Radiative transport limit for the random Schrödinger equation with long-range correlations

Gomez, Christophe

doi:10.1016/j.matpur.2012.02.007

Cited by 19 publications

(39 citation statements)

References 38 publications

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“…Equation (1) then describes energy transport in a certain macroscopic limit via asymptotics of the Wigner transform [15,23,30]. See [7,10,24,28,16] for more details on the link between (1) and wave propagation in random media. In the present work, we investigate the regularity of the solutions of (1) and two asymptotic regimes.…”

Section: Introductionmentioning

confidence: 99%

Radiative Transfer with Long-Range Interactions: Regularity and Asymptotics

Gomez¹,

Pinaud²,

Ryzhik³

2017

Multiscale Model. Simul.

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International audienceThis work is devoted to radiative transfer equations with long-range interactions. Such equations arise in the modeling of high frequency wave propagation in random media with long-range dependence. In the regime we consider, the singular collision operator modeling the interaction between the wave and the medium is conservative, and as a consequence wavenumbers take values on the unit sphere. Our goals are to investigate the regularizing effects of grazing collisions, the diffusion limit, and the peaked forward limit. As in the case where wavenumbers take values in R^{d+1}, we show that the transport operator is hypoelliptic, so that the solutions are infinitely differerentiable in all variables. Using probabilistic techniques, we show as well that the diffusion limit can be carried on as in the case of a regular collision operator, and as a consequence that the diffusion coefficient is non-zero and finite. We finally consider the regime where grazing collisions are dominant

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Section: Introductionmentioning

confidence: 99%

Radiative Transfer with Long-Range Interactions: Regularity and Asymptotics

Gomez¹,

Pinaud²,

Ryzhik³

2017

Multiscale Model. Simul.

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“…This result means that −s , with s = 1/(2κ) < 1, is the first propagation scale on which the random perturbations become significant, and induces a random phase modulation on the propagating wave. In [22,Theorem 2.2] the author studies the loss of coherence of φ on the propagation scale −1 (s = 1), and shows that the Wigner transform (3) of φ converges in probability for the weak topology on L 2 (R 2d ) to the unique solution of a deterministic radiative transfer equation similar to (4). In other words, the wave decoherence happening on this propagation scale ( −1 ) does not depend on the particular realization of the random potential.…”

Section: Introductionmentioning

confidence: 99%

Wave Decoherence for the Random Schrödinger Equation with Long-Range Correlations

Gomez

2013

Commun. Math. Phys.

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In this paper, we study the loss of coherence of a wave propagating according to the Schrödinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze the decoherence phenomena is a properly rescaled Wigner transform of the solution of the random Schrödinger equation. We exhibit anomalous wave decoherence effects at different propagation scales.

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“…The main result of this paper is the following theorem, that states the convergence of the pulse (20). Theorem 2.1 (Convergence result).…”

Section: Resultsmentioning

confidence: 95%

“…Theorem 2.1 (Convergence result). The family (p ε L ) ε∈(0,1) , defined by (20), converges in law in the space C 0 ((−∞, +∞), L 2 (R 2 )) ∩ L 2 ((−∞, +∞) × R 2 ) to a limit given by…”

Section: Resultsmentioning

confidence: 99%

Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

Gomez

Pinaud

2017

Arch Rational Mech Anal

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This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the whitenoise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional Itô-Schrödinger equation, that is a Schrödinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence.where c 0 is the background velocity (constant for simplicity), and the random field V (z, x), with a stationary covariance, models fluctuations around c 0 in the slab (0, L z ) × R 2 . The parameters σ and l c represent the amplitude and the correlation length of the fluctuations. The main assumption on V is that it satisfies the long-range property in the z-direction, which is translated mathematically into a bounded non-integrable autocorrelation function which decreases at infinity only as Preliminaries and main resultsThroughout this work, we will use the following conventions for the Fourier transform:f denotes the Fourier transform w.r.t. the variable t as in (6), andf that w.r.t. t and x, f (ω, κ) = 1 (2π) 3 f (t, x)e i(ωt+κ·x) dtdx with f (t, x) = f (ω, κ)e −i(ωt+κ·x) dωdκ.

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Radiative transport limit for the random Schrödinger equation with long-range correlations

Cited by 19 publications

References 38 publications

Radiative Transfer with Long-Range Interactions: Regularity and Asymptotics

Radiative Transfer with Long-Range Interactions: Regularity and Asymptotics

Wave Decoherence for the Random Schrödinger Equation with Long-Range Correlations

Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

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