Floquet Theory of Classical Relaxation in Time-Dependent Field

Abstract

The anomalous transport of particles in the presence of a time-dependent field is considered in the framework of a comb model. This turbulent-like dynamics consists of inhomogeneous time-dependent advection along the x-backbone and Brownian motion along the y-side branches. This geometrically constrained transport leads to anomalous diffusion along the backbone, which is described by a fractional diffusion equation with time-dependent coefficients. The time periodic process leads to localization of the transport and a particular form of relaxation. The analytical approach is considered in the framework of the Floquet theory, which is developed for the fractional diffusion equation with periodic in time coefficients. This physical situation is considered in detail and analytical expressions for both the probability density function and the mean squared displacement are obtained. The new analytical approach is developed in the framework of the fractional Floquet theory that makes it possible to investigate a new class of anomalous diffusion in the presence of time periodic fields.

Keywords

• fractional diffusion equation
• Floquet theory
• fractional Schrödinger equation
• Mittag-Leffler function
• dilatation operator
• Green function