2.57 Fall 2004 – Lecture 17
1
2.57 NanotoMacro Transport Processes
Fall 2004
Lecture 17
In last lecture, we discussed the Nparticle distribution function. We have obtained
(
)
(
)
)
n
(
)
n
(
)
n
(
)
n
(
N
,
,
t
f
Systems
.
No
p
r
p
r
∆
∆
=
in a small volume of the phase space,
∆
r
(n)
∆
p
(n)
,
where
∆
r
(n)
=
∆
r
1
∆
r
2
…
∆
r
N
=
∆
r
(1)
∆
r
(2)
…
∆
r
(n)
and
∆
p
(n)
=
∆
p
1
∆
p
2
…
∆
p
N
=
∆
p
(1)
∆
p
(2)
…
∆
p
(n)
.
The time evolution of f
(N)
(t,
r
(n)
,
p
(n)
) in the phase space is governed by the
Liouville
equation
,
0
p
f
p
r
f
r
t
f
n
1
i
)
i
(
)
N
(
)
i
(
n
1
i
)
i
(
)
N
(
)
i
(
)
N
(
=
∑
∂
∂
×
+
∑
∂
∂
×
+
∂
∂
=
=
±
±
which can be derived based on the fact that the traces of systems in the ensemble do not
intersect.
Note: The number of the degree of freedom in the phase space is normally very big. For
3D cases, it is 6N
A
=6
×
6.02E23.
The Liouville equation involves a huge number of variables, which makes its impractical
in terms of the boundary and initial conditions, as well as analytical and numerical
solutions. One way to simplify the Liouville equation is to consider one particle in a
system. This is a representative particle having coordinate
r
1
and momentum
p
1
, each of
the vectors has m components, i.e., mdegrees of freedom.
We introduce a
oneparticle
distribution function
by averaging the Nparticle distribution function over the rest (N
1) particles in the system,
r
(i)
p
(i)
ensemble
at t=0
Flowline
control
volume
r
(i)
r
(i)
+
∆
r
(i)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2.57 Fall 2004 – Lecture 17
2
(
)
(
)
N
N
p
p
r
r
p
r
p
r
d
d
d
d
,
,
t
f
)!
1
N
(
!
N
,
,
t
f
2
2
)
n
(
)
n
(
)
N
(
1
1
)
1
(
•
•
•
∫
•
•
•
∫
•
•
•
−
=
.
For simplicity in notation, we will drop the subscript 1 and understand (
r
,
p
) as the
coordinates and momenta of one particle. Since f
(N)
(t,
r
(n)
,
p
(n)
) represents the number
density of systems having generalized coordinates (
r
(n)
,
p
(n)
) in the ensemble, the one
particle distribution function represents number density of systems having (
r
,
p
),
f(t,
r
,
p
)d
3
r
d
3
p
=number of systems in d
3
r
d
3
p
.
With the introduction of the averaging method to obtain the oneparticle distribution
function, one can start from the Liouville equation and carry out the averaging over the
space and momentum coordinates of the other (N1) particles.
This procedure leads to
c
t
f
f
dt
d
f
dt
d
t
f
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
∇
•
+
∇
•
+
∂
∂
p
r
p
r
where the subscripts (
r
and
p
) in the gradient operators represent the variables of the
gradient. The scattering term
c
f
t
∂
⎛
⎞
⎜
⎟
∂
⎝
⎠
will be discussed in details later in this lecture. The
above equation is the
Boltzmann equation
or Boltzmann transport equation.
Note: The derivative
d
dt
r
has the meaning of velocity, while
d
dt
p
denotes force.
Consider the collision process between two particles as shown in the above figure. After
the collision, the energy and the velocity of each particle may change. Clearly, the
collision is a timedependent process. The rigorous way of dealing the collision process is
to solve the corresponding timedependent Schrödinger equation for the combined system
made of both particles. This is, however, usually very complicated and not practical.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 WRIGHT,J
 Photon, Fundamental physics concepts, Reciprocal lattice, Boltzmann equation

Click to edit the document details