Vlasov equations

If Boltzmann's equation (3.11) is solved in situations where ${\bf E}$ and ${\bf B}$ are known external fields then it is a linear differential equation. However in a plasma, governed by the set of equations (3.11) - (), one must solve for self-consistent ${\bf E}$ and ${\bf B}$fields. The equations that describe how charge and current densities affect the magnetic and electric fields (Maxwell's equations) must also be considered. The interdependent nature of the particle and field interactions is illustrated in Figure . The velocity of a particle injected into a plasma will change under the influence of ${\bf E}$ and ${\bf B}$ fields. These forces are different for electrons and ions, inducing currents, which in turn alter the fields. When equations (3.11) - (), are solved in a self-consistent manner, with the collisional term in Boltzmann's equation equal to zero, they are referred to as the Vlasov equations. Equations (3.11) - () are a system of nonlinear integro-differential equations. They provide the basis for both kinetic theory (treated in later lectures) and fluid theory.

  

Figure: Flow chart illustrating the nonlinear interactions between particles and electromagnetic fields in a plasma.