Diffusion and turbulence in phase-space and formation of phase-space vortices

TL;DR

This paper extends a fluid-like model of phase-space, introducing a modified Vlasov equation to describe anisotropic diffusion and the formation of phase-space vortices driven by turbulent-like flows, revealing solitary wave modes.

Contribution

It presents a new modified Vlasov equation for anisotropic diffusion in phase-space and analyzes vortex formation due to turbulence, including derivation of a vorticity transport equation with solitary wave solutions.

Findings

Growth of phase-space vortices due to turbulent-like flow.

Derivation of a Schamel-KdV vorticity transport equation.

Identification of solitary modes in phase-space vorticity waves.

Abstract

In this work, the recently introduced fluid-like treatment of the phase-space has been further extended and some interesting outcomes have been presented. A modified form of the Vlasov equation has been presented which describes the diffusion of the phase-space density. This anisotropic diffusion is analysed and the flow of the phase-space probability field is shown. Growth of phase-space vortices is then shown due to increased turbulent-like flow, which is marked by the dominating inertial flow above the diffusive flow. The nature of this flow is judged by using a parameter for the phase-space. It is then shown that the formation of phase-space vortices is due to growth of turbulent-like flow in the phase-space. On the bases of the diffusion parameters, the vorticity field transport of the hydrodynamic phase-space is studied and a Schamel-KdV form of the vorticity transport equation is derived, suggesting solitary modes of the phase-space vorticity waves.