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

Abstract

We consider anomalous diffusion of a particle moving in a tilted periodic potential in the presence of Lévy noise and nonlinear friction. Using Monte Carlo simulations, we have found some interesting characteristics of diffusion in such a nonlinear system: when the noise intensity is weak and the external force is close to the critical value at which local minima of the potential just vanish, the nonmonotonic behavior of the effective diffusion index and the superballistic diffusion are observed. This is due to the bimodal nature of the velocity distribution, and thus the test particles exist in either a running state or a long-tailed behind state in the spatial coordinate; the latter is disintegrated into small pieces of the probability peaks. We provide a relation between the group diffusion coefficient and the phase diffusion coefficient. It is shown that the distance between the above two-state centers increasing with time plays the definitive role in the superballistic group diffusion.

Anomalous correlated Lévy flight induced by coexistence of correlation and dissipative nonlinearity

“…Considering the existence of large jumps of the LF, introducing nonlinear friction instead of the linear Stokes friction is a natural and convenient way to study the LF. Hence, the diffusion behavior of the inertial LF driven by the nonlinear friction, considering both the force-free potential and the tilted periodic potential, was studied in our previous work [8]. The results in this work indicate that after introducing nonlinear friction, the inertial LF exhibits normal diffusion in the force-free potential, but displays superdiffusion once again in the tilted periodic potential, and even the superballistic diffusion phenomenon [8,11].…”

“…which gives rise to the velocity potential Φ(v) = γ 0 2 v 2 + γ 2 4 v 4 . Correspondingly, the asymptotic velocity PDF is W (v) ∼ |v| −(μ+3) , |v| → ∞, which decays faster than the Lévy-stable law, and leads to a natural cutoff in the velocity distribution [8,9]. The velocity distribution of the particle in the force-free potential is depicted in figure 1.…”

“…The velocity distribution of the particle in the force-free potential is depicted in figure 1. As a consequence, the MSD of the particle increases linearly with time, i.e., the scaling form can read as x 2 (t) ∼ 2B μ t, and B μ is the diffusion coefficient (figure 1 in reference [8]).…”

“…The long tailed property of its distribution implies that the mean square displacement (MSD) diverges, even in the harmonic potential [1,2]. To overcome this dilemma, several methods are applied when studying the LF, i.e., implementation of a cutoff in the Lévy driving noise [3,4], confining the particles to steeper anharmonic potentials [5][6][7], and introducing dissipative nonlinear friction [8][9][10].…”

Giant Enhancement of Diffusion in a Tilted Egg-Carton Potential

“…To analyze the enhancement phenomenon of diffusion, we apply a relation between the group and phase diffusion coefficients in the one-dimensional case. [18] Assume that the spatial distribution of the particle along the 𝑥 direction is composed of a number of subdistributions 𝑊 𝑖 (𝑥, 𝑡), i.e.,…”