) exp(Φ0 (t, x) − Φ(t, x, y)), j ε ⇀ ˜j :=
Z
Rd
two-scale
(v ⊗ χ(t, x, y, v)) .(∇x ρ + ρ∇x Φ0 )(t, x),
two-scale
where Φ0 and χ are defined in (2) and (10) respectively and ρ is the weak limit of ρε . Proof of Lemma
5.7. Let us denote g ε = (f ε − ρε m)/ε. Integrating (B0ε ) with respect to vdv, we obtain Z Z 2 ε ε (6) ε ∂t j + ε∇x . (v ⊗ v)g dv + d∇x ρε = −dρε ∇x Φε + vQ0, σε (g ε )dv Rd
Rd
7
∞ Multiplying this equation by a test function εψ(t, x, x/ε) where ψ ∈ [D(R∗+ × Rdx ; C# (Y ))]d and integrating with respect to dtdx yields to Z Z 1 (7) ερε (∇x .ψ)ε + (∇y .ψ)ε − ∇x Φε .(ψ)ε dtdx = §ε , d ε R+ Rd
where ε
§ :=
−ε
Z
Z
−ε
Z
Z
(8)
R+ Rd
R+
Z 2 ε ε (∂t ψ)ε j +
ε
Rd
(v ⊗ v)g dv (ε∇x ψ + ∇y ψ)ε dtdx
v Q0, σε (g ε )(ψ)ε dtdxdv.
R2d
By analyzing (2), (3) and (4), one can show that §ε converges to zero. Moreover, Φε is regular enough to get ε∇x Φε −→ ∇y Φ
two-scale strong.
Therefore, by passing to the limit in (7), the two-scale limit ρ˜ of ρε satisfies Z +∞Z Z ∞ (Y ))]d . ρ˜ [∇y .ψ − ∇y Φ.ψ] dtdxdy = 0, ∀ ψ ∈ [D(]0, +∞[×Rdx; C# Rd
0
Y
An integration by parts gives: d d ρ˜ ∈ L∞ (R+ loc × Rx × Ry )
and ∇y ρ˜ = −ρ˜∇y Φ,
1,∞ d wich in turn implies ρ˜ ∈ L∞ (R+ loc × R ; W# (Y )) and
ρ˜(t, x, y) = ρ(t, x) exp(Φ0 (t, x) − Φ(t, x, y)) where ρ (the average of ρ˜) is the weak limit of ρε . Let now ψ ε be a test function belonging to D(R+ × R2d ). The weak formulation associated to (B0ε ) associated with (5) − (9) reads Z 1 ε ε ε ε ε f ε∂t ψ + v∇x ψ − (∇x Φ)ε .∇v ψ − (∇y Φ)ε .∇v ψ dtdxdv ε R+ ×R2d (9) Z Z 1 + Q0,σε (f ε )ψ ε dtdxdv + ε f0ε (x, v)ψ ε (0, x, v)dxdv = 0. ε R+ ×R2d 2d R Choosing ψ ε (t, x, v) :=
εϕ(t, x, xε , v)
x
m(v)e−Φ(t,x, ε )
∞ with ϕ ∈ D(R+ × Rdx ; C# (Y ) × L∞ (Rdv )) and ϕ/m(v) ∈ L∞ , and using the property ε
[v.∇x − ∇x Φε ∇v ](m(v)e−Φ
(t,x)
) = 0,
the above formulation becomes Z Z fε (ε∂ ϕ + εϕ∂ Φ + v.∇ ϕ − ∇ Φ.∇ ϕ) + f0ε ψ ε (0) t t x x v ε ε −Φ R+ ×R2d m(v)e R2d (10)
−
Z
R+ ×R2d
1 = ε
Z
gε (v.∇y ϕ − ∇y Φ.∇v ϕ − Q0, σε (ϕ))ε m(v)e−Φε
R+ ×Rd
Z ∇y .
Rd
vϕdv
ε
ρε e Φ ε
8
where the notation (.)ε means that we take expression between brackets at point (t, x, x/ε, v) and we used f ε = ρε m(v) + ε g ε . Let us denote D# and E# Z d ∞ ∞ d D# = D(R+ × R ; C (Y ) × L (R )), E = ϕ ∈ D / ∇ . vϕdv = 0 # # y t x # m v Rd
and pass to the limit only for ϕ ∈ E# in order to overcome the difficulty due to the right hand side of (10). We will show later that this choice allows to describe the limit equation. From Lemma 5.7, Estimate (4) and thanks to two-scale strong convergence of Φε and σε , one can deduce ε
eΦ f ε m(v)
⇀
2d If we denote g˜ ∈ L2 (R+ × Y, loc × R
dtdxdydv m(v) )
ε
g ε eΦ m(v)
ρ(t, x)eΦ0 (t,x) ,
⇀
two-scale.
the two-scale limit of g ε , then
g˜ eΦ g˜ eΦ0 (t,x) := , m(v) MΦ
two-scale
Q0, σ0 (˜ g eΦ ) , m(v)
two-scale.
and ε
Q0, σε (g ε eΦ ) m(v) We deduce from (10) " Z
ρe
R+ ×Y ×Rd
Φ0
⇀
# (L∗0, σ0 ϕ)˜ g eΦ0 (v∇x ϕ − ∇x Φ.∇v ϕ) − dtdydv = 0, MΦ
∀ϕ ∈ E# .
which means that v∇x (ρeΦ0 ) +
L0, σ0 (˜ g eΦ0 ) ⊥ ∈ E# . MΦ
The following lemma proven by Poupaud and Goudon, characterizes the orthogonal of E# . ′ ⊥ ′ Lemma 5.8. [31]. Let T ∈ D# . Then, T belongs to E# if and only if there exists S ∈ D# such that ∇v S = 0 and T = v.∇y S. ′ According to this lemma, there exists S ∈ D# satisfying ∇v S = 0 and
v∇x (ρeΦ0 ) +
L0, σ0 (˜ g eΦ0 ) = v.∇y S. MΦ
Multiplying this equation by MΦ , combined with (10) imply L0, σ0 (˜ geΦ0 + ∇x (ρeΦ0 )χ − SMΦ ) = 0 and then g˜eΦ0 = MΦ S − (∇x ρ + ρ∇x Φ0 )eΦ0 (t,x) χ(t, x, y, v). Since j ε (t, x) =
Z
vg ε (t, x, v)dv,
Rd
the two-scale limit of j ε is given by the formula: Z ˜j(t, x, y) = − (v ⊗ χ(t, x, y, v))dv .(∇x ρ + ρ∇x Φ0 )(t, x). Rd
9
Integrating this equation on the unit cell, we obtain j ε ⇀ J(ρ) := −D(t, x). [∇x ρ(t, x) + ρ(t, x)∇x Φ0 (t, x)] where D(t, x) is defined by (11). Enf of the Proof of Theorem 2.1: Conclusion To complete the proof of Theorem 2.1, we pass to the limit in (5) with test function ϕ(t, x, v) not depending on y. This leads to the weak formulation of the Drift-Diffusion equation (5). Corollary 5.9. The diffusion matrix D(t, x) is positive definite. Proof. Let define by Ms the symmetric part of the matrix D. For all i, j ∈ {1, 2, . . . , n} Msi,j =
1 2
Z
Y ×Rd
(L0, σ0 (χi )χj + χi L0, σ0 (χj ))
dydv MΦ
where L0, σ0 (χi ) = vi MΦ (y, v) and L0, σ0 is defined by (6). Let η ∈ Rd and U = show, thanks to entropy inequality, that s
(M η, η) = where ρU =
R
Rd
Z
Y ×Rd
L0, σ0 (U )U
d X
χi ηi . It is simple to
i=1
dydv ≥ CkU − ρU mk2L2 MΦ MΦ
U dv. Therefore, if (Ms η, η) = 0 then U belongs to N (L0, σ0 ) and for all v ∈ Rd L0, σ0 (U ) = (η, v)MΦ = 0
which implies that η = 0 and M is definite positive.
6. Formal analysis of the Fermi-Dirac statistics. This section is devoted to the formal derivation of the fluid model corresponding to the Fermi-Dirac model (α = 1). We are concerned with the limit (ε → 0) of the solution f ε of ε ∂t f ε + 1 TΦε (f ε ) = Q1, σε2(f ) , ε
(1)
(x, v) ∈ R2d
ε
f ε (x, v, 0) = f0ε (x, v)
where Φε and Q1, σε are given by (8) and Q1, σε (f ) =
Z
Rd
σε (x, v, v ′ )(m f (v ′ )(1 − f ) − m(v ′ )f (1 − f (v ′ )))dv ′ .
We assume instead of (A1): (A’1)
0 ≤ f0ε ≤ 1 and f0ε is uniformly bounded in L1 (R2d ).
Φ and σε satisfy assumptions (A2) and (A3) respectively. It is well known that under the natural assumption (A’1), the system (1) has a weak solution satisfying kf ε (t)kL1 (R2d ) = kf0ε kL1 (R2d ) and 0 ≤ f ε ≤ 1. Thus, one can deal with the two-scale convergence for f ε .
6.1. Fluid system. Assume that f ε and σε admit (1) and (2) as an asymptotic expansion. One can show that Q1, σε (f ε ) = Q1, σ0 (f0 ) + ε (Q1, σ1 (f0 ) + DQ1, σ0 (f0 )f1 ) (2)
+ε2 DQ1, σ0 (f0 )f2 + Q0, σ2 (f0 ) + DQ1, σ1 (f0 )f1 +
Z
Rd
+...
σ0 f1 f1 (v ′ )(m(v ′ ) − m)dv ′
where DQ1, σi is the linearized of Q1, σi : DQ1, σi (f )(g) = Q0, σi (g) +
Z
Rd
σi (f (v ′ )g + f g(v ′ ))(m(v ′ ) − m)dv ′ . 10
Inserting the asymptotic expansion (1) in (1), thanks to (2), an identification of coefficients with same power on ε leads to (3)
L1, σ0 f0 = 0,
(4)
(v.∇y − ∇y Φ.∇v − DQ1, σ0 (f0 ))f1 = −v.∇x f0 + ∇x Φ.∇v f0 + Q1, σ1 (f0 ), (v.∇y − ∇y Φ.∇v − DQ1, σ0 (f0 ))f2 = −∂t f0 − v.∇x f1 + ∇x Φ.∇v f1 + Q0, σ2 (f0 )
(5)
+DQ1, σ1 (f0 )f1 +
Z
Rd
σ0 f1 f1 (v ′ )(m(v ′ ) − m)dv ′
where ε
2
L1, σ0 = v.∇y − ∇y Φ.∇v − Q1, σ0 .
The solution f is bounded in L . Assume that f ε has (1) as an asymptotic expansion. Then, it converges in two-scale to the leading term f0 . Letting ε goes to zero, we wish to describe the equation satisfied by the average of f0 . This is the homogenized fluid model. To obtain this model, we shall prove the existence and uniqueness of the solution of (3) and then study the cell problem associated with the linearized operator. Namely, a necessary solvability condition of (4) and (5) provides Drift-Diffusion model. First equation. We consider the function (6)
H(f ) =
f . (1 − f )MΦ
Then L1, σ0 satisfies Lemma 6.1. 1. Entropy inequality:
Z
Y ×Rd
(7)
1 = 2
Z
L1, σ0 (f )H(f )dydv = −
Y ×R2d
Z
Q1, σ0 (f )H(f )dydv
Y ×Rd
σ0 (1 − f )(1 − f (v ′ ))MΦ m(v ′ ) H(f (v ′ )) − H(f )
2. The null-space N (L1, σ0 ) is (8)
N (L1, σ0 ) = {F (µ(t, x) + Φ0 (t, x) − Φ(t, x, y)),
2
dydvdv ′ ≥ 0.
¯ µ ∈ R}
where F (η) is the Fermi-Dirac function associated with the chemical potential η: F (η) =
1 1 + exp(|v|2 /2 − η)
¯ ∀η ∈ R.
Proof. Multiplying (3) by H(f ) and integrating gives
Z
L1, σ0 (f )H(f )dydv = −
Y ×Rd
Z
Q1, σ0 (f )H(f )dydv
Y ×Rd
since (v.∇y − ∇y Φ.∇v )(MΦ ) = 0. Using micro-reversibility property of the cross-section σ0 , we have
Z
2
L1, σ0 (f )H(f ) =
Y ×Rd
Z
Y ×R2d
σ0 (1 − f )(1 − f (v ′ ))MΦ m(v ′ )[H(f (v ′ )) − H(f )]2 dydvdv ′ ≥ 0.
This implies if L1, σ0 (f ) = 0 then H(f ) doesn’t depend on v. Therefore, ∀ y ∈ Y,
f (y) =
h MΦ (y) ∈ N (Q1, σ0 ) (1 + h MΦ (y))
where h is independent on v. Rewrite (3), gives 0 = (v.∇y − ∇y Φ.∇v )f = v.∇y h 11
MΦ (1 + h MΦ )2
which implies that h does not depend on y. Hence f= where µ(t, x) = log
h(t, x)MΦ = [1 + exp(|v|2 /2 − µ(t, x) − Φ0 + Φ(t, x, y)]−1 . (1 + h(t, x)MΦ )
h(t,x) (2π)d/2
.
As a consequence of this lemma, the solution f0 of (3) is a Fermi-Dirac distribution associated with the chemical potential µ(t, x) + Φ0 (t, x) − Φ(t, x, y). ¯ Let us now define for all µ(t, x) ∈ R
F (µ(t, x), y, v) =
(9)
1 , 1 + e|v|2 /2−µ(t,x)+Φ(t,x,y)
L(µ, σ) = v.∇y − ∇y Φ.∇v − DQ1, σ (F (µ)),
In all the sequel, F (µ) = F (t, x, y, v) will denotes the Fermi-Dirac function associated with µ(t, x) − Φ(t, x, y). The following proposition will provide an equation on the averaged of f0 = F (µ + Φ0 ). ¯ and σ a cross section satisfying (4). Then, L(µ, σ) is maximal Proposition 6.2. Let µ(t, x) ∈ R monotone on LF (1−F )(µ) identified to L2# (Y ; L2 (Rd )) with the inner product (10)
hf, giF (1−F )(µ) =
Z
Y ×Rd
fg dydv. F (µ)(1 − F (µ))
1. The null-space of L(µ) is equal to
N (L(µ, σ)) = R F (µ)(1 − F (µ))
if µ ∈ R
N (L(−∞, σ)) = R MΦ
(11)
2. R(L(µ, σ)) =
N (L(+∞, σ)) = R MΦ−1 .
g ∈ LF (1−F )(µ) ;
Z
g(y, v)dydv = 0
Y ×Rd
tion if and only if f and g ∈ R(L(µ, σ)).
and L(µ, σ)(f ) = g has only one solu-
As a consequence, there exists one solution h(µ) ∈ [R(L(µ, σ))]d of (12)
L(µ, σ)(h(µ, σ)) = vF (µ)(1 − F (µ)).
Moreover, (13)
Π(µ, σ) =
Z
Y ×Rd
v ⊗ h(µ, σ0 , y, v)dydv
is positive definite for all µ ∈ R and Π(±∞, σ) = 0. This proposition will be proved (partially) in the last paragraph. Let us continue the formal derivation and deduce from (4) and (5) an equation on ρ=
Z
F (µ + Φ0 )dydv.
Y ×Rd
To do this, we replace in (4) f0 by F (µ + Φ0 ). We obtain L(µ + Φ0 , σ0 )f1
=
−∇x (µ + Φ0 ).vF (µ + Φ0 )(1 − F (µ + Φ0 )).
In view of (12), there exists f1 ∈ [R(L(µ + Φ0 , σ0 )) ∩ D(L(µ + Φ0 ))]d and satisfying f1 = −∇x (µ + Φ0 ).h(µ + Φ0 , σ0 ). Therefore, equation (5) becomes L(µ + Φ0 , σ0 )(f2 ) = −∂t F (µ + Φ0 ) + ∇x . [v ⊗ h(µ + Φ0 , σ0 )∇x (µ + Φ0 )] Integrating this equation with respect to dydv leads to ∂t ρ − ∇x [Π(µ + Φ0 , σ0 )∇x (µ + Φ0 )] = 0 12
and if f0 is the two-scale limit of f0ε , then ρ is a solution of the following drift-diffusion system
∂t ρ(t, x) + ∇x .J(µ + Φ0 ) = 0 J(µ + Φ ) = −Π(µ + Φ , σ )(t, x).∇ (µ + Φ )(t, x) 0 0 0 x 0 Z f0 (x, y, v)dydv. ρ(0, x) =
(14)
Y ×Rd
6.2. Spectral analysis of L(µ). This subsection is devoted to the proof of proposition 6.2. The outline is exactly the same as in the Boltzmann statistics. We shall only explain how to prove the monotony of the linearized operator L(µ). We begin by writing DQ1,σ (F (µ)) in the following form DQ1,σ (F (µ))(r)
=
K(µ)(r) − λ(µ)r
where λ(µ) and K(µ) are given by λ(µ) =
Z
σ[F ′ (µ)m + (1 − F (µ)′ )m′ ]dv ′
Rd
and
Z
K(µ)(r) =
Rd
σ[F (µ)m′ + (1 − F (µ))m]r ′ dv ′ .
Using property of σ, one can deduce
(1 − F ′ )m′ (1 − F )m ′ m′ m )r − ( + )r dv ′ −DQ1,σ (F (µ))(r) = σF (µ)F (µ) ( + ′ ′ F F F F FF′ Rd
Z
′
Let denote, for µ ∈ R, G(µ) =
(1 − F ′ )m′ m + , F FF′
a simple computation gives √ (1 − F (µ))m eµ˜ = e−µ˜ , G(µ) = and e−µ˜ = e−µ+Φ /( 2π)d F (µ) F (µ)(1 − F (µ)) and −DQ1,σ (F (µ)) satisfies (for all µ ∈ R) (15)
−2
Z
Rd
DQ1,σ (F (µ))(r)r = e−µ˜ F (1 − F )(µ)
Using the property (16)
ZZ
R2d
σF (µ)F (µ)′ G(µ, v)r − G(µ, v ′ )r ′
2
dvdv ′ .
(v.∇y − ∇y Φ.∇v ) [F (µ)(1 − F (µ))] = 0,
we deduce easily the monotony of L(µ) : (17)
Z
Y ×Rd
L(µ)(r) r dydv =− F (µ)(1 − F (µ))
Z
DQ1,σ (F (µ))(r) r
Y ×Rd
dydv ≥ 0. F (µ)(1 − F (µ))
and by the same argument as proposition 3.1, L(µ) is maximal monotone. Moreover, if L(µ)(r) = 0, then r = k(t, x, y)F (µ)(1 − F (µ)) ∈ DQ1,σ (F (µ)). Writing L(µ)(r), we obtain L(µ)(kF (µ)(1 − F (µ))) = ∇y k.vF (µ)(1 − F (µ)) = 0. ¯ N (L(µ)) = R F (µ)(1 − F (µ)). Therefore, for µ ∈ R, The proof of the solvability condition of L(µ)f = g: (18)
Z
g(y, v)dydv = 0. Y ×Rd
is exactly the same as in the first case. It is based on the entropy inequality which allows to deduce that under this condition all solution of (19) εf ε + L(µ)f ε = g.
is bounded in LF (1−F )(µ) (as ε goes to zero). The positivity of the diffusion matrix is a consequence of the monotony of L(µ). We refer to [29] for further details. 13
6.3. Concluding remarks. Two Drift-Diffusion models are derived from Semiconductors Boltzmann equations, corresponding to a given electrostatic potential varying on the scale of the mean free path of particles. When the collision operator is linear, the proof of convergence is rigorous. In the nonlinear case, the Fermi-Dirac statistics, only the formal derivation and Existence and uniqueness of solutions of the cell problem are studied. We think that these results can be improved but some serious difficulties must be overcome . It seems that rigorous proof for the nonlinear case needs a compactness argument like two-scale averaging lemma. The difficulty here arises from the fact that the double scale limit of f ε has a double scale dependence which is not the case for periodic homogenization for nonlinear elliptic partial differential equations [13, 47] etc. Indeed, in the elliptic case, the limit varies only on the macroscopic scale and ellipticity estimates often provides uniform estimates on the gradient of the solution. In a forthcoming paper however, the second author tackles this problem by taking advantage of the explicit knowledge of the double scale dependence of the limit solution. We illustrate this difficulty by explaining why the Hilbert expansion method fails in proving the convergence. In order to avoid technicality (like linearization and boundary layers) we consider the linear case in the whole space setting. In analogy to [12, 37, 48], we write the Boltzmann equation for the remainder (20)
r ε (t, x, v) := f ε (t, x, v) − f0 (t, x,
x x x , v) − εf1 (t, x, , v) − ε2 f2 (t, x, , v) ε ε ε
and try to prove the convergence of r ε in L1 . In this case, f0 and f1 are given by (8) and (9) respectively and f2 is a solution of (5), which can be written under the following form −L0, σ0 f2
=
(∂t ρ)MΦ + ∇x .[(v ⊗ χ).D−1 J(ρ)] + [∇x Φ ⊗ D−1 J(ρ)].∇v χ +Q0,σ1 (f1 ) + ρ∂t MΦ .
where χ, D and J(ρ) are given by (10) (11) and (12) respectively. This implies f2 = χ1 + χ2 + χ3 + ρ(t, x)χ4 where χ1 , χ2 and χ3 are respectively the solutions (on component wise) of −L0, σ0 χ1 = (∂t ρ)MΦ + ∇x .[(v ⊗ χ)D−1 J(ρ)], −L0, σ0 χ2 = [∇x Φ(t, x, y) ⊗ D−1 J(ρ)].∇v χ, −L0, σ0 χ3 = Q0,σ1 (f1 ) and χ4 ∈ D(L0, σ0 ) ∩ R(L0, σ0 ) the solution of −L0, σ0 χ4 = ∂t (MΦ ). The remainder r ε defined by (20) is then a solution of the following scaled Boltzmann equation : (21)
TΦε (r ε ) = S ε = −ε{(∂t f1 )ε + ε(∂t f2 )ε + v.(∇x f2 )ε − (∇x Φ.∇v f2 )ε }
where TΦε is given by (5). The L1 −estimate associated to this equation is kr ε (t)kL1 (Ω) ≤ kr ε (t = 0)kL1 (Ω) +
Z
t 0
kS ε (s)kL1 (Ω) ds.
According to this estimate, the convergence of r ε towards zero "needs" an approximation of the right hand side of (21) in L1 norm. In view of expressions of f1 and f2 , this property requires Sobolev regularities on the solution χ of
(
L0, σ0 (χ) := v.∇y χ − ∇y Φ(t, x, y).∇v χ − Q(χ) = vMΦ , χ:
Y-periodic on y.
Due to the coupling between variables y and v in the transport part of L0, σ0 , it seems to be difficult to have more than L2MΦ −regularity on χ. Acknowledgments. The authors acknowledge support from the European IHP network No. RNT2 2001 349 entitled "HYperbolic and Kinetic Equations : Asymptotics, Numerics, Analysis" and from the CNRSDGRST bilateral project No. 13904 entitled "Modèles classiques et quantiques de transport électronique : analyse asymptotique et simulations". 14
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