# Subdiffusion in an external force field

## TL;DR

This paper investigates how external forces influence subdiffusive processes modeled by subordinated Langevin equations, analyzing statistical properties and deriving Fokker-Planck equations for non-stationary, non-ergodic, aging systems.

## Contribution

It introduces a comprehensive analysis of external force effects on subdiffusion using subordinated Langevin equations, including statistical measures and Fokker-Planck equations, highlighting different force interactions.

## Key findings

- External forces affect subdiffusion differently depending on their timing and nature.
- Statistical quantities reveal non-stationarity, non-ergodicity, and aging in the processes.
- Derived Fokker-Planck equations describe the dynamics under various external forces.

## Abstract

The phenomena of subdiffusion are widely observed in physical and biological systems. To investigate the effects of external potentials, say, harmonic potential, linear potential, and time dependent force, we study the subdiffusion described by subordinated Langevin equation with white Gaussian noise, or equivalently, by the single Langevin equation with compound noise. If the force acts on the subordinated process, it keeps working all the time; otherwise, the force just exerts an influence on the system at the moments of jump. Some common statistical quantities, such as, the ensemble and time averaged mean squared displacement, position autocorrelation function, correlation coefficient, generalized Einstein relation, are discussed to distinguish the effects of various forces and different patterns of acting. The corresponding Fokker-Planck equations are also presented. All the stochastic processes discussed here are non-stationary, non-ergodicity, and aging.