Chemistry Reference
In-Depth Information
Hence, an overall model has to describe the temporal behavior of all included particles
of very different kinds, that is, atoms and molecules in ground and excited states, ions
in ground and excited states, and electrons. Depending on the types of particles, their
interaction with each other and with electromagnetic fields is also very different. In
a first step, all these differences are ignored by considering more generally a many-
particle system where external forces act on the particles and where interactions of
the particles are possible.
9.1.1 P
LASMA AS A
M
ANY
P
ARTICLE
S
YSTEM
Considering a system of
N
particles, a unique description of this system is given by all
the coordinates
,
N
. Generally, a system
of 3
N
partial differential equations of second order for the coordinate components has
to be solved for given initial conditions of all particles to get the complete behavior
of the system—a visionary task. However, the complete information about position
and momentum of every single particle is not of interest in the understanding and
modeling of plasma-chemical systems. In fact, the desired information concerns the
mean or thermodynamic behavior of the system. Consequently, a statistical approach
is required here.
In order to deduce a statistical description, the distribution of
N
particles in the
6
N
-dimensional phase space is considered. In this space, a time-dependent
N
-particle
distribution function (density function) ρ
x
k
and momenta
p
k
of the particles
k
=
1,
...
can be defined. It represents the
probability to find the particle
k
at a given time
t
and at the position
(
x
k
,
p
k
,
t
)
x
k
with the
momentum
p
k
and is consequently normalized to unity according to
ρ
(
x
k
,
p
k
,
t
)
d
3
N
xd
3
N
p
=
1.
(9.1)
The average value
...
of a quantity
A
(
x
k
,
p
k
)
depending on all coordinates and
momenta of the particles is given by
A
A
(
t
)
=
(
x
k
,
p
k
)
ρ
(
x
k
,
p
k
,
t
)
d
3
N
xd
3
N
p
.
(9.2)
For closed systems, the density function has to be constant along the trajectory in the
6
N
-dimensional phase space according to the normalization (9.1). This requirement
immediately leads to the basic equation for ρ, the continuity equation [2-5]
∂
3
N
d
ρ
dt
=
∂
ρ
∂
x
k
∂
ρ
x
k
∂
∂
p
k
∂
ρ
p
k
∂
t
+
t
+
=
0,
(9.3)
∂
∂
t
k
=
1
named after J. Liouville who deduced this fundamental equation in the nineteenth
century.
Despite its simple structure, the Liouville equation (9.3) is a very complicated
partial differential equation that cannot be solved for the large number of plasma
