  Single-file diffusion behaves as normal diffusion at small time and as
anomalous subdiffusion at large time. These properties can be described by
fractional Brownian motion with variable Hurst exponent or multifractional
Brownian motion. We introduce a new stochastic process called Riemann-Liouville
step fractional Brownian motion which can be regarded as a special case of
multifractional Brownian motion with step function type of Hurst exponent
tailored for single-file diffusion. Such a step fractional Brownian motion can
be obtained as solution of fractional Langevin equation with zero damping.
Various types of fractional Langevin equations and their generalizations are
then considered to decide whether their solutions provide the correct
description of the long and short time behaviors of single-file diffusion. The
cases where dissipative memory kernel is a Dirac delta function, a power-law
function, 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 diffusion
behaves as normal diffusion, we also consider the possibility of the process
that begins as ballistic motion.
