Modeling Single-File Diffusion by Step Fractional Brownian Motion and Generalized Fractional Langevin Equation

摘要

Single-file diffusion behaves as normal diffusion at small time and asanomalous subdiffusion at large time. These properties can be described byfractional Brownian motion with variable Hurst exponent or multifractionalBrownian motion. We introduce a new stochastic process called Riemann-Liouvillestep fractional Brownian motion which can be regarded as a special case ofmultifractional Brownian motion with step function type of Hurst exponenttailored for single-file diffusion. Such a step fractional Brownian motion canbe obtained as solution of fractional Langevin equation with zero damping.Various types of fractional Langevin equations and their generalizations arethen considered to decide whether their solutions provide the correctdescription of the long and short time behaviors of single-file diffusion. Thecases where dissipative memory kernel is a Dirac delta function, a power-lawfunction, and a combination of both of these functions, are studied in detail.In addition to the case where the short time behavior of single-file diffusionbehaves as normal diffusion, we also consider the possibility of the processthat begins as ballistic motion.