% GPP1 poll
%

\documentstyle[12pt,equations,my_commands,times]{article}

% Margins etc. can be set with the following commands:

\setlength{\oddsidemargin}{0.0in} %left margin -3/4 inch.
%%\setlength{\evensidemargin}{0.25in} %right margin -3/4 inch?
\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{-1.0in} %top margin -1.375 inch.
\setlength{\textheight}{9.3in}

\setlength{\parindent}{0.0in}
\setlength{\parskip}{1em}

\hyphenation{Magneto-hydro-dynamics}

% ??:
%\renewcommand{\baselinestretch}{1.5}	% to double space
\sloppy         % don't be as fussy as TeX about line breaks
\raggedbottom   %

\begin{document}

% \pagestyle{empty}

{\Large \bf
\center

% Class Poll for GPP-I\\
% \vspace{0.1in}
}
Subj: Class Poll for GPP-I Topics\\
From: Greg Hammett\\
To: GPP-I Students

As I prepare some upcoming lectures, I thought I would do a quick poll of
the students to gauge what your background is on certain topics.  I'm doing
this so I can aim the lectures at the right level.  My intent is to start
with the basics, and go on to derive fairly important widely-used forms of
the Fokker-Planck equation.  If anyone is a little rusty on something, I am
happy to review some foundations at the beginning of a derivation.  But if
everyone would be bored if I cover some of these topics, I'd like to know
ahead of time.  So could you please read through the following 5 questions
and rank how comfortable you are with each on a 1-10 scale, where: 1=never
heard of it, 3=I've learned a little about this, but not enough to be real
confident or quick about it, i.e., I'd still appreciate a review of this
topic, 6 = I know this topic and could do it with a little thought or
looking at some old notes, 10=I know this very well and have solved several
homework-type problems on it.  There is nothing to be gained by inflating
your rankings (I'm expecting ranks of 1-4 on most of these questions). At
this stage in your training there are inevitable gaps of some topics you
haven't learned much about yet.  Please email your answers to
hammett@pppl.gov.  Thanks.


A type-set version of this is at {\tt
w3.pppl.gov/\~{}fisch/homework/poll.pdf} or {\tt poll.ps}.

1. How familiar are you with the concept of a random walk process (also
known as Brownian motion, etc.)?  I.e., if every time step $\Delta t$ a
particle takes a random step to the left or right of size $\pm \Delta x$,
do you know how to work out the diffusion coefficient
$$
D={(\Delta x)^2 \over 2 \Delta t}
$$

2. Do you know how to show that the RMS (root mean square) displacement of
a particle undergoing a random walk increases with time proportional to
$\sqrt{t}$?

3. Have you worked with the convection-diffusion equation
$$
{\partial n(x,t) \over \partial t } + {\partial \over \partial x} (n u) = D
{\partial^2 n \over \partial x^2}
$$
Do you know how to relate the convection velocity $u$ and diffusion
coefficient $D$ to a random walk process?

4. Do you know how to show that, if the initial condition for the
convection-diffusion equation is $n=\delta(x)$, and if $u$ and $D$ are
constants, the solution will be a Gaussian that spreads out and shifts in
time?

5. Do you know how to derive a Fokker-Planck equation from a general Markov
transition probability function?  I.e., in cases where the random walk
diffusion coefficient $D=D(x)$ varies with $x$, do you know how to derive
the proper Fokker-Planck equation of the form
$$
{\partial n(x,t) \over \partial t} = -{\partial \over \partial x} (n u) +
{\partial^2 \over \partial x^2} (D n)?
$$

6. Do you know how to derive and use the multi-dimensional Fokker-Planck
equation
$$
{\partial n\over \partial t} = -{\partial \over \partial x_i} \left[ { \langle
\Delta x_i \rangle \over \Delta t} n \right] +
{\partial^2 \over \partial x_i \partial x_j} \left[{\langle \Delta x_i
\Delta x_j \rangle \over 2 \Delta t} n \right]
$$
\end{document}
