[Paper Review] Self-Averaging in Scaling Limits for Random High-Frequency Parabolic Waves

Albert Fannjiang|arXiv (Cornell University)|May 30, 2003

Microwave Imaging and Scattering Analysis7 references2 citations

TL;DR

This paper establishes self-averaging in scaling limits of random high-frequency parabolic waves in disordered media, demonstrating that under general conditions on the random refractive index, spatial diversity leads to deterministic behavior governed by transport equations—excluding the white-noise limit. The key contribution is proving statistical stability in the radiative transfer and diffusion limits via rigorous scaling analysis.

ABSTRACT

Abstract. We consider several types of scaling limits for the Wigner equation of the parabolic waves in random media, the limiting cases of which include the radiative transfer limit, the diffusion limit and the white-noise limit. We show under fairly general assumptions on the random refractive index field that any significant amount of spatial diversity (thus excluding the white-noise limit) leads to statistical stability or self-averaging in the sense that the limiting law is deterministic and is governed by various transport equations depending on the specific scaling involved. The celebrated Schrödinger equation

Motivation & Objective

• To analyze scaling limits of the Wigner equation for parabolic waves in random media.
• To determine under what conditions the limiting wave behavior becomes statistically stable (self-averaging).
• To identify the transport equations governing the limiting behavior in the radiative transfer and diffusion limits.
• To exclude the white-noise limit from self-averaging, as it lacks spatial diversity.
• To establish that general random refractive index fields lead to deterministic macroscopic laws under appropriate scaling.

Proposed method

• Analyzes the Wigner transform of the parabolic wave equation in random media with general random refractive index fields.
• Applies scaling limits to derive asymptotic equations in the high-frequency, small-correlation-length regime.
• Uses moment methods and weak convergence techniques to study the limiting behavior of the Wigner distribution.
• Identifies conditions under which the limiting law becomes deterministic, implying self-averaging.
• Derives transport equations—radiative transfer and diffusion—depending on the specific scaling regime.
• Excludes the white-noise limit by requiring non-degenerate spatial diversity in the refractive index field.

Experimental results

Research questions

• RQ1 Under what conditions does the Wigner distribution of parabolic waves in random media converge to a deterministic limit?
• RQ2 How does spatial diversity in the refractive index field affect the emergence of self-averaging in wave propagation?
• RQ3 Which transport equations govern the limiting behavior in the radiative transfer and diffusion limits?
• RQ4 Why does the white-noise limit fail to exhibit self-averaging, and how does it differ from other scaling regimes?
• RQ5 To what extent do general random refractive index fields lead to universal deterministic behavior in the macroscopic limit?

Key findings

• Self-averaging occurs in all scaling limits except the white-noise limit, provided there is sufficient spatial diversity in the refractive index field.
• The limiting wave behavior is governed by deterministic transport equations, including the radiative transfer and diffusion equations.
• The deterministic limit arises due to statistical stability induced by spatial averaging over random inhomogeneities.
• The white-noise limit is excluded from self-averaging because it lacks spatial correlation structure, leading to non-deterministic fluctuations.
• The results hold under fairly general assumptions on the random refractive index, including ergodicity and mixing conditions.
• The Wigner distribution converges weakly to a deterministic solution of a transport equation in the scaling limit.