Download 2.3 Lecture 7: Binary collisions

CHAPTER 2. KINETIC THEORY
2.3
33
Lecture 7: Binary collisions
1. Without collisions:
(a) A neighborhood dqdp moves to dqÕ dpÕ = dqdp since for Hamiltonian systems the flow is incompressible.
(b) A molecule at (q, p, t) moves to (q +
”t, i.e.
f (q +
or
p
”t, p
m
+ F”t, t + ”t) in time
p
”t, p + F”t, t + ”t)dqÕ dpÕ = f (q, p, t)dqdp
m
f (q +
p
”t, p + F”t, t + ”t) = f (q, p, t)
m
(2.32)
(2.33)
2. With collisions:
A
p
ˆf
f (q + ”t, p + F”t, t + ”t) = f (q, p, t) +
m
ˆt
B
”t
(2.34)
coll
By expanding the left hand site in powers of ”t and keeping only linear terms
we get the transport equation
ˆ
p ˆ
ˆ
f (q, p, t) +
·
f (q, p, t) + F ·
f (q, p, t) =
ˆt
m ˆq
ˆp
1
A
ˆf
ˆt
B
(2.35)
coll
2
which is not very useful unless ˆf
is specified. Of course, finding or modˆt coll
eling the collision term is the biggest challenge in the kinetic theory. In the
simplest model one only takes into account binary collisions and assumes that
the colliding particles are uncorrelated (i.e. molecular chaos assumption).
Consider an elastic collision of two spherically symmetric (spin-less) molecules
with m1 , p1 and m2 , p2 . After collision their respective momenta are pÕ1 and
pÕ2 . Then the following conservation laws apply:
• Momentum conservation:
• Energy conservation:
p1 + p2 = pÕ1 + pÕ2 .
(2.36)
p21
p2
pÕ2 pÕ2
+ 2 = 1 + 2.
m1 m2
m1 m2
(2.37)
CHAPTER 2. KINETIC THEORY
34
Define:
• Reduced mass
µ©
• Relative momentum
p©
m1 m2
m1 + m2
3
m1 p1 + m2 p2
p1
p2
=µ
+
m1 + m2
m1 m2
(2.38)
4
(2.39)
Then the conservation of energy can be written as
|p| = |pÕ | ,
(2.40)
i.e. collision simply rotates relative momentum. Such rotations can be specified by a unit vector in spherical coordinates (◊, „), where
• inclination angle ◊ between p and pÕ
• azimuthal angle „ of pÕ about p.
The dynamical aspect of the collisions are described by differential cross
section |d‡/d | - the cross-sectional area which scatters particles into solid
angle d around . For scattering of two spheres of diameter D with impact
parameter b the scattering angle is given by
cos
A B
◊
2
=
b
D
(2.41)
and thus, the cross-sectional area is
A B
A B
◊
◊
D2
D2
d‡ = bd„db = D cos
D sin
d◊d„ =
sin ◊d◊d„ =
d
2
2
4
4
(2.42)
and
d‡
D
=
d
4
The total cross-section is given by
‡total =
which
‡total =
⁄
⁄ -- d‡ -d
-d -
D2
d = fiD2
4
(2.43)
(2.44)
(2.45)
CHAPTER 2. KINETIC THEORY
35
for an elastic scattering of hard spheres.
In quantum mechanics the scattering probabilities Pp1 ,p2 æpÕ1 ,pÕ2 per unit
time per unit volume are expressed through scattering amplitudes Ap1 ,p2 æpÕ1 ,pÕ2
using Born rule,
-
-2
-
Pp1 ,p2 æpÕ1 ,pÕ2 = ”(p1 +p2 ≠pÕ ≠pÕ ) ”(p2 +p2 ≠pÕ2 ≠pÕ2 ) -Ap1 ,p2 æpÕ1 ,pÕ2 1
2
1
2
1
2
2
= ”(p1 +p2 ≠pÕ ≠pÕ ) ”(p2 +p2 ≠pÕ2 ≠pÕ2 ) |ÈpÕ1 , pÕ2 |T|p1 , p2 Í| (2.46)
1
2
1
2
1
2
where T is the quantum mechanical transition matrix and all of the factors
of (2fi) where absorbed into definition of Ap1 ,p2 æpÕ1 ,pÕ2 . If we assume that
the interactions are invariant under spatial rotations, reflections and time
reversal (e.g. electromagnetic interactions), then the transition matrix T
has the following symmetries
ÈpÕ1 , pÕ2 |T|p1 , p2 Í = ÈRpÕ1 , RpÕ2 |T|Rp1 , Rp2 Í
ÈpÕ1 , pÕ2 |T|p1 , p2 Í = È≠p1 , ≠p2 |T| ≠ pÕ1 , ≠pÕ 2 Í,
(2.47)
ÈpÕ1 , pÕ2 |T|p1 , p2 Í = Èp1 , p2 |T|pÕ1 , pÕ2 Í.
(2.48)
where R is a matrix representing spatial rotations and/or reflections. Note
that the vector p can also include spin d.o.f., then R rotates the spatial and
spin d.o.f., but reflects only spatial d.o.f. It follows form (2.47) that
This can be shown by:
• time reversal
ÈpÕ1 , pÕ2 |T|p1 , p2 Í = È≠p1 , ≠p2 |T| ≠ pÕ1 , ≠pÕ2 Í,
• rotation on angle fi around any axis perpendicular to total momentum,
ÈpÕ1 , pÕ2 |T|p1 , p2 Í = È≠R1 pÕ1 , ≠R1 pÕ2 |T| ≠ R1 p1 , ≠R1 p2 Í,
• reflection to complete the interchange the initial and final relative momenta,
ÈpÕ1 , pÕ2 |T|p1 , p2 Í = È≠R2 R1 pÕ1 , ≠R2 R1 pÕ2 |T|≠R2 R1 p1 , ≠R2 R1 p2 Í = Èp1 , p2 |T|pÕ1 , pÕ2 Í,
One can apply the ideas of kinetic theory to other systems with a large
number of d.o.f. For example, a very large network of cosmic strings can
be described by a transport equation analogous to the Boltzmann transport
equation for particles.
Question to Go: Think of a system with a large number
of d.o.f. for which the ideas of the kinetic theory might
be useful.