Semi-classical limit for the varying-mass Schrödinger equation with random inhomogeneities

1 Introduction

The Schrödinger equation with varying mass has gained great attention in solid state physics, and been successfully used to to study electronic properties of semiconductor hetero-structures [27, 24, 11, 29]

. For example, it can describe localized defects in crystalline media, which may yield bound states localized to the defect. It is also related to describe non-compact line defect (edge) perturbations, e.g., by interpolating two-dimensional honeycomb structures via domain walls

[13, 12]. Edge modes may be produced by such perturbations, which propagate in the direction parallel to the edge and localize transverse to the edge.

We are interested in deriving the asymptotic limit of the following varying-mass Schrödinger equation

i ε \partial_{t} u^{ε} (t, x) + \frac{1}{2} ε^{2} \nabla_{x} \cdot (m (t, x) \nabla_{x} u^{ε} (t, x)) = 0,

(1.1)

where $t > 0$ , $x \in R^{d}$ with $d \geq 1$ and $ε ≪ 1$ is the rescale Planck constant. The varying mass $m_{0}$ , which can be time-dependent, is assume to be random and highly oscillatory, with a given covariance matrix in time and space. We shall assume that $u^{ε}$ decays fast enough at infinity to validate all the derivations. One goal of the paper is to show that in the $ε \to 0$ regime, the Wigner transform of the solution converges to a special radiative transfer equation.

The problem is motivated by a fact that simulating (1.1) is extremely challenging in the semi-classical regime ( $ε ≪ 1$ ). The challenges are two-folded. In deterministic regime (meaning $m (t, x)$ is a deterministic highly oscillatory function in $(t, x)$ ), standard numerical solvers require to the small wavelength of $u^{ε} (t, x)$ to be resolved, for example, a mesh size and time step of order $o (ε)$ is required when finite difference methods are used [22, 23]. The time-splitting spectral method [4, 6] can improve the mesh size to be of order $O (ε)$ , however, has limitations to compute the Schrödinger equation with varying mass. A bigger problem comes from the randomness in $m$ . Since only the covariance of $m$

is given, numerically one has to find many realizations and compute the deterministic Schrödinger equation before finding the ensemble mean/variance of the solution. The number of realizations, however, increases as

ε \to 0

, as details in the random fluctuation become more and more important.

Disregarding the challenges from the stochasticity, merely for the deterministic system, alternative approaches have been developed. These included the WKB-type methods, e.g., Gaussian beam methods [18, 17] and frozen Gaussian approximation [15, 19, 20]. The idea is to apply the WKB-type ansatz

u^{ε} (t, x) = A (t, x) exp (\frac{i S (t, x)}{ε}),

and derive the eikonal equation for $S (t, x)$ and transport-like equation for $A (t, x)$ , where both $S (t, x)$ and $A (t, x)$ are functions of large scale, i.e., $ε$ -independent. To our best knowledge, no such types of methods have been applied to efficiently solve (2.1) in the literature yet. And even for standard Schrödinger equation with random potential term, the application of the methods have not been fully understood.

In the paper, we shall systematically derive asymptotic equations for (2.1) by the Wigner transform [14], which is a main tool in semiclassical theory parallel to the WKB-type methods mentioned above. The literature on deriving the asymptotic equations for wave propagation in random media [2, 3, 1, 26, 21, 10, 9, 8, 7] is rich, most of which started with the Schrödinger equation with constant mass, and the randomness and high oscillations are introduced through the potential term. When it is the effective mass term that is random and highly oscillatory, the process of the derivation is rather similar but much more delicate, as will be detailed later in our paper. As a proof of concept, we numerically verify the derived radiative transfer equation by carefully computing and comparing its solution to the one of VMSE (1.1), and show that the two solutions agree.

The rest of the paper is organized as follows. To better illustrate the derivation, we start with a simpler case with $m_{0}$ being deterministic and only spatially dependent, and derive the limiting radiative transfer equation by the Wigner transform in Section 2. In Section 3, we systematically introduce the derivation of the limiting equation for the varying-mass Schrödinger equation (1.1) with random heterogeneities. We present our numerical validation in Section 4 and make conclusive remarks in Section 5.

2 Wigner transform of VMSE in the deterministic setting

As a preparation, we first investigate the semi-classical limit for (1.1) with deterministic mass in this section, which is to consider:

i ε \partial_{t} u^{ε} (t, x) + \frac{1}{2} ε^{2} \nabla_{x} \cdot (m_{0} (x) \nabla_{x} u^{ε} (t, x)) = 0 .

(2.1)

The varying mass $m_{0}$ is a real function of $x$ . It is assumed to be deterministic and time-independent. We study the Cauchy problem, i.e., the $u^{ε}$ is assumed to decay at infinite. $u^{ε}$ is a complex function and we typically care about its physical observables such as the energy density $ρ^{ε}$ and the energy flux $J^{ε}$ , defined respectively as

ρ^{ε} (t, x) = | u^{ε} (t, x) |^{2}, J^{ε} (t, x) = ε Im (¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ u^{ε} (t, x) \nabla_{x} u^{ε} (t, x)) .

Wigner transform is a technique explored in [25] for the Schrödinger equation with random potential, and has been demonstrated as a very powerful tool for investigating the semi-classical limit. It defines a function on the phase space:

W^{ε} (t, x, k) = \frac{1}{(2 π)^{d}} \int_{R^{d}} e^{i k y} u^{ε} (t, x - \frac{ε}{2} y)^{ε} (t, x + \frac{ε}{2} y) d y .

(2.2)

Here $^{ε}$ is the complex conjugate of $u^{ε}$

. This definition is essentially the Fourier transform of

⟨ x - \frac{ε}{2} y | u ⟩ ⟨ u | x + \frac{ε}{2} y ⟩

on the $y$ variable.

The Wigner transform loses phase information: if $u^{ε}$ is perturbed to $u^{ε} e^{i S}$

, the Wigner transform is kept the same. However, the physical observables can be recovered, namely, the first and second moments of

W^{ε}

provide the energy density and the energy flux:

\int_{R^{d}} W^{ε} (t, x, k) d k = ρ^{ε} (t, x), \int_{R^{d}} k W^{ε} (t, x, k) d k = J^{ε} (t, x) .

(2.3)

It is not guaranteed that $W^{ε}$ is positive, and thus it does not serve directly as the energy density on the phase space. By plugging in the Schrödinger equation, we derives the equation satisfied by $W^{ε}$ in the following Lemma.

Let $u^{ε}$ satisfy the VMSE (2.1), then its Wigner transform (2.2) satisfies:

\partial_{t} W^{ε} + \frac{1}{ε} Q_{1}^{ε} W^{ε} + Q_{2}^{ε} W^{ε} = ε Q_{3}^{ε} W^{ε},

(2.4)

where the three operators are defined to be:

Q_{1}^{ε} W^{ε} = \frac{| k |^{2}}{2} \int_{R^{d}} \frac{e^{i p x}}{(2 π)^{d}} {~ m}_{1} (t, p) i [W^{ε} (t, x, k - \frac{ε}{2} p) - W^{ε} (t, x, k + \frac{ε}{2} p)] d p

(2.5)

Q_{2}^{ε} W^{ε} = \frac{k}{2} \cdot \int_{R^{d}} \frac{e^{i p x}}{(2 π)^{d}} {~ m}_{1} (t, p) [\nabla_{x} W^{ε} (t, x, k - \frac{ε}{2} p) + \nabla_{x} W^{ε} (t, x, k + \frac{ε}{2} p)] d p

(2.6)

	$Q_{3}^{ε} W^{ε} =$	$\frac{1}{8} \int_{R^{d}} \frac{e^{i p x}}{(2 π)^{d}} {~ m}_{1} (t, p) i [Δ_{x} W^{ε} (t, x, k - \frac{ε}{2} p) - Δ_{x} W^{ε} (t, x, k + \frac{ε}{2} p)] d p$		(2.7)
		$+ \frac{1}{8} \int_{R^{d}} \frac{e^{i p x}}{(2 π)^{d}} {~ m}_{1} (t, p) i \| p \|^{2} [W^{ε} (t, x, k - \frac{ε}{2} p) - W^{ε} (t, x, k + \frac{ε}{2} p)] d p .$		(2.7)

where the Fourier transform of $m_{0}$ in space is defined by

{~ m}_{0} (t, p) = \int_{R^{d}} e^{- i p z} m_{0} (t, z) d z .

(2.8)

The proof is direct derivation. Notice that

\partial_{t} W^{ε} = \frac{1}{(2 π)^{d}} \int_{R^{d}} e^{i k y} \partial_{t} u^{ε}^{ε} d y + \frac{1}{(2 π)^{d}} \int_{R^{d}} e^{i k y} u^{ε} \partial_{t}^{ε} d y,

(2.9)

we have, plugging in (2.1):

$\partial_{t} W^{ε} =$	$\frac{i ε}{2 (2 π)^{d}} \int_{R^{d}} e^{i k y} \nabla_{x} \cdot (m_{0} (x - \frac{ε}{2} y) \nabla_{x} u^{ε} (x - \frac{ε}{2} y))^{ε} (x + \frac{ε}{2} y) d y$	(2.10)
	$- \frac{i ε}{2 (2 π)^{d}} \int_{R^{d}} e^{i k y} \nabla_{x} \cdot (m_{0} (x + \frac{ε}{2} y) \nabla_{x}^{ε} (x + \frac{ε}{2} y)) u^{ε} (x - \frac{ε}{2} y) d y$
$:=$	$\frac{i ε}{2 (2 π)^{d}} M_{1} - \frac{i ε}{2 (2 π)^{d}} M_{2} .$

Since the two terms $M_{1}$ and $M_{2}$ are conjugate with $y \to - y$ for the second term, we only study the first one. With integration by parts:

$M_{1} =$	$\frac{2}{ε} \int_{R^{d}} [\nabla_{y} (e^{i k y}) \cdot \nabla_{x} u^{ε} (x - \frac{ε}{2} y)] m_{0} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y) d y$	(2.11)
	$+ \frac{2}{ε} \int_{R^{d}} e^{i k y} m_{0} (x - \frac{ε}{2} y) \nabla_{x} u^{ε} (x - \frac{ε}{2} y) \cdot \nabla_{y}^{ε} (x + \frac{ε}{2} y) d y$
$:=$	$I_{1} + I_{2} .$

We treat the $I_{1}$ and $I_{2}$ respectively in the following. Perform integration by parts again to $I_{1}$

$I_{1} =$	$\frac{4}{ε^{2}} \int_{R^{d}} Δ_{y} (e^{i k y}) m_{0} (x - \frac{ε}{2} y) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$	(2.12)
	$+ \frac{4}{ε^{2}} \int_{R^{d}} \nabla_{y} (e^{i k y}) \cdot \nabla_{y} m_{0} (x - \frac{ε}{2} y) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$
	$+ \frac{2}{ε} \int_{R^{d}} [\nabla_{y} (e^{i k y}) \cdot \nabla_{x}^{ε} (x + \frac{ε}{2} y)] m_{0} (x - \frac{ε}{2} y) u^{ε} (x - \frac{ε}{2} y) d y$
$:=$	$I_{11} + I_{12} + I_{13} .$

Note that $I_{1}$ and the last term $I_{13}$ can be combined so that a complete $x -$ gradient of $u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)$ is available, namely one arrives at a formula for $I_{1}$

$I_{1} =$	$\frac{1}{2} I_{1} + \frac{1}{2} (I_{11} + I_{12} + I_{13}) = \frac{1}{2} (I_{1} + I_{13}) + \frac{1}{2} (I_{11} + I_{12})$	(2.13)
$=$	$\frac{2}{ε^{2}} \int_{R^{d}} Δ_{y} (e^{i k y}) m_{0} (x - \frac{ε}{2} y) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$
$+$	$\frac{2}{ε^{2}} \int_{R^{d}} \nabla_{y} e^{i k y} \cdot \nabla_{y} m_{0} (x - \frac{ε}{2} y) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$
$+$	$\frac{1}{ε} \int_{R^{d}} \nabla_{y} (e^{i k y}) \cdot \nabla_{x} [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] m_{0} (x - \frac{ε}{2} y) d y .$

For $I_{2}$ in (2.11), integration by parts against $\nabla_{x} u^{ε} (x - \frac{ε}{2} y)$ produces

$I_{2} =$	$\frac{4}{ε^{2}} \int_{R^{d}} [\nabla_{y} (e^{i k y}) \cdot \nabla_{y}^{ε} (x + \frac{ε}{2} y)] m_{0} (x - \frac{ε}{2} y) u^{ε} (x - \frac{ε}{2} y) d y$	(2.14)
$+$	$\frac{4}{ε^{2}} \int_{R^{d}} e^{i k y} [\nabla_{y} m_{0} (x - \frac{ε}{2} y) \cdot \nabla_{y}^{ε} (x + \frac{ε}{2} y)] u^{ε} (x - \frac{ε}{2} y) d y$
$+$	$\int_{R^{d}} e^{i k y} m_{0} (x - \frac{ε}{2} y) u^{ε} (x - \frac{ε}{2} y) Δ_{x}^{ε} (x + \frac{ε}{2} y) d y := I_{21} + I_{22} + I_{23} .$

On the other hand, integration by parts against $\nabla_{y}^{ε} (x + \frac{ε}{2} y)$ gives

$I_{2} =$	$\frac{4}{ε^{2}} \int_{R^{d}} [\nabla_{y} (e^{i k y}) \cdot \nabla_{y} u^{ε} (x - \frac{ε}{2} y)] m_{0} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y) d y$	(2.15)
$+$	$\frac{4}{ε^{2}} \int_{R^{d}} e^{i k y} [\nabla_{y} m_{0} (x - \frac{ε}{2} y) \cdot \nabla_{y} u^{ε} (x - \frac{ε}{2} y)]^{ε} (x + \frac{ε}{2} y) d y$
$+$	$\int_{R^{d}} e^{i k y} m_{0} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y) Δ_{x} u^{ε} (x - \frac{ε}{2} y) d y := I_{21}^{'} + I_{22}^{'} + I_{23}^{'} .$

Note that $I_{21}$ and $I_{21}^{'}$ can be combined after another integration by parts

	$I_{21} + I_{21}^{'} =$	$- \frac{4}{ε^{2}} \int_{R^{d}} \nabla_{y} (e^{i k y}) \cdot \nabla_{y} m_{0} (x - \frac{ε}{2} y) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$		(2.16)
	$-$	$\frac{4}{ε^{2}} \int_{R^{d}} m_{0} (x - \frac{ε}{2} y) Δ_{y} (e^{i k y}) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y .$		(2.16)

$I_{22}$ and $I_{22}^{'}$ can be combined similarly

	$I_{22} + I_{22}^{'} =$	$- \frac{4}{ε^{2}} \int_{R^{d}} \nabla_{y} (e^{i k y}) \cdot \nabla_{y} m_{0} (x - \frac{ε}{2} y) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$		(2.17)
	$-$			(2.17)

Hence using (2.14)-(2.17) and the trick in (2.13), one derives the formula for $I_{2}$ in (2.11)

$I_{2} =$	$\frac{1}{4} (I_{21} + I_{22} + I_{23}) + \frac{1}{2} I_{2} + \frac{1}{4} (I_{21}^{'} + I_{22}^{'} + I_{23}^{'})$	(2.18)
$=$	$(\frac{1}{4} I_{23} + \frac{1}{2} I_{2} + \frac{1}{4} I_{23}^{'}) + \frac{1}{4} (I_{21} + I_{21}^{'}) + \frac{1}{4} (I_{22} + I_{22}^{'})$
$=$	$\frac{1}{4} \int_{R^{d}} e^{i k y} m_{0} (x - \frac{ε}{2} y) Δ_{x} [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$
$-$	$\frac{2}{ε^{2}} \int_{R^{d}} \nabla_{y} (e^{i k y}) \cdot \nabla_{y} m_{0} (x - \frac{ε}{2} y) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$
$-$	$\frac{1}{ε^{2}} \int_{R^{d}} m_{0} (x - \frac{ε}{2} y) Δ_{y} (e^{i k y}) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$
$-$

Finally from (2.13) and (2.18), one gets

$M_{1} =$	$\frac{1}{4} \int_{R^{d}} e^{i k y} m_{0} (x - \frac{ε}{2} y) Δ_{x} [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$	(2.19)
	$+ \frac{1}{ε} \int_{R^{d}} m_{0} (x - \frac{ε}{2} y) \nabla_{y} (e^{i k y}) \cdot \nabla_{x} [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$
	$+ \frac{1}{ε^{2}} \int_{R^{d}} m_{0} (x - \frac{}{ε} 2 y) Δ_{y} (e^{i k y}) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$
	$- \frac{1}{ε^{2}} \int_{R^{d}} e^{i k y} Δ_{y} m_{0} (x - \frac{ε}{2} y) [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y$
$:=$	$T_{1} + T_{2} + T_{3} + T_{4} .$

All the $T_{i}$ terms can be explicitly expressed by the Wigner transform (2.2). In particular:

$T_{1}$	$= \int_{R^{d}} e^{i p x} {~ m}_{0} (p) Δ_{x} W^{ε} (x, k - \frac{ε}{2} p) d p,$	(2.20)
$T_{2}$	$= \int_{R^{d}} e^{i p x} {~ m}_{0} (p) i k \cdot \nabla_{x} W^{ε} (x, k - \frac{ε}{2} p) d p,$
$T_{3}$	$= \int_{R^{d}} - \| k \|^{2} e^{i p x} {~ m}_{0} (p) W^{ε} (x, k - \frac{ε}{2} p) d p,$
$T_{4}$	$= ε^{2} \int_{R^{d}} \| p \|^{2} e^{i p x} {~ m}_{0} (p) W^{ε} (x, k - \frac{ε}{2} p) d p .$

We use $T_{1}$ as an example to show this. Recalling:

Δ_{x} W^{ε} (x, k) = \frac{1}{(2 π)^{d}} \int e^{i k y} Δ_{x} [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y,

we have

	$\int_{R^{d}} e^{i p x} {~ m}_{0} (p) Δ_{x} W^{ε} (x, k - \frac{ε}{2} p) d p$	(2.21)
$=$	$\frac{1}{(2 π)^{d}} \int \int \int e^{i p x} e^{- i p z} m_{0} (z) e^{i (k - \frac{ε}{2} p) y} Δ_{x} [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d z d p d y,$
$=$	$\int_{R^{d}} e^{i k y} m_{0} (x - \frac{ε}{2} y) Δ_{x} [u^{ε} (x - \frac{ε}{2} y)^{ε} (x + \frac{ε}{2} y)] d y = \frac{1}{4} T_{1},$

where we used the fact that

δ (x) = \frac{1}{(2 π)^{d}} \int_{R^{d}} e^{i x z} d z, and \frac{1}{(2 π)^{d}} \int \int f (x) e^{i x z} d z d x = f (0) .

(2.22)

Using (2.20), we get

$M_{1} =$	$\frac{1}{4} \int_{R^{d}} e^{i p x} {~ m}_{0} (p) Δ_{x} W^{ε} (x, k - \frac{ε}{2} p) d p$	(2.23)
	$+ \frac{1}{ε} \int_{R^{d}} e^{i p x} {~ m}_{0} (p) i k \cdot \nabla_{x} W^{ε} (x, k - \frac{ε}{2} p) d p$
	$+ \int_{R^{d}} \| p \|^{2} e^{i p x} {~ m}_{0} (p) W^{ε} (x, k - \frac{ε}{2} p) d p$

By the conjugate argument, one gets, setting $p \to - p$ :

$M_{2} =$	$\frac{1}{4} \int_{R^{d}} e^{i p x} {~ m}_{0} (p) Δ_{x} W^{ε} (x, k + \frac{ε}{2} p) d p$	(2.24)
	$- \frac{1}{ε} \int_{R^{d}} e^{i p x} {~ m}_{0} (p) i k \cdot \nabla_{x} W^{ε} (x, k + \frac{ε}{2} p) d p$
	$+ \int_{R^{d}} \| p \|^{2} e^{i p x} {~ m}_{0} (p) W^{ε} (x, k + \frac{ε}{2} p) d p$

Finally, substitute (2.23) and (2.24) into (2.10), and we arrive at the Wigner equation in (2.4).

[Formal] With certain regularity, the formal $ε \to 0$ limit of (2.4) is a transport equation:

\partial_{t} W + m_{0} (x) k \cdot \nabla_{x} W - \frac{| k |^{2}}{2} \nabla_{x} m_{0} (x) \cdot \nabla_{k} W + O (ε^{2}) = 0 .

(2.25)

The trajectory of particles follows:

˙ x = k m_{0} (t, x), ˙ k = - \frac{| k |^{2}}{2} \nabla_{x} m_{0} (t, x),

(2.26)

and

| k |^{2} m_{0} (x) = const .

(2.27)

To prove the limiting equation (2.25), one only needs to derive the limiting behavior of $Q_{i} W^{ε}$ . Indeed, formally, assuming $W^{ε} \in C^{2}$ :

\frac{1}{ε} Q_{1}^{ε} W^{ε} = - \frac{| k |^{2}}{2} \int_{R^{d}} \frac{e^{i p x}}{(2 π)^{d}} {~ m}_{1} (t, p) i p \cdot \nabla_{k} W^{ε} (t, x, k) d p + O (ε^{2}) = - \frac{| k |^{2}}{2} \nabla_{x} m_{0} \cdot \nabla_{k} W^{ε} + O (ε^{2}),

and that

Q_{2}^{ε} W^{ε} = k \cdot \int_{R^{d}} \frac{e^{i p x}}{(2 π)^{d}} {~ m}_{1} (t, p) \nabla_{x} W^{ε} (t, x, k) d p + O (ε^{2}) = m_{0} k \cdot \nabla_{x} W^{ε} + O (ε^{2}) .

One then arrives at (2.25) by plugging them in the original equation (2.4). To show (2.27) with $m_{0} = m_{0} (x)$ , one simply takes the time derivative along the trajectory:

\frac{d}{d t} [| k (t) |^{2} m_{0} (x (t))] = 2 k \cdot ˙ k m_{0} + | k |^{2} \nabla_{x} m_{0} \cdot ˙ x = 0,

according to the trajectory equation (2.26), and thus the quantity $| k |^{2} m_{0}$ is conserved along the trajectory.

This proposition essentially guarantees the positivity of the solution to the limiting equation (2.25).

3 Semi-classical limit for VMSE with random perturbation

We consider the VMSE where the effective mass involves random perturbation, namely:

i ε \partial_{t} u^{ε} + \frac{1}{2} ε^{2} \nabla_{x} \cdot (m^{ε} (t, x) \nabla_{x} u^{ε}) = 0,

(3.1)

where the effective mass is

m^{ε} (t, x) = m_{0} (t, x) + \sqrt{ε} m_{1} (t / ε, x / ε) .

(3.2)

While the leading order $m_{0}$ is assumed to be deterministic and smooth, we allow the random perturbation $m_{1} (t, x)$ to present small scales at $ε$ . Furthermore we assume it is mean-zero and stationary in both $t$ and $x$ with the correlation function $R (t, x)$ :

R (t, x) = E [m_{1} (s, z) m_{1} (t + s, x + z)] \forall x, z \in R^{d} and t, s \in R .

(3.3)

Taking the Fourier transform of the function in both time and space, one has:

^R (ω, p) = \int_{R^{d + 1}} e^{- i ω s - i p z} R (s, z) d s d z,

(3.4)

then it is straightforward to show:

E [{~ m}_{1} (τ, p) {^m}_{1} (ω, q)] = (2 π)^{d} e^{- i ω τ}^R (ω, p) δ (p + q),

(3.5)

and

^R (- ω, p) =^R (ω, p), and^R (ω, - p) =^R (ω, p) .

We dedicate this section to the derivation of the semi-classical limit of the equation above. We will show that In the zero limit of $ε$ , the Wigner transform of $u^{ε}$ , the solution to the VMSE (2.1) with varying random mass (3.2), solves the radiative transfer equation:

\partial_{t} W + m_{0} k \cdot \nabla_{x} W - \frac{k^{2}}{2} \nabla_{x} m_{0} \cdot \nabla_{k} W = \frac{1}{(2 π)^{d}} \int_{R^{d}} \frac{1}{4} (p \cdot k)^{2}^R (\frac{m_{0}}{2} (p^{2} - k^{2}), p - k) [W (p) - W (k)] d p .

(3.6)

In view of (2.4) in Lemma 2, noting that $m_{0} \to m_{0} + \sqrt{ε} m_{1}$ , the Wigner equation (3.1) is transformed to

\partial_{t} W^{ε} + \frac{1}{ε} Q_{1}^{ε} W^{ε} + Q_{2}^{ε} W^{ε} + \frac{1}{\sqrt{ε}} P_{1}^{ε} W^{ε} + \sqrt{ε} P_{2}^{ε} W^{ε} = ε Q_{3}^{ε} W^{ε} + \frac{1}{\sqrt{ε}} P_{3}^{ε} W^{ε},

(3.7)

where the operators $Q_{i}^{ε}$ are defined in (2.5)-(2.7), and $P_{i}^{ε}$ are their counterparts defined by $m_{1}$ :

P_{1}^{ε} W^{ε} = \frac{| k |^{2}}{2} \int_{R^{d}} \frac{e^{i p ξ}}{(2 π)^{d}} {~ m}_{0} (τ, p) i [W^{ε} (k - \frac{1}{2} p) - W^{ε} (k + \frac{1}{2} p)] d p,

(3.8)

P_{2}^{ε} W^{ε} = \frac{k}{2} \cdot \int_{R^{d}} \frac{e^{i p ξ}}{(2 π)^{d}} {~ m}_{0} (τ, p) [\nabla_{x} W^{ε} (k - \frac{1}{2} p) + \nabla_{x} W^{ε} (k + \frac{1}{2} p)] d p,

(3.9)

and

P_{3}^{ε} W^{ε} =

\frac{ε^{2}}{8} \int_{R^{d}} \frac{e^{i p ξ}}{(2 π)^{d}} {~ m}_{0} (τ, p) i [

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