Chemistry Reference
In-Depth Information
Liouville equation
(
N
-particle distribution function [density function])
Reduction to 1-particle
distribution function
f
(Kinetic theory)
α
Binary collisions
short range
Long-range interaction,
small deflection
No collisions
Boltzmann equation
Fokker-Planck equation
Vlasov equation
3
Integration over
d
v
Macroscopic balances
(Hydrodynamics)
FIGURE 9.2
Hierarchy of equations in plasma physics.
point, the different interactions of the particles with each other and with particles
of another kind have to be inspected in more detail. As mentioned in Section 9.2,
the interaction processes have to be treated as quantum mechanical problems in
principle [6].
In the framework of the classical kinetic description of gases, a large number of
particles move rapidly and randomly, where interactions among the particles beyond
a small distance are negligible. The velocity of a particle and its position are assumed
to be uncorrelated. As soon as the impact ranges of particles overlap, a collision
process occurs at which the influence of external fields is neglected in the particle
interaction region. Because the interaction radius is much smaller than the mean free
path of the particles and because the collision time is sufficiently small, the particle
motion during the collision process and the simultaneous interaction of more than
two particles can be neglected. Therefore, it is sufficient to determine the changes
in velocities and states of the particles in a binary collision process. In addition,
the mean free path of the particles is much smaller than the vessel dimensions, so
that the influence of the wall can remain unconsidered. From a strict point of view,
this approach is applicable to collision processes between neutral particles as well
as between a charged and a neutral particle, while the interaction between charge
carriers is long-range and collective in nature.
In order to derive the kinetic equation according to the classical kinetic theory,
which yields the desired one-particle distribution function
f
α
for the particles
of species α, first the motion of the particles under the action of an external field
without collisions is considered. This motion over a sufficiently small time step
(
x
,
v
,
t
)
t
results in a change of the space and velocity coordinates and thus the phase space
element according to
F
a
m
α
ˆ
−→ ˆ
x
−→
x
=
x
+
v
t
,
v
v
=
v
+
t
,
d
3
xd
3
v
−→
d
3
xd
3
ˆ
v
,
ˆ
(9.7)
