If Boltzmann's equation (3.11) is solved in situations
where
and
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*
and 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
and
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.