# Transition of multi-diffusive states in a biased periodic potential

## TL;DR

This paper investigates how particles in a biased periodic potential transition through multiple diffusive states under super-Ohmic damping, revealing complex regimes including hyper-diffusion and collapse, with implications for understanding anomalous diffusion.

## Contribution

It introduces a frequency-dependent damping model with colored noise, analyzing multi-diffusive transitions and the concept of equivalent velocity traps in hyper-diffusive systems.

## Key findings

- Sequential four diffusive regimes identified: thermalization, hyper-diffusion, collapse, and restoration.
- First escape time from locked to running state follows an exponential distribution.
- Reformation of ballistic diffusion considered without exhibiting collapsed diffusion.

## Abstract

We study a frequency-dependent damping model of hyper-diffusion within the generalized Langevin equation. The model allows for the colored noise defined by its spectral density, assumed to be proportional to $\omega^{\delta-1}$ at low frequencies with $0<\delta<1$ (sub-Ohmic damping) or $1<\delta<2$ (super-Ohmic damping), where the frequency-dependent damping is deduced from the noise by means of the fluctuation-dissipation theorem. It is shown that for super-Ohmic damping and certain parameters, the diffusive process of the particle in a titled periodic potential undergos sequentially four time-regimes: thermalization, hyper-diffusion, collapse and asymptotical restoration. For analysing transition phenomenon of multi-diffusive states, we demonstrate that the first exist time of the particle escaping from the locked state into the running state abides by an exponential distribution. The concept of equivalent velocity trap is introduced in the present model, moreover, reformation of ballistic diffusive system is also considered as a marginal situation, however there does not exhibit the collapsed state of diffusion.