#### Date of Degree

5-2018

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Program

Physics

#### Advisor

Azriel Z. Genack

#### Committee Members

Victor A. Gopar

Alexander A. Lisyansky

Tobias Schäfer

So Takei

Azriel Z. Genack

#### Subject Categories

Condensed Matter Physics | Other Statistics and Probability | Statistical, Nonlinear, and Soft Matter Physics

#### Keywords

random media, wave transport, Anderson localization, random matrix theory, Lévy flights, stochastic processes

#### Abstract

This thesis is a study of wave transport inside random media using random matrix theory. Anderson localization plays a central role in wave transport in random media. As a consequence of destructive interference in multiple scattering, the wave function decays exponentially inside random systems. Anderson localization is a wave effect that applies to both classical waves and quantum waves. Random matrix theory has been successfully applied to study the statistical properties of transport and localization of waves. Particularly, the solution of the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation gives the distribution of transmission.

For wave transport in standard one dimensional random systems in which the average number of scatterers per unit length is a constant, we get the ensemble average of the logarithm of transmission scales linearly with system length *L*, ⟨ln *T*⟩=-*L*/*l*, where *l* is the mean free path. The average transmission scales exponentially with *L* for deeply localized systems, ⟨*T*⟩∝exp(-*L*/2*l*). We have also investigated the statistics of intensity inside the random systems and obtained the analytical expression of average intensity ⟨*I*(*x*)⟩. In addition, we find the average of the logarithm of intensity falls linearly with depth *x*, ⟨ln *I*(*x*)⟩=-*x*/*l*.

We have explored the statistics of anomalous wave transport inside random systems with Lévy disorder, in which waves perform Lévy flights. We find the average logarithm of transmission scales as a power law with system length, ⟨ln *T*⟩∝-*L*^*α*, and the mean transmission scales as a power law for large *L*, ⟨*T*⟩→*L*^(-*α*), where α is the stability parameter of the *α*-stable distribution associated with the Lévy flights. We have also investigated the statistics of intensity in the interior of the random systems for Lévy flights of waves. We obtain an analytical expression for the average intensity and find the average of the logarithm of intensity falls as a power law with depth *x*, ⟨ln *I*(*x*)⟩∝-*x*^*α*.

We have also studied the impact of internal and edge reflection on the statistics of wave transport in random media. We find that the statistics of transmission is independent of the location of the reflector. When a reflector is present in a random system, the average of the logarithm of transmission is shifted by ln *Γ*, ⟨ln *T*⟩=-*L*/*l*+ln *Γ*, where* Γ* is the transmission coefficient of the reflector. The parameter Γ can be found in term of the statistics of transmission by *Γ*=1-[1-(2⟨1/*T*⟩-1) e^2⟨ln*T* ⟩ ]^2. Thus, the transmission of the reflector can be obtained from the information of transmission measured outside the system. We have also investigated the statistics of intensity inside random media and calculated the average intensity, which depends on the location of the reflector. The average logarithm of intensity is not affected by the presence of reflector before the reflector, and shifts by ln *Γ* beyond the reflector.

#### Recommended Citation

Ma, Xujun, "Standard and Anomalous Wave Transport Inside Random Media" (2018). *CUNY Academic Works.*

https://academicworks.cuny.edu/gc_etds/2619

#### Included in

Condensed Matter Physics Commons, Other Statistics and Probability Commons, Statistical, Nonlinear, and Soft Matter Physics Commons