Modeling single-file diffusion with step fractional Brownian motion and a generalizedfractional Langevin equation

Abstract

Single-file diffusion behaves as normal diffusion at small time and as subdiffusion at largetime. These properties can be described in terms of fractional Brownian motion withvariable Hurst exponent or multifractional Brownian motion. We introduce a newstochastic process called Riemann–Liouville step fractional Brownian motionwhich can be regarded as a special case of multifractional Brownian motion with astep function type of Hurst exponent tailored for single-file diffusion. Such a stepfractional Brownian motion can be obtained as a solution of the fractional Langevinequation with zero damping. Various kinds of fractional Langevin equations and theirgeneralizations are then considered in order to decide whether their solutions provide thecorrect description of the long and short time behaviors of single-file diffusion. Thecases where the dissipative memory kernel is a Dirac delta function, a power-lawfunction and a combination of these functions are studied in detail. In addition tothe case where the short time behavior of single-file diffusion behaves as normaldiffusion, we also consider the possibility of a process that begins as ballistic motion.