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Phys761/F10/Hassam/Problem 1
At t=0, a neutral gas is configured so that
f(x,
v
,0) = (2
π
v
th
2
)
-3/2
exp[-v
2
/2v
th
2
] [n
0
+ n
1
sin(2
π
x/L)],
where v
th
is the thermal speed and n
1
< n
0
.
Assuming that L <<
λ
, where
λ
is the mean
free path, and that gravity can be neglected, the collisionless Boltzmann equation
∂
f/
∂
t +
v.
∂
f/
∂
x
=
0
applies.
1.
Calculate the density n(x,0).
Make a sketch of n(x,0) in x-y space. Make a sketch
of f(x,v
x
,0) in x-v
x
space (looks like baguettes).
2.
Solve the PDE to find f(x,
v
,t) for all t.
The PDE can be solved by the method of
characteristics;
or you may use the fact that f(x-v
x
t,v
y
,v
z
) is a solution – prove
this. Make a sketch of the f(x,v
x
,t) baguettes at some t in x-v
x
space.
3.
Calculate n(x,t).
What happens to the ripples in x as t goes to infinity?

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