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AS551 General Plasma Physics 1 \\
November 13, 1998 \\
\vspace{0.1in}
{\bf Problem Set (due Monday Nov. 23)}
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{\bf 1. Solution of a diffusion equation.}  Following Nov. 11, 1998
lecture notes.  p.17.  Starting with a diffusion equation,
$$
{\partial n \over \partial t} = D { \partial^2 n \over \partial x^2}
$$
with $D =$constant, show that if the initial condition is
$n(x,0)=\delta(x-x_0)$ that the solution $n(x,t)$ will be a Gaussian
distribution whose width $\sigma = \sqrt{2 D t}$.  [It is easiest to do
this by Fourier transforming in $x$.]

{\bf 2. Properties of a convection-diffusion equation.}  Do the homework
problem on p.18 of the Nov. 11, 1998 lecture notes.  I already give the
proof there that $\partial N/\partial t = 0$.  You should apply a similar
method to prove that
$$
{\partial \langle x \rangle \over \partial t} = u
$$
and
$$
{\partial \over \partial t} \langle (x-\langle x \rangle)^2 \rangle = 2 D
$$

{\bf 3. Random walk}.  Consider a simple random walk where every time step
$\Delta t$ each particle takes a random step of $\pm \Delta x$.  Start with
1 million particles at x=0.  On the first step, half of the particles will
move left $-\Delta x$ and half will move right $+\Delta x$. On the second
step, half of the particles at $x=+\Delta x$ will move back to $x=0$, and
half will move to $x=+2 \Delta x$, etc.  Repeat this process by hand for a
total of 6 steps, and for each time step, plot the resulting \# of
particles vs. x.  What is the average $x^2$ of all the particles?  What is
the maximum $x$ of any particles?  What fraction of the particles made it
that far?

{\bf 4. Slowing down rate.}  Starting with the Eq. at the bottom of p.11 of
the Nov. 9, 1998 notes, which after correcting a typo is
$$
{\langle \Delta v_\alpha \rangle \over \Delta t}
= -{4 \pi e_\alpha^2 e_\beta^2 \log \Lambda \over m_\alpha M v_\alpha^3}
\vec v_\alpha \int_0^{v_\alpha} d v_\beta 4 \pi v_\beta^2 f_\beta(v_\beta)
$$
Assuming that $f_\beta$ is a Maxwellian, derive the formula for the slowing
down rate $\nu_s^{\alpha \backslash \beta}$ as given on p.31 of the NRL plasma
formulary.

{\bf 5. Pitch-angle scattering in the Lorentz collision operator.}  Do
problem 13.1 in Goldston and Rutherford.

\newpage

{\bf 6. Fokker-Planck Collision Operator.}  The general form of the
Fokker-Planck collision operator is
$$
\left. {\partial f_\alpha (\vec v, t) \over \partial t} \right|_{coll} 
= 
-{ \partial \over \partial \vec v} \cdot \left[ f_\alpha {\langle \Delta
\vec v \rangle \over \Delta t} \right] 
+ { \partial \over \partial \vec v} { \partial \over \partial \vec v} :
\left[ f_\alpha {\langle \Delta \vec v \Delta \vec v \rangle \over 2 \Delta
t} \right]
$$
For general distributions $f_\beta$ of particles being collided with, the
coefficients of this Fokker-Planck equation are given by the Rosenbluth
potentials on p.35 of the NRL formulary.  If $f_\beta$ are all Maxwellian,
then the diffusion tensor becomes diagonal for the reasons give on p.12 of
the Nov. 12, 1998 notes.  Then the collision operator reduces to the form:
$$
\left. {\partial f_\alpha(\vec v, t) \over \partial t} \right|_{coll} 
= 
{ \partial \over \partial \vec v} \cdot \left[ f_\alpha \nu_s \vec v
\right]
+ { \partial \over \partial \vec v} { \partial \over \partial \vec v} :
\left[ f_\alpha \left( {1 \over 2 } \nu_\Par \vec v \vec v
+ {1 \over 4} \nu_\perp (v^2 \vec{\vec 1} - \vec v \vec v) \right) \right]
\eql{1}
$$
where $\nu_s$, $\nu_\Par$, and $\nu_\perp$ are the slowing down rate, the
parallel diffusion rate, and the perpendicular (or transverse) diffusion
rate.  These are given by summing the expressions on p.31 of the NRL
formulary over field particles $\beta$.  For example,
$$
\nu_s = \sum_\beta \nu_s^{\alpha \backslash \beta}
$$
We derived a collision operator like \eqr{1} rigorously for e-i collisions
(in the $m_e/m_i=0$ limit where $\nu_\Par=0$).  For this problem you are to
assume it is true in general (with $\nu_\Par$, etc., as given on p.31 of
the NRL). Show that \eqr{1} can be written as
$$
\left. {\partial f_\alpha(\vec v, t) \over \partial t} \right|_{coll} 
= 
- \sum_\beta
{\partial \over \partial \vec v} \cdot \vec J^{\alpha \backslash \beta}
$$
where the phase-space flux of particles  $\vec J^{\alpha \backslash \beta}$
is given on p. 36 of the NRL formulary.

Hint: some useful identities for this problem, which you should prove,
include
$$
{\partial g(v) \over \partial \vec v} = {\partial g \over \partial v }
{\vec v \over v}
$$
and
$$
{\partial \over \partial \vec v} \cdot \left[ g(v) (v^2 \vec{\vec 1} - \vec
v \vec v) \right]
=
g(v) {\partial \over \partial \vec v} \cdot \left[ (v^2 \vec{\vec 1} - \vec
v \vec v) \right] = -g 2 \vec v
$$
for any arbitrary function $g(v)$ which depends only on $v=|\vec v|$.


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