This equation is a semi-classical quantum Vlasov equation. However, unlike the classical case that Df/Dt [not equal to] 0 owing to the RHS quantum h-term in Eq.(4), the distribution function f is not preserved, except for linear electric fields that leads to a vanishing h-term due to [[partial derivative].sup.3][phi]/[partial derivative][x.sup.3] = 0, along with the classical characteristic equations of the electrons:
In the absence of an external magnetic field, [B.sub.0], replacing f with [f.sub.0] + [f.sub.1] and [phi] = [[phi].sub.1] in the Wigner-Poisson equation, Eq.(6), offers the linearized semi-classical quantum Vlasov equation in which Eq.(5) is kept unchanged:
Christlieb, "A conservative high order semi-Lagrangian WENO method for the Vlasov equation," Journal of Computational Physics, vol.
Crouseilles applied the semi-Lagrangian method to Vlasov equations. This approach easily achieved high-order time accuracy and positivity [17,18].
Bostan, Gyrokinetic Vlasov equation
in three dimensional setting.
Shoucri presents the application of the method of characteristics for the numerical solution of hyperbolic-type partial differential equations, with particular attention to the Vlasov equation
and other equations pertinent to plasma physics.
Nov 26-28 First International Workshop on the Theory and Applications of the Vlasov Equation
. Nancy, France.
Individual topics include hydrodynamic limits of kinetic models, collisionless plasmas and the Vlasov Maxwell system, irreversible behaviors in Vlasov equation
and many-body Hamiltonian dynamics in terms of Landau damping, chaos and granularity, guiding center theory, variational formulation of exact and reduced Vlasov-Maxwell equations, general gyrokinetic theory (including an article with applications in magnetic confinement research in plasma plastics) kinetic to fluid descriptions in plasmas, nonlocal closures in long mean free path regimes, modeling quantum plasmas and inelastic kinetic theory in terms of the granular gas.
A more general class of stochastic systems is governed by McKean- Vlasov equations
in which the coefficients are not only functions of the state but also of the probability measure induced by the state itself.