Environmental Engineering Reference
In-Depth Information
Transitions of individual particles between states result from collisions with oth-
er particles. We denote by
W
(
n
1
,
n
2
,...,
n
i
. . .) the probability that
n
1
particles are
found in the first state,
n
2
particles are found in the second state,
n
i
particles are
found in the
i
th state, and so on. We wish to calculate the number of possible re-
alizations of this distribution. First, we take the
n
1
particles for the first state from
the total number of particles
N
.Thereare
N
!
C
n
1
N
D
(
N
n
1
)!
n
1
!
ways to do this. Next, we select
n
2
particles corresponding to the second state from
the remaining
N
n
1
particles. This can be done in
C
n
2
N
n
1
ways. Continuing this
operation, we find the probability distribution to be
N
!
.
Y
i
W
(
n
1
,
n
2
,...
n
i
,...)
D
const
(
n
i
!) ,
(1.36)
where const is a normalization constant. By this formula, Boltzmann introduced
the equipartition law [44-47], which is the basis of statistical mechanics. We note
the contradiction between the description of an ensemble of identical particles
within the framework of statistical mechanics and the dynamical description [48,
49]. The principle of detailed balance is valid in the dynamical description of this
ensemble, which means that reverse of time
t
t
requires the development of
this particle ensemble in the inverse direction, that is, particles move strictly along
the same trajectories in the inverse direction. The equipartition law requires the
development of the ensemble to its equilibrium state. This contradiction will be
removed if the trajectories of particles are became slightly uncertain.
!
1.2.2
The Boltzmann Distribution
Let us determine the most probabl
e n
umber of particles
n
i
that are to be found in
a state
i
assuming a large number
n
i
1 of particles in each state a
nd
requiring
that the probability
W
a
s well as its lo
ga
rithm have maxima at
n
i
D
n
i
.Wethen
introduce
dn
i
1, and expand the value ln
W
over the interval
dn
i
near the maximum of this value. Using the relation
D
n
i
n
i
, assume that
n
i
dn
i
ln
n
m
!
Z
n
m
D
1
ln
n
!
D
dx
ln
x
,
1
we have (
d
/
dn
)(ln
n
!)
ln
n
.
On the basis of this relation, we obtain from (1.36)
ln
W
(
n
1
,
n
2
,...,
n
i
...)
D
ln
W
(
n
1
,
n
2
,...
n
i
,...)
X
i
X
i
dn
i
/(2
n
i
) .
ln
n
i
dn
i
(1.37)
