DMS Graduate Student Seminar

Speaker: Prof. Evdokiya (Eva) Kostadinova (Physics Department, Auburn University) 

Title: Fractional Laplacian Spectral Approach (FLS) to Anomalous Diffusion in Complex Systems

Abstract: Fractional-power operators have received much attention due to their wide application in modeling anomalous diffusion. Anomalous diffusion is a microscopic process that leads to non-linear mean squared displacements and non-Gaussian distributions caused by non-local interactions, correlations, memory effects, and/or stochasticity in complex systems. Here we explore the combination of spectral theory and fractional calculus to analyze anomalous diffusion in complex systems. In this method, processes driving anomalous diffusion are represented by a Hamiltonian operator with a fractional Laplacian kinetic energy term and a random disorder potential energy term. The spectrum of possible energy states for the system operator is studied in an infinitely-dimensional Hilbert space. Specifically, the identification of a continuous spectrum is interpreted as transport, energy mixing, or onset of turbulence in the system at hand. The main advantage of the spectral method is the ability to map a non-linear problem in phase space to a linear problem in Hilbert space, thus utilizing linear algebra and avoiding the need to assume boundary conditions. 

Work supported by NSF-PHY-2308742, NSF-PHY-2308743, NSF-PHY-2440328, EPSCoRFTPP OIA-2148653, DE-SC0023061, DE-SC0024547, DE-SC0024681, DE-SC0021185, DE-SC-0019176, DE-FG02-05ER54809, and DE-FC02-04ER54698.