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

Abstract

In this work, the recently introduced fluid-like treatment of the phase-space (Lobo and Sayal in PP 31:092301, 2024), termed as the phase-space hydrodynamic model, 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 diffusive 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.