Compressed asymptotic analysis of a quantum transport equation
Chiara Manzini & Giovanni Frosali
University of Florence, chiara.manzini@unifi.it & giovanni.frosali@unifi.it
Wigner-BGK equation
∂tw + v · ∇xw −Θ[V ]w = −ν(w − weq), x, v ∈ IRd, t > 0 (1)
• w(x, v, t) ∈ IR quasi-distribution function, w(t = 0) = w0,
• V external potential enters via Θ[V ]w(x, v) = i
~(2π)d2 Z
δV (x, ~η)Fvw(x, η)eiηvdη with δV (x, ~η) := V (x + ~η/2) − V (x − ~η/2), Fv Fourier tr.
• interaction with the environment: after a time 1/ν the system will relax toweq(quantum corrected thermal eq.)
weq(x, v) :=³ m 2π~
´d e−βH
( 1 + ~2
"
−β2 8m
d
X
r=1
∂2V
∂x2r+ + β3
24m
d
X
r=1
µ ∂V
∂xr
¶2
+β3 24
d
X
r,s=1
vrvs ∂2V
∂xr∂xs
#
+ O(~4) )
where H(x, v) := mv2/2+V (x) is the Hamiltonian, β ≡ 1/kT , with T (constant) temperature and k Boltzmann constant.
Formulation of the problem
• Parametrize ([Gardner 94]) the equilibrium function via n(x, t) ≡ n[w](x, t) :=
Z
w(x, v, t) dv ≈ Z
weq(x, v) dv (2) then the Wigner thermal equilibrium function reads
weq(x, v) = n(x, t)n
F (v) + ~2F(2)(x, v)o
+ O(~4), with F (v) = (βm/2π)d/2e−βmv2/2 and F(2) O(~2)-correction.
• Rescale (1) such that tV
t0
≈ tC
t0
≈ǫ, with tV, tC, t0potential, collision and system characteristic times. Thus,
ǫ∂tw +ǫv · ∇xw −Θ[V ]w = −ν (w − weq), (3)
• Let Xk := L2(IR2d, (1 + |v|2k)dx dv; IR), Xkv := L2(IRd, (1 +
|v|2k)dv; IR). Eq. (3) in abstract form reads ǫ∂tw =ǫSw +Aw +Cw, lim
t→0+kw(t) − w0kXk= 0 (4) with Su := −v· ∇x, Aw :=Θ[V ]w, Cw:=−(ν w − Ω w),
Ωw(x, v) = νn[w](x)h
F (v) + ~2F(2)(x, v)i .
Preliminary result ((4) with ǫ= 0)
If V ∈ Wk,∞ with 2k > d,A+C∈ B(Xk) (resp.ly, B(Xkv)).
∀ x ∈ IRd fixed, ker(A+C) := {cM (v), c ∈ IR} ⊂ Xkv with
M (x, v) := νF−1
( FvF (η) ν − iδV (x, η)
Ã
1 − β~2 24m2
d
X
r,s=1
ηrηs
∂2V (x)
∂xsxs
!)
For all h ∈ Xkv, (A+C)u = h has a solution if and only if Z
h(v) dv = 0 .
Compressed expansion [Mika&Banasiak 95]
Xk= {α(x)M (x, v), α ∈ Xkx} ⊕
½ f ∈ Xk
¯
¯
¯
¯ Z
f (v)dv = 0
¾
Define the projections Pf = MR f(v)dv and Q = I − P.
w = Pw + Qw =: ϕ + ψ ≡ Mn[w]+ ψ By operating formally with P and Q on both sides of (4),
∂tϕ = PSPϕ + PSQψ (5)
∂tψ = QSPϕ + QSQψ + (1/ǫ) Q(A+C)Qψ with initial conditions ϕ(0) = ϕ0 = Pf0, ψ(0) = ψ0 = Qf0. Split ϕ and ψ into the sums of “bulk” and initial layer parts
ϕ(t) = ¯ϕ(t) + ˜ϕ (t/ǫ) , ψ(t) = ¯ψ(t) + ˜ψ (t/ǫ) . Let ¯ϕ unexpanded and expand ˜ϕ, ¯ψ, ˜ψ as
ψ(t) = ¯¯ ψ0(t) +ǫψ¯1(t) +ǫ2ψ¯2(t) + . . .
Eqs. (5) for the bulk part terms up to the orderǫ2 become
∂tϕ¯ = PSP ¯ϕ + PSQ ¯ψ0+ǫPSQ ¯ψ1 (6) 0 = Q(A+C)Q ¯ψ0
0 = QSP ¯ϕ+ Q(A+C)Q ¯ψ1 (7)
Auxiliary problems
• To solve (7) with unknown ¯ψ1, solve (A+C)D1 = −v· ∇xM + M
Z
v· ∇xM dv, (A+C)D2 = M
µ
−v + Z
vM dv
¶
with unknowns D1, (D2)i∈ ker P. Hence (6) becomes
∂n
∂t = −∇x·
· n
Z
vM dv +ǫ µZ
v ⊗ D2dv· ∇xn + n Z
vD1dv
¶¸
.
• The initial value comes from the analysis of the initial layer.
• The coefficients can be expressed in terms of the moments of M R vM(x, v) dv , R v ⊗ vM(x, v) dv
Quantum drift-diffusion equation (high-field)
∂n
∂t = 1
νm∇· (n∇V ) + ǫ
νβm∇· ∇n
+ ǫ
ν3m2∇· [∇V ⊗ ∇V∇n + n2 (∇ ⊗ ∇) V ∇V + n∆V ∇V] + ǫβ~2
12νm2∇· [(∇ ⊗ ∇) V ∇n + n∇· (∇ ⊗ ∇) V ]
(cf.[Poupaud 92]). By (2)∇ log n = −β∇V + O(~2), thus
∂n
∂t− 1
νm∇· (n∇V )− ǫ
νβm∇· ∇n− ǫ
ν3β2m2∇·µ (∇ ⊗ ∇n)∇n n
¶
+ ǫ~2 12νm2
µ
∇4n − ∇ ⊗ ∇∇n ⊗ ∇n n
¶
= 0 .
Special thanks to Elidon Dhamo (University of M¨unster) & Francesco Mugelli (University of Florence) for technical support.
