Environmental Engineering Reference
In-Depth Information
1.2
Statistics of Atomic Particles in Excited and Weakly Ionized Gases
1.2.1
Distribution Function of a System of Identical Particles
The subject of this topic is a weakly ionized gas that consists of a large number
of atoms or molecules and a small admixture of electrons and ions. Each of its
components is a system of many weakly interacting identical particles, and our
first task is to represent the distribution of particles in these systems for various
parameters, and apply to a system of many identical particles the laws of statistical
physics. In this case we deal with the distribution function of free particles
f
(
x
)
over a parameter
x
, so for a macroscopic uniform system of particles
f
(
x
)
dx
is the
number density of particles with the value of this parameter between
x
and
x
dx
.
Within the framework of statistical physics [40, 41], we expound the distribution
function for each parameter twofold using the ergodic theorem [42, 43]. In the first
case, we deal with an ensemble of a large number of particles, and the distribution
function determines the average number particles with a value of this parameter
in an indicated range. In the second case, one test particle is observed, and a given
parameter of this particle varies in time as a result of interaction of this particle
with other particles. Then the distribution function characterizes the relative time
when a given parameter is found in an indicated range.
We start from the energy distribution for weakly interacting particles in a closed
system. In an ensemble of a large number of particles, each of the particles is in
one of a set of states described by the quantum numbers
i
. The goal is to find the
average number of particles that are found in one of these states. For example, if
we have a gas of molecules, the problem is to find the molecular distribution of
its vibrational and rotational states. We shall consider problems of this type below.
Consider a closed system of
N
particles, so this number does not change with time,
as well as the total energy of particles. Denoting the number of particles in the
i
th
state by
n
i
, we have the condition of conservation of the total number of particles
in the form
C
X
N
D
n
i
.
(1.34)
i
Since the ensemble of particles is closed, that is, it does not exchange by energy
with other systems, we require conservation of the total energy
E
of the particles,
where
X
E
D
ε
i
n
i
,
(1.35)
i
where
i
is the energy of a particle in the
i
th state. In the course of evolution of
the system, an individual particle can change its state, but the average number of
particles in each state stays essentially the same. Such a state of a closed system is
called a state of thermodynamic equilibrium.
ε
