Environmental Engineering Reference
In-Depth Information
Although we did not consider the dynamics for evolution of the ensemble of
identical particles, it is clear that this ensemble tends to the equilibrium state with
time. This nowmeans that if the distribution function for particles differs from the
equilibrium one at the beginning, it becomes the equilibrium one through a long
time, and for the gaseous system in the absence of external fields the equilibrium
distribution function is given by (3.9). Formally, one can find the contradiction
between this statement and the principles of particle mechanics. Indeed, let the
ensemble of particles develops from a nonequilibriumstate to the equilibriumone.
Let us reverse time, and then according to the principle of detailed balance the
system would develop from the equilibrium state to a nonequilibrium state. This
contradictionmeans there is an additional principle of statistical mechanics [15-17]
that corresponds to a small shift of phases for collision events. From this we also
conclude that the kinetics of an ensemble of identical particles includes additional
assumptions which are valid for systems of many particles.
3.1.4
Collision Integral for Electrons in a Gas
The specifics of electron-atom collisions in a gas follow from the small ratio of
the electron mass
m
e
to the mass
M
of an atom. Even if the electron momentum
experiences a large change as a result of a collision with an atom, the electron ener-
gy varies little. Therefore, the velocity distribution of electrons is nearly symmetric
with respect to directions of electron motion [18-21]. This allows one to represent
the distribution function in the form of expansion over spherical harmonics [22-
26], which results from the small ratio
m
e
/
M
and is the basis of strict kinetics for
evolution of electrons in a gas [27]. Therefore, we have for the electron distribution
function [12, 27]
f
(
v
)
D
f
0
(
)
C
v
x
f
1
(
) ,
(3.10)
v
v
where the
x
-axisisinthedirectionoftheelectricfield
E
.
The kinetic equation (3.4) now has the form
e
m
e
@
f
@
v
D
I
col
(
f
),
where
E
is the electric field strength. Assuming the number density of electrons
N
e
to be small compared with the atom number density
N
a
, we find that the presence
of electrons in a gas does not affect the Maxwell distribution function
(
v
a
)forthe
atoms, and the electron-atom collision integral has a linear dependence on the
distribution function
f
(
v
), so the electron-atom collision integral
I
ea
has the form
'
I
ea
(
f
)
D
I
ea
(
f
0
)
C
I
ea
(
v
x
f
1
).
Let us integrate this equation over the angle
between the directions of the elec-
tron velocity and the electric field strength while accounting for (3.10) and also
θ
