Group superballistic diffusion: bimodal velocity inducing coexistence of two states in a corrugated plane

Correlated continuous-time random walk in the velocity field: the role of velocity and weak asymptotics

Within the framework of a space-time correlated continuous-time random walk model, anomalous diffusion of particles moving in the velocity field is studied in this paper. The weak asymptotic form ω(t) ∼ t^-(1+α), 1 < α < 2 for large t, is considered to be the waiting time distribution. The analytical results reveal that the diffusion in the velocity field, i.e., the mean squared displacement, can display a multi-fractional form caused by dispersive bias and space-time correlation. The numerical results indicate that the multi-fractional diffusion leads to a crossover phenomenon in-between the process at an intermediate timescale, followed by a steady state which is always determined by the largest diffusion exponent term. In addition, the role of velocity and weak asymptotics is discussed. The extremely small fluid velocity can characterize the diffusion by a diffusion coefficient instead of diffusion exponent, which is distinctly different from the former definition. In particular, for the waiting time displaying a weak asymptotic property, if the anomalous part is suppressed by the normal part, a second crossover phenomenon appears at an intermediate timescale, followed by a steady normal diffusion, which implies that the anomalies underlying the process are smoothed out at large timescales. Moreover, we discuss that the consideration of bias and correlation could help to avoid a possible not readily noticeable mistake in studying the topic concerned in this paper, which may be helpful in the relevant experimental research.

Correlated continuous-time random walk in a velocity field: Anomalous bifractional crossover

The diffusion of space-time correlated continuous-time random walk moving in the velocity field, which includes the fluid flowing freely and the fluid flowing through porous media, is investigated in this paper. Results reveal that it presents anomalous diffusion merely caused by space-time correlation in the freely flowing fluid, and the bias from the velocity field only supplies a standard advection, which is verified by the corresponding generalized diffusion equation which includes a standard advection term. However, the diffusion in the fluid flowing through porous media, i.e., the mean squared displacement, can display a bifractional form of which one originates from space-time correlation and the other one originates from dispersive bias caused by sticking of the porous media. The fractional advection term emerging in the corresponding generalized diffusion equation confirms the results. Moreover, the coexistence of correlation and dispersive bias result in crossover phenomenon in-between the diffusive process at an intermediate timescale, but just as the definition of diffusion, the one owning the largest diffusion exponent always prevails at large timescales. However, since the two fractional diffusions originate from a different mechanism, even if it owns the smaller diffusion exponent, that one can dominate the diffusion if it fluctuates much stronger than the other one, which no longer obeys the previous conclusion.

Transition of multidiffusive states in a biased periodic potential

We study a frequency-dependent damping model of hyperdiffusion within the generalized Langevin equation. The model allows for the colored noise defined by its spectral density, assumed to be proportional to ω^{δ-1} at low frequencies with 0<δ<1 (sub-Ohmic damping) or 1<δ<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, hyperdiffusion, collapse, and asymptotical restoration. For analyzing transition phenomenon of multidiffusive 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 an equivalent velocity trap is introduced in the present model; moreover, reformation of ballistic diffusive system is also considered as a marginal situation but does not exhibit the collapsed state of diffusion.