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

TL;DR

This paper introduces a new stochastic process called Riemann-Liouville step fractional Brownian motion to model single-file diffusion, capturing its transition from normal to anomalous subdiffusive behavior over time.

Contribution

It proposes a novel step fractional Brownian motion model and explores fractional Langevin equations with various memory kernels to accurately describe single-file diffusion dynamics.

Findings

The new process models both short and long-time diffusion behaviors.

Fractional Langevin equations with different kernels can replicate observed diffusion properties.

The model accounts for ballistic initial motion in single-file diffusion.

Abstract

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.