Random wave equation: statistical
stability and the diffusion regime
Lenya Ryzhik, Stanford
Tomasz Komorowski, Poland
Guillaume Bal, Columbia
Stefano Olla, Paris
Outline:
1. The wave field for the wave equation in a random medium
2. Diffusion approximation for time-dependent random media
Supported by AFOSR
1
Overall goal: understand behavior of waves and particles in
weakly random media – long distances and large times.
Applications: (1) forward problems – when and what to measure, (2) inverse problems.
Questions: (i) effective macroscopic models, (ii) correctors
to such models, (iii) getting away from scale separation to
an extent possible
Today’s talk: (1) the wave profile in the Fourier domain
as a random process, (2) diffusion approximation in timedependent random media
2
Wave equation in a random medium:
1 ∂ 2p
− ∆p = 0.
2
2
c (x) ∂t
Weak random fluctuations:
√
c(x) = c0(1 + εµ(x)), ε 1 – mean square size of the fluctuations.
Mean-zero: hµ(x)i = 0
Spatially homogeneous: hµ(x)µ(y)i = R(x − y).
”Correct” (non-trivial effects) propagation distance and time:
L ∼ l/ε, T = c0l/ε,
l – correlation length of the medium.
3
Probing a random medium with a pulse
Central wave length λ, medium correlation length l:
λ l – effective medium approximation, poor resolution,
weak scattering
λ l – random geometric optics, strong scattering, no ballistic information
λ ∼ l – radiative transport regime, weak ballistic information
Goals: (i) penetration and (ii) resolution.
For penetration: low frequency signal.
For resolution: high frequency.
Balance: radiative transport regime.
4
Radiative transport regime for waves
In the rescaled variables:
∂ 2p
2
−
c
0
∂t2
√
x
1 + εµ( ) ∆p = 0.
ε
!
Initially: p(0, x) = p0(x/ε), or p(0, x) = A(x)eiS(x)/ε, or other
oscillatory data
p(t, x) is very oscillatory – look at |p(t, x)|2
The Wigner transform (canceling the rapid phase):
W (t, x, k) =
Z
εy
εy
dy
ik·y
e
p t, x −
p̄ t, x +
2
2 (2π)d
!
!
5
The radiative transport equation:
Z
∂W
2
0)(W (t, x, k 0) − W (t, x, k))dk 0
+ c0k̂ · ∇xW = c2
|k|
R̂(k
−
k
0
∂t
0
|k|=|k |
The ballistic part: W (0)(t, x, k) = W0(x − c0k̂, k)e−Σ(k)t
Total cross-section: Σ(k) =
Z
2
c2
|k|
R̂(k
0
|k|=|k0|
− k0)dk0
Exponential decay in time (but low frequencies scatter less).
What can we say about the field itself?
6
A simplified model: a randomly forced wave equation
√
x
1 ∂ 2p
Replace
− 1 + εµ( ) ∆p = 0
2
2
c0 ∂t
ε
1 ∂ 2p
− ∆p = F (t, x),
by
2
2
c0 ∂t
!
with a random force F (t, x).
Requirements on the force:
(1) local in time and space,
(2) energy conservation,
(3) momentum conservation
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A discrete model (Komorowski, Olla, Ryzhik’12): a random
exchange of momentum so that the total kinetic energy and
the total momentum are conserved: add to each triple of
adjacent particles, a diffusion on the corresponding circle of
constant energy and momentum:
dqy (t) = py (t)dt
dpy (t) =
∆qy − ω02qy
dt + dηy (εt),
∆qy = qy+1 + qy−1 − 2qy – the discrete Laplacian.
dηy (εt) – energy and momentum preserving
(weak) noise.
v
The dispersion relation ω(k) =
q
α̂(k) =
u 2
uω
t 0
2
+ 4 sin2(πk).
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The complex wave function (in the Fourier domain):
ψ̂(t, k) := ω(k)q̂(k, t) + ip̂(t, k)
|ψ̂(t, k)|2 – the energy density in the mode space.
dψ̂(t, k) = −iω(k)ψ̂(t, k)dt + idη̂(εt, k).
The average energy: the limit
t
lim E|ψ̂ , k |2 = Ē (t, k)
ε→0
ε
!
is the solution of the radiative transport equation
∂tĒ (t, k) =
Z
R(k, k0)
h
Ē
t, k0
− Ē (t, k) dk0.
i
9
For the wave function: remove deterministic fast phase:
ψ̃ (ε)(t, k) := eiω(k)t/εψ̂(t/ε, k).
The limit (in law and pointwise in k):
lim ψ̃ (ε)(t, k) = ψ̃(t, k) – an Ornstein-Uhlenbeck process:
ε→0
q
β̂(k)
ψ̃(t, k)dt + R(t, k)dwk (t),
dψ̃(t, k) = −
4
Z
with explicit β̂(k), R(t, k) = Ē(t, k0)R(k, k0)dk0,
wk (t) – independent Brownian motions. Explicit solution
1 β̂(k)t
−4
ψ̃(t, k) = e
ψ̂(0, k) +
Z t
q
−1
β̂(k)(t−s)
e 4
R(s, k)dwk (s).
0
The radiative transport solution follows:
Ē(t, k) =
1 β̂(k)t
−2
e
|ψ̂(0, k)|2
+
Z t
−1
e 2 β̂(k)(t−s)R(s, k)ds
0
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Summarize: after phase conjugation the wave field obeys a
stochastic differential equation in the Fourier domain – this
gives point-wise statistics in the Fourier space.
What about self-averaging? Pointwise:
q
β̂(k)
dψ̃(t, k) = −
ψ̃(t, k)dt + R(t, k)dwk (t)
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But in the sense of distributions: simple attenuation
ψ̃(t, k) = e−β̂(k)t/4ψ̂(0, k) in probability
That is, for a smooth test function η(k):
Z
η(k)ψ̃(t, k)dk →
Z
e−β̂(k)t/4ψ̂(0, k)dk.
Simple phase conjugation and local averaging in the Fourier
domain removes stochasticity.
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Contrasting with 1D:
1. In higher dimensions the limit is a process in time – time
correlations do not disappear as in 1D.
2. In higher dimensions averaging over the Fourier modes
kills stochasticity.
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Diffusion approximation in time-dependent random media:
Komorowski, Ryzhik’13. RTE in a time-independent medium:
Z
∂ W̄
+ k · ∇xW̄ = R(k2 − p2, p − k)(W̄ (t, x, p) − W̄ (t, x, k))dp.
∂t
In a time-dependent medium:
Z
∂ W̄
+ k · ∇xW̄ =
R(p − k)(W̄ (t, x, p) − W̄ (t, x, k))dp.
|k|=|p|
∂t
Spohn’77, Erdös-Yau’01, Bal-Papanicolaou-R’02, ...
Expected long time limit: energy equipartition over the energy shell and diffusion approximation:
∂a
W̄ (t, x, k) = a(t, x, |k) and
= D(|k|)∆xa.
∂t
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For time-independent media: Keller-Larsen’70s energy
equipartition over energy shells, diffusion approximation.
Time-dependent media: the energy shell is the whole space
– all modes are coupled. Perfect energy equipartition is impossible!
Where does the energy go?
High modes?
Low modes?
Oscillate between high and low?
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In a Markovian and isotropic medium:
Z
∂ W̄
+ k · ∇xW̄ = R(k2 − p2, p − k)(W̄ (t, x, p) − W̄ (t, x, k))dp.
∂t
Extreme cases: (1) white noise in time randomness R(ω, p) =
R̂(|p|), or (2) time-independent, R(ω, p) = R̂(|p|)δ(ω).
White noise: the long time limit of W (t, x, k) – Fokker-Planck
∂W
+ k · ∇xW = D∆k W – momentum diffusion
∂t
Time-independent randomness: Fokker-Planck
∂W
+ k · ∇xW = D∇k · [(I − k̂ ⊗ k̂)∇k W ], momentum diffuses
∂t
on the sphere {|k| = const}
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Komorowski, R.’13: away from these two extremes, randomness is neither frozen nor white noise.
Theorem. (Intermediate
time scales)
!
t x k
= W̄ (t, x, k),
lim W 3 , 4 ,
ε↓0
ε ε ε
∂ W̄
D
where
+ k · ∇xW̄ (t, x, k) = ∇ ·
I − k̂ ⊗ k̂ · ∇k W̄ (t, x, k)
∂t
|k|
with an explicit D.
On the long but not too long time scale t ∼ O(ε−3), the limit
is as in time-independent media – no energy is pumped into
the system!
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−4.)
Theorem. (Really
long
time
scales
t
∼
ε
!
t
x k
lim W 4 , 9/2 ,
= W̄ (t, x, |k|), where W̄ (t, x, `) = V̄ (t, x, `1/4)
ε↓0
ε ε
ε
and
`5/4Ωd
∂tV̄ (t, x, `) = MV̄ (x, `) +
∆xf (x, `),
b(d − 1)
and MV̄ (x, `) := dc∂`V̄ (x, `) + 4c`∂`2V̄ (x, `), where Ωd is the
surface area of the unit sphere in Rd.
When d = 2, or 3 the momentum is recurrent and enters
0 infinitely many times – energy oscillates between low and
high levels. Energy is alternatively pumped and dissipated by
time-dependent random media.
17
Open problems
1. Wave field statistics for the true wave equation
2. Applications to probing and pulse shaping
3. Effect of long range correlations (in progress)
4. Diffusion approximation for dispersive waves
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