Anomalous diffusion in a randomly modulated velocity field

Abstract

This paper proposes a simple model of anomalous diffusion, in which a particle moves with the velocity field induced by a single "dipole" (a doublet or a pair of source and sink), whose moment is modulated randomly at each time step. A motivation to introduce such a model is that it may serve as a toy model to investigate an anomalous diffusion of fluid particles in turbulence. We perform a numerical simulation of the fractal dimension of the trajectory using periodic boundary conditions in two and three dimensions. For a wide range of the dipole moment, we estimate the fractal dimension of the trajectory to be 1.7 − −1.9 (2D) and 2.4 − −2.7 (3D).1 There are also literature discussing the relationship between turbulence and SLE using random fields [20,21,22], and it is expected that there may be a correspondence between these and our model.

Analytical Study of Anomalous Diffusion by Randomly Modulated Dipole

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This paper derives the Fokker-Planck (FP) equation for a particle moving in potential by a randomly modulated dipole. The FP equation describes the anomalous diffusion observed in the companion paper [1] and breaks the conservation of the total probability at the singularity by the dipole. It also shows anisotropic diffusion, which is typical in fluid turbulence.

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“…Eq. ( 1) was proposed in the companion paper [1] as a toy model that captures some features of the fluid turbulence. The authors performed a numerical simulation of the random process, revealing that the particle's trajectory has a fractal dimension 2.4 ∼ 2.7 in D = 3 and 1.7 ∼ 1.9 in D = 2.…”

“…It motivates us to study (1) analytically. In this paper, we first derive the Fokker-Planck (FP) equation for the random process using the path-integral formalism [2] (section 2).…”

“…We solve the Fokker-Planck equation by combining Bessel functions (section 3). 1 It has an unusual feature that the total probability is not invariant due to the singularity at the origin. It corresponds to a phenomenon that the particle, which gets too close to the origin, jumps because of the strong force from the dipole.…”

“…It corresponds to a phenomenon that the particle, which gets too close to the origin, jumps because of the strong force from the dipole. The trajectory resembles the Lévy flight [4] and 1 We note that the exact solution in [3] is a special one where the particle starts from the origin r = 0. For our purse, we need solutions where the particle starts somewhere else r = 0. has a nontrivial fractal dimension in three dimensions.…”