This is a series of research projects that aims to incorporate newly-developed mathematical techniques in the study of exotic transport phenomena in physical systems characterized by disorder, non-local interactions, and correlated effects. Specifically, we focus on the application of spectral theory and fractional calculus to important phenomena, such as turbulence, streaming instability, and conductivity in low-dimensional materials. Our work is a generalization of the famous theory of Anderson localization introduced in 1958 by Philip Anderson, for which he received the Nobel Prize in Physics in 1977. While the original question addressed by Anderson was focused on localization due to disorder, our work aims to identify the physical systems where delocalization can exist, despite the presence of disorder. Anderson ended his Nobel Prize speech with a reference from Lewis Carroll, emphasizing how complex, yet fascinating a simple transport question can be.

“Now, here, you see, it takes all the running you can do, to keep in the same place.” ~ Lewis Carroll, Alice Through the Looking Glass

The spectral approach developed by our collaborator, Dr. Conni Liaw, is a mathematical technique which determines the existence of a continuous component in the spectrum of the system’s Hamiltonian. In our previous work, we have provided a physical interpretation of the spectral method and applied to the study of conductivity in two-dimensional (2D) materials of various geometry and the investigation of waves in 2D dusty plasma crystals. A significant outcome of this work was theoretically confirming the existence of a metal-to-insulator transition in honeycomb lattices with substitutional disorder, such as graphene doped with hydrogen. In the present study, we combine the spectral approach with a fractional Laplacian technique that models anomalous diffusion in correlated media. Since the combined mathematical model is applicable to the study of transport in complex systems (including multiphase flows, strongly coupled systems, and plasmas), it can open the door to a wonderland of scientific discoveries.