Environmental Engineering Reference
In-Depth Information
the phase space, referred to as a phase point. The evolution of the system state with
time is completely described by the motion or trajectory of the phase point through
phase space. The trajectory of the point is expressed by equations of motion of the
N
bodies. Integration of such a large system of equations is not feasible, and statistical
methods are usually used.
Define the distribution function
f
p
υ
,
f
p
υ
(
r
,
v
,
t
)
d
r
d
v
(1.41)
(the number of particles in the system that have phase points in d
r
d
v
around
r
and
v
at time
t
).
=
Here
v
and
r
have components
υ
i
and
r
i
(
i
=
1
,
2
,...,
l
)
, respectively, d
v
=
d
υ
1
,
d
υ
2
,...,
d
υ
l
,andd
r
=
d
r
1
,
d
r
2
,...,
d
r
l
. By this definition, we have
···
f
p
υ
(
,
,
)
=
.
r
v
t
d
r
d
v
N
(1.42)
all
r
,
v
The ensemble average of any function
ψ
(
r
,
v
)
of the position
r
andvelocity
v
of the
system is defined by
1
N
ψ
=
···
v
ψ
(
r
,
v
,
t
)
f
p
υ
d
r
d
v
.
(1.43)
all
r
,
The assumption that the particles do not interact with each other leads to (Carey 1999)
d
f
p
υ
d
t
=
,
0
(1.44)
whis is called the
Liouville equation
. It indicates that if we follow the particlesin
a volume element along a flow line in phase space without collisions, the distribution
is conserved.
f
p
υ
(
r
+
d
r
,
v
+
d
v
,
t
+
d
t
)=
f
p
υ
(
r
,
v
,
t
)
.
(1.45)
If collisions occur, the distribution
f
p
υ
will change over a time interval d
t
by an
amount (
∂
f
p
υ
/
∂
t
)
coll
d
t
, and therefore
f
p
υ
(
r
+
d
r
,
v
+
d
v
,
t
+
d
t
)
−
f
p
υ
(
r
,
v
,
t
)=(
∂
f
p
υ
/
∂
t
)
coll
d
t
,
(1.46)
which is equivalent to
∂
f
p
υ
(
r
+
d
r
,
v
+
d
v
,
t
+
d
t
)
−
f
p
υ
(
r
,
v
,
t
)
f
p
υ
∂
=
coll
.
(1.47)
d
t
t
The Taylor expansion of
f
p
υ
(
r
+
d
r
,
v
+
d
v
,
t
+
d
t
)
at the point
(
r
,
v
,
t
)
yields
1
∂
f
p
υ
1
∂
f
p
υ
∂υ
j
υ
j
+
∂
f
p
υ
∂
l
j
l
j
∑
d
r
j
+
∑
d
d
t
+
higher order terms
∂
=
=
∂
r
j
t
f
p
υ
∂
=
coll
,
(1.48)
d
t
t
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