Transition of multi-diffusive states in a biased periodic potential

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.