Vlasov–Fokker–Planck model

A kinetic equation combining collisionless Vlasov transport in a self-consistent field with Fokker–Planck terms for collisional diffusion and friction.

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Vlasov equation: A kinetic equation describing collisionless evolution of a particle distribution under forces, including self-consistent fields. Its transport term carries particles through phase space under the model’s collective force.
One-particle distribution function: A function giving the density of particles over position and velocity at a specified time. It is the unknown whose transport and collisional relaxation the model describes.
Plasma physics: The study of ionized matter and its collective electromagnetic behavior. Plasma applications combine long-range self-consistent fields with collisional relaxation.
Collisionless Vlasov equation: The kinetic equation for particles moving under self-consistent forces without a collision term. It retains collective transport but omits the collisional relaxation added here.
Fokker–Planck equation: An equation describing probability or particle distributions evolving through drift and diffusion. Its drift and diffusion terms add collisional effects to Vlasov transport.
Mean-field theory: An approximation that replaces many interacting degrees of freedom with an averaged field. The Vlasov part represents collective interactions through an averaged self-consistent field.
Charged-particle beam: A directed group of charged particles transported and controlled by electromagnetic fields. Beam distributions can spread and relax through Coulomb scattering while generating collective fields.
Boltzmann equation: A kinetic equation describing particle distributions through streaming and a collision integral. Its collision integral treats discrete encounters more generally than a Fokker–Planck approximation.
Poisson equation: A partial differential equation relating a potential to its source distribution. It often determines the self-consistent electrostatic potential from the particle density.
Collision operator: A mathematical term in a kinetic equation that models changes caused by particle collisions. The Fokker–Planck contribution is one particular approximation to collisional evolution.