PoissonVlasov

Poisson–Vlasov, also called the Vlasov–Poisson system, is a kinetic model used to describe the evolution of a collisionless plasma or a self-gravitating system under a self-consistent electrostatic (or gravitational) field. It couples a Vlasov equation for the distribution function f(t, x, v) in phase space with Poisson’s equation for the electrostatic or gravitational potential Φ(t, x). For a single charged species with charge q and mass m, the system can be written as ∂f/∂t + v·∇_x f + (q/m)(−∇Φ)·∇_v f = 0, together with ∇^2 Φ = −ρ/ε0, where ρ(x, t) = ∫ q f(x, v, t) dv. In multispecies plasmas, the total charge density is ρ = ∑_s q_s ∫ f_s dv, and each species f_s satisfies its own Vlasov equation with the same potential Φ. The Poisson–Vlasov equations form the electrostatic limit of the more general Vlasov–Maxwell system; when applied to gravity, the sign of the Poisson equation changes, giving a Vlasov–Poisson gravity model with ρ = ∫ f dv.

The model is used in plasma physics to study wave–particle interactions, screening, and nonlinear structures in collisionless regimes, as well as in astrophysics to model large-scale, self-gravitating stellar systems. Typical analyses consider either the whole space or periodic domains, with initial data f0(x, v) and appropriate boundary conditions for Φ. Mathematically, the system is well studied, with results on existence and properties of solutions in three dimensions under various assumptions, and it serves as a foundational framework for numerical methods such as particle-in-cell and semi-Lagrangian schemes.