Environmental Engineering Reference
In-Depth Information
The condition that this quantity is maximal is
X
ln
n
i
dn
i
D
0 .
(1.38)
i
In addition to this equation, we take into account the relations following from (1.34)
and (1.35) which account for the conservation of the total number of particles and
their total energy:
X
dn
i
D
0 ,
(1.39)
i
X
ε
i
dn
i
D
0 .
(1.40)
i
Equations (1.38), (1.39), and (1.40) allow us to determine the average number
of particles in a given state. Multiplying (1.39) by
ln
C
and (1.40) by 1/
T
,where
C
and
T
are characteristic parameters of this system, and summing the resulting
equations, we have
X
ln
n
i
T
dn
i
ε
i
ln
C
C
D
0.
i
Becausethisequationisfulfilledforany
dn
i
, one can require that the expression
in the parentheses is equal to zero. This leads to the following expression for the
most probable number of particles in a given state:
C
exp
T
.
ε
i
n
i
D
(1.41)
This formula is the Boltzmann distribution.
We now determine the physical nature of
C
and
T
in (1.41) that follows from the
additional equations (1.34) and (1.35). From (1.34) we have
C
P
i
exp(
N
.
This means that the value
C
is the normalization constant. The energy parameter
T
is the particle temperature and characterizes the average energy of a particle. Below
we express this parameter in energy units and hence we will not use the dimen-
sioned proportionality factor - the Boltzmann constant
k
ε
i
/
T
)
D
10
16
erg/K -
D
1.38
10
16
erg (see
as is often done. Thus, the Kelvin is the energy unit, equal to 1.38
Appendix B).
We
c
an prove that at large
n
i
the probability of observing a significant deviation
from
n
i
is small. According to (1.37) and (1.38) the probability
W
near its maxima
is
W
(
n
1
,
n
2
,...
n
i
,...)exp
"
#
.
X
i
n
i
)
2
2
n
i
(
n
i
W
(
n
1
,
n
2
,...
n
i
...)
D
Fr
om this
it
follows th
at
a significant shift of
n
i
from its average value
n
i
is
j
n
i
1/
(
n
i
)
1/2
.Sin
ce
n
i
n
i
j
1,
th
erelativeshiftofthenumberofparticlesinone
(
n
i
)
3/2
. Thus, the observed number of particles in a
given state differs little from its average value.
j
n
i
n
i
j
/
n
i
state is small (
