We present the Quantum Vlasov or Wigner equation as a "phase space" presentation of quantum mechanics that is close to the classical Vlasov equation, but where the "distribution function" $w(x,v,t)$ will in general have also negative values.
We discuss the relation to the classical Vlasov equation in the semi-classical asymptotics of small Planck's constant, for the linear case [2] and for the nonlinear case where we couple the quantum Vlasov equation to the Poisson equation [4, 3, 5] and [1].
Recently, in some sort of "inverse semiclassical limit" the numerical concept of solving Schrödinger-Poisson as an approximation of Vlasov-Poisson attracted attention in cosmology, which opens a link to the "smoothed Schrödinger/Wigner numerics" of Athanassoulis et al. (e.g. [6]).[-]

Dark matter (DM) constitutes almost 85% of all mass able to cluster into gravitationally bound objects. Thus it has played the determining role in the origin and evolution of the structure in the universe often referred to as the Cosmic Web. The dark matter component of the Cosmic Web or simply the Dark Matter Web is considerably easier to understand theoretically than the baryonic component of the web if one assumes that DM interacts only gravitationally. One of the major differences between the DM and baryonic webs consists in the multi stream structure of the DM web. Thus it allows to use three diagnostic fields that do not present in the baryonic web: the number of streams field in Eulerian space, the number of flip flops field in Lagrangian space, and the caustic structure in the both. Although these characteristics have been known for a long time their systematic studies as fields started only a few years ago. I will report new recent results of numerical studies of the three fields mentioned above and also discuss the features of the DM web they have unveil.[-]

Gyrokinetic simulation is considered to be an essential tool to study turbulent transport driven by micro-scale instabilities in tokamak plasmas. It is roughly categorized into two approaches; delta-$f$ local and full-$f$ global approaches. In full-$f$ approach, both turbulent transport and profile evolutions are solved self-consistently under the power balance between external heat source/sink. In this talk, we address (A) numerical technique to treat such full-$f$ gyrokinetic Vlasov-Poisson equations [1] and (B) characteristics of global ion-scale turbulence and transport barrier [2]. We also discuss (C) the role of stable modes in collisionless or weakly collisional plasmas [3].[-]

We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond [1, 3]. First we establish the convergence towards the Euler equation under several assumptions on the energy and on the norms of the initial data. Then we analyze the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density of particles with a moving point charge, characterized by a Dirac mass in the phase-space.[-]

In this talk, I will present ColDICE[1, 2], a publicly available parallel numerical solver designed to solve the Vlasov-Poisson equations in the cold case limit. The method is based on the representation of the phase-space sheet as a conforming, self-adaptive simplicial tessellation whose vertices follow the Lagrangian equations of motion. In this presentation, I will mainly focus on describing the underlying algorithm and its practical implementation, as well as showing a few practical examples demonstrating its capabilities.[-]