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\begin{document}
\title{Notes for Astrophysical Sciences 554:\\
Irreversible Processes Plasmas}
\author{G. W. Hammett, et.al.\\
{\it Princeton University Plasma Physics Laboratory}\\
{\it P.O. Box 451, Princeton, NJ 08543 USA} }
%\date{}
\maketitle
\begin{abstract}
These are some notes to supplement my lectures for the second half of this
course. Another major resource (from which I am heavily borrowing) is
Prof. Krommes' extensive notes at
\href{ftp://ftp.pppl.gov/pub/krommes/AST-554/notes.dvi}{ftp://ftp.pppl.gov/pub/krommes/AST-554/notes.dvi}.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simple 1-D Krook Illustration of the Chapman-Enskog Method for
Deriving Transport Coefficients}
(I actually did the next section on Rosenbluth potentials first, but in
these notes I'll start with this simpler topic first.)
\subsection{Almost-trivial case: Krook model conserving only particles}
We will eventually do a more complete treatment of the
Braginskii-Chapman-Enskog equations for a plasma with magnetic fields, but
we start with this very simple (or even trivial) limit to illustrate the
essential ideas with a minimum of algebra.
Start with a simple 1-D kinetic equation for a gas (no magnetic or electric
field) with a simple number-conserving Krook operator (also known as a
BGK model or Bhatnagar-Gross-Krook model):
\[
{\partial f \over \partial t}
+ v {\partial f \over \partial x} = - \nu (f - f_M \int dv f)
\]
[In my senior year in college in 1980, I took an applied math course from
Prof. Krook, who was at that time a friendly elderly man. Which is not to
say he wasn't friendly as a young man also.] Here $f_M=\exp(-v^2/(2
v_t^2))/\sqrt{2 \pi v_t^2}$ is a stationary Maxwellian with unit density.
This Krook model collision operator causes $f$ to relax to a stationary
Maxwellian of a fixed temperature, i.e., it doesn't conserve momentum or
energy. This is actually an appropriate model for a case where $f$ is for
some trace species which is colliding with some background thermal bath of
particles. The momentum lost by these trace particles goes into the much
larger bath of background particles, with which the trace particles
eventually come into thermal equilibrium. We consider this as a ``trace
species'' so that the momentum or energy lost from the trace species
(a.k.a. `test-particles'') causes a negligible change in the momentum and
energy of the background bath species.
(?? A somewhat more standard notation would be to absorb the density into
the definition of $f_M = n \exp(-v^2/(2 v_t^2))/\sqrt{2 \pi v_t^2}$, with
$n=\int dv f$. I could rewrite all the notes to use this form.)
Integrating the kinetic equation over all velocity gives:
\[
{\partial n \over \partial t}
+ {\partial (n u)\over \partial x} = 0
% - \nu (\int dv f - \int dv f) = 0
\]
where $n = \int dv f$, and $n u = \int dv f v$.
To calculate how the density $n$ evolves in time, we need to know the flow
$u$. We could operate on the kinetic equation with $\int dv v$ and get an
equation for $\partial (n u ) / \partial t$, but that would require knowing
the $\int dv f v^2$ moment. This gives rise to an infinite chain of moment
equations, i.e. to a ``closure problem'' which occurs in many contexts.
In the high collisionality limit, the Chapman-Enskog procedure provides a
rigorous asymptotic method for truncating the chain of equations and
deriving a closure for a higher moment in terms of lower moments. To make
the ordering expansion we will be doing a little clearer, introduce the
expansion parameter $\epsilon$ into the kinetic equation as
\be
{\partial f \over \partial t}
+ v {\partial f \over \partial x} = - {\nu \over \epsilon} (f - f_M \int dv f)
\label{1dkk}
\ee
and expand $f=f_0 + \epsilon f_1 + \ldots$. [The parameter $\epsilon$ is
just used to help keep track of the order of various terms, and in the end
we can set $\epsilon$ to 1.] To lowest order as $\epsilon \rightarrow 0$,
we have:
\[
0 = - {\nu \over \epsilon} (f_0 - f_M \int dv f)
\]
so $f_0 = f_M n = f_M(v) n(x,t)$. Note that the velocity dependence of
$f_0$ is determined, but the density is an arbitrary function of time and
space at this order, and will not be determined until higher order
equations. (Also note that I've kept the full $f$ (instead of expanding
it) in the velocity integral in the collision operator, so $n$ contains the
full density. If I had just used $f_0$, then the density would still be
arbitrary, but higher order components might contain density also?? Using
the full $f$ as I've done here is a better analog to the particle conserving
properties of the full collision operators involving conservative
derivatives. Perhaps I am belaboring this point.)
The next order equation is
\[
{\partial f_0 \over \partial t}
+ v {\partial f_0 \over \partial x} = - \nu f_1
\]
substituting $f_0=F_M(v) n$ and using the fluid equation $\partial n /
\partial t = - \partial ( n u) / \partial x$, we get
\be
-f_M {\partial (nu) \over \partial x} + f_M v {\partial n \over \partial x}
= - \nu f_1
\label{1d-krook-a}
\ee
However,
\bea-non
n u & = & \int dv v f \nonumber \\
& = & \int dv v (f_0 + \epsilon f_1 + \ldots ) \\
& = & 0 + \epsilon \int dv v f_1 + \ldots
\eea-non
and so the $n u$ term in \eqr{1d-krook-a} is higher order (i.e.,
the lowest order equation for the density is $\partial n / \partial t = 0 +
{\cal O}(\epsilon)$. Then \eqr{1d-krook-a} simplifies to lowest order to
give
\[
- \nu f_1 = f_M v {\partial n \over \partial x}
\]
This is sometimes called the correction equation. With a more rigorous
integro-differential collision operator, it is more complex to invert the
collision operator to solve for $f_1$, though with the Krook model used
here the inversion is trivial. We can now solve to find the particle flux
\bea-non
n u &=& \int dv v f_1 \cr
&=& - {1 \over \nu} \int dv f_M v^2 {\partial n \over \partial x} \cr
&=& - D {\partial n \over \partial x} \cr
\eea-non
where the diffusion coefficient
\[
D = { v_t^2 \over \nu} = \nu \lambda_{\rm mfp}^2
\]
has the proper dimensions of a random-walk diffusion coefficient, and can
be clearly understood from random-walk arguments where the step size is
$\lambda_{\rm mfp}$ and the step time is $1/\nu$. [The mean free path is
defined by $\lambda_{\rm mfp} = v_t/\nu$.]
Now that all of the higher moments (in this case just $nu$) have been
completely specified in terms of lower moments ($nu = - D \partial n /
\partial x$), we have a closed set of equations (just a single equation
actually) that can be solved to find the evolution of density over time:
\[
{\partial n \over \partial t} = D {\partial^2 n \over \partial x^2}
\]
It is important to remember the fundamental ordering assumptions being used
here: time scales slow compared to a collision frequency $(\partial /
\partial t )/ C \ll 1$ and mean free paths short compared to gradient
scale lengths $(v_t \partial / \partial x ) / \nu \sim \lambda_{\rm mfp}/L
\ll 1$. In a strong magnetic field, the restrictions on the gradients
perpendicular to the field can be relaxed somewhat, and only have to long
compared to the gyroradius, not the mean free path. You should read the
section on p. 38-39 of the NRL, which is a nice summary of the assumptions
in Braginskii.
\subsection{Extension to a momentum-conserving Krook operator}
Now extend the procedure of the previous section to include a particle and
momentum conserving Krook operator:
%
\be
{\partial f \over \partial t} + v {\partial f \over \partial x} = C(f) =
-{\nu \over \epsilon} ( f - n f_M )
\label{1dkin}
\ee
%
where $n(x,t) = \int dv v f$ and $f_M$ is now a shifted Maxwellian
\[
f_M = {1 \over \sqrt{ 2 \pi v_t^2}} e^{-{(v-u(x,t))^2 \over 2 v_t^2}}
\]
and $n u = \int dv f v$ so that $n f_M$ contains the same amount of
momentum and particles as $f$. (?? Could add a sketch of $f(v)$ and
$f_0(v)$, showing that they contain the same density and average momentum,
and that $f_1(v)=f(v)-f_0(v)$ has no net density or momentum.) The first
two fluid moment equations of \eqr{1dkin} are
\[
{\partial n \over \partial t} + {\partial (n u) \over \partial x} = 0
\]
%
\[
m n {du \over dt} = m n \left( {\partial u \over \partial t} + u {\partial
u \over \partial x} \right) = - {\partial p \over \partial x}
\]
we keep 2 moment equations (for particle and momentum density) in this
section, while in the previous section we only kept 1 equation because in
this section our collision operator has two conserved quantities while in
the previous section it had only 1. It is not until the pressure moment
that this collision operator can start to have any effect, which is
calculated by the ``correction equation'' for $f_1$ below and used to
provide a closure for $p$. (This is related to the projection operation
that Krommes talks about: evaluating the collision operator $g = C f$ leads
to a $g$ which has no density or momentum, i.e., it is a projection onto a
a subspace of all possible $g(v)$. In other problems with similarities to
this, some people refer to annihilation operators.)
As before, expanding $f = f_0 + \epsilon f_1$, to lowest order we have
$f=f_0 = n f_M$. To next order we have
\[
{\partial f_0 \over \partial t} + v {\partial f_0 \over \partial x} =
-\nu f_1
\]
Use
\[
{\partial f_0 \over \partial t} = {1 \over n} {\partial n \over \partial t}
f_0 + {(v-u) \over v_t^2} {\partial u \over \partial t} f_0
\]
Substituting the fluid equations for $\partial n /\partial t$ and $\partial
u / \partial u$, and using the lowest order expression for the pressure $p
= \int dv f m (v-u)^2 = \int dv f_0 m (v-u)^2 = n T_0$ (where $T_0=m
v_t^2$), after a little bit of algebra one can get a closed ``correction
equation'' which defines $f_1$ in terms of gradients of $n$ and $u$.
Plugging this expression for $f_1$ in to find the next order corrections to
the pressure, and carrying out a bit of algebra (I'm getting tired of
typesetting the algebra), in particular using the useful i.d. in 1 dimension
\[
\langle v^{2n} \rangle \equiv {1 \over \sqrt{2 \pi v_t^2}} \int dv e^{-{v^2
\over 2 v_t^2}} v^{2n} = v_t^{2n} (2n-1)!!
\]
(where the double factorial $n!! = n (n-2) (n-4) \cdots (3) (1)$ for odd
$n$), eventually leads to the result:
\[
p = \int dv m (v-u)^2 (f_0 + \epsilon f_1)
= n T_0 - \epsilon n m {2 v_t^2 \over \nu} {\partial u \over \partial x}
\]
and the final momentum equation is
\[
m n {du \over dt} = - T_0 {\partial n \over \partial x} + {\partial \over
\partial x} \left( n m \eta_0 {\partial u \over \partial x} \right)
\]
where the viscosity $\eta_0 = 2 v_t^2 / \nu$ again makes senses as a
random walk diffusion coefficient. But note that the density equation is
still
\[
{\partial n \over \partial t} + {\partial (n u) \over \partial x} = 0
\]
i.e., there is no diffusion directly in this equation. There is particle
transport, but it is hidden self-consistently in the higher moment
equations.
To study the magnitude of the transport associated with this viscosity, one
can linearize the equations for small amplitude perturbations of the form
$n = n_0 + n_1(t) \cos k x$, and find that the mode frequency is
\[
\omega = { - i \eta_0 k^2 \pm \sqrt{-\eta k^2 + 4 v_t^2 k^2} \over 2}
\]
Thus the density gradients decay away in time, as they would for diffusion,
but the damping is combined with sound wave phenomena.
\subsection{Higher-order corrections, ``Burnett equations''}
**This section is advanced or speculative material that can/should be
skipped.**
Here we try to carry out this procedure to second order in $\epsilon$. We
run into troubles, which might be related to difficulties in the higher
order ``Burnett equations'' that are sometimes discussed in the literature.
The steps I take here I'm guessing are conceptually similar to the
derivation of the Burnett equations, though here we use a simple Krook
collision operator that only conserves particles.
We go back to \eqr{1dkk}, with the simple Krook model that conserves only
particles, and make the substitution $f=f_0 + \epsilon f_1 + \epsilon^2 f_2
+ \ldots$ (except in the last $\int dv f \equiv n$ term of the Krook
collision operator, which we keep to all orders to simplify some matters).
Use the lowest order result that $f_0 = f_M(v) n(x,t)$, and we are left
with
%
\be {\partial (f_0 + \epsilon f_1 +
\ldots) \over \partial t} + v {\partial (f_0 + \epsilon f_1 + \ldots )
\over \partial x} = - \nu (f_1 + \epsilon f_2 + \ldots)
\ee
%
Substituting $f_0=n f_M$ and using $\partial n / \partial t = - \partial (n
u) / \partial x = - \partial (\int dv f v) / \partial x$, gives through
first order in $\epsilon$:
%
\be
-f_M {\partial \over \partial x} \int dv v \epsilon f_1 + \epsilon
{\partial f_1 \over \partial t} + v {\partial n \over \partial x} f_M +
\epsilon v {\partial f_1 \over \partial x} = - \nu (f_1 + \epsilon f_2 +
\ldots)
\label{1dkkmix}
\ee
%
The terms independent of $\epsilon$ give the results $f_1 = - (v/\nu) F_M
\partial n / \partial x$ and the first order flux $(n u)_1 = \int dv v f_1
= - D \partial n / \partial x$, as we got in the previous section.
Substituting this expression for $f_1$ into the terms in \eqr{1dkkmix}
proportional to $\epsilon^1$ give the result
%
\be
\nu^2 f_2 = v f_M {\partial^2 n \over \partial t \partial x} + (v^2 -
v_t^2) f_M {\partial^2 n \over \partial x^2}
\ee
%
Integrating to find the second order flux gives $(nu)_2 = \int dv v f_2 =
(v_t/\nu)^2 \partial^2 n / \partial t \partial x$, and using this in the
closure for the density equation gives:
%
\be
{\partial n \over \partial t} = D {\partial^2 n \over \partial x^2} -
{v_t^2 \over \nu^2} {\partial \over \partial t} {\partial^2 n
\over \partial x^2}.
\label{Burnett-bad}
\ee
%
With a little rearranging to
%
\be
\left(1 + {v_t^2 \over \nu^2} {\partial^2 \over \partial x^2}\right)
{\partial n \over \partial t} = D {\partial^2 n \over \partial x^2}
\ee
%
And we see that this just doesn't look well behaved. For example,
Fourier-transform in x:
%
\be
{\partial n_k \over \partial t} = - {D k^2 \over 1 - \lambda_{\rm mfp}^2
k^2} n_k
\ee
%
There is a singularity at $k \lambda_{\rm mfp} =1$, and modes with $k
\lambda_{\rm mfp} > 1$ are actually unstable (with the growth rate $\nu$ at
large $k$). One might try to ad-hoc patch up these equations by noting
that, to the extent that $k^2 \lambda_{\rm mfp}^2 \ll 1$ is the fundamental
Chapman-Enskog ordering assumption, then one can approximate the
denominator in this expression by
%
\be
{\partial n_k \over \partial t} = - D k^2 (1 + \lambda_{\rm mfp}^2 k^2) n_k
\label{Burnettk-bad}
\ee
%
which after Fourier transforming back leads to a better behaved
hyperdiffusion term:
%
\be
{\partial n \over \partial t} = D {\partial^2 n \over \partial x^2} - D
\lambda_{\rm mfp}^2 {\partial^4 n \over \partial x^4}
\ee
%
I suppose another way of trying to rationalize this is to say it is
equivalent to just substituting the lowest order result $\partial n /
\partial t = D \partial^2 n / \partial x^2$ into the last term of
\eqr{Burnett-bad}. Still, the fact we had to make such modifications seems
a bit strange. The papers mentioned in the following discussion might help
clarify the situation.
To summarize and comment on the higher order corrections:
At zeroth order (ignoring the transport coefficients), the fluid equations
are the dissipationless ``Euler equations''. The Chapman-Enskog procedure
is usually carried out to first order in $\nu/\omega$ or $\lambda_{\rm
mfp}/L$, yielding the standard viscosities etc. that are in the
Navier-Stokes fluid equations (or in Braginskii's fluid equations for
plasmas). Attempts to work to second order in Kn$= \lambda_{\rm mfp}/L$
(the ``Knudson number''), lead to what are usually called the Burnett
equations or variations thereof. Some papers have reported fundamental
difficulties with the Burnett equations (such as violation of the
second-law of thermodynamics due to negative dissipation or a heat flux in
an isothermal gas), or problems with the higher-order boundary conditions
required, while other papers have reported improved results with the
Burnett equations in some regimes or ways to fix the Burnett equations (see
for example D.W. Mackowski, et.al., Phys. Fluids 11, 2108 (1999),
``Comparison of Burnett and DSMC predictions of pressure distributions and
normal stress in one-dimensional, strongly nonisothermal gases'', and
references therein, or ``Numerical Simulation of the flow around a flying
vehicle with high speed at high altitude'', K.L. Guo and G.S. Liaw,
(\href{http://library.redstone.army.mil/hsvsim/papers/hsc013.pdf}{http://library.redstone.army.mil/hsvsim/papers/hsc013.pdf})
and the papers by Balakrishnan and others they cite, and other recent
papers).
Recent papers have been written on these topics, motivated in part by the
growing importance of longer mean-free-path effects in various applications
(such as micromachinery, space shuttle reentry, thermal transport in
the vapors used for crystal growth, etc.).
The Chapman-Enskog procedure is an asymptotic expansion, and, as is well
known in the theory of asymptotics, asymptotic series are not necessarily
guaranteed to converge as more terms are added at a fixed value of the
expansion parameter (it is only guaranteed that a finite number of terms in
the expansion will converge to the right answer in the limit as the
expansion parameter is taken to its limit). In fact, sometimes keeping
more higher-order terms causes the approximation to get worse if the
expansion parameter isn't small enough. (Give a simple example??:)
Instead, it seems to me that an expansion procedure is needed that can
robustly interpolate between the short and long mean-free-path limits. The
Landau-fluid closure models that I and others have worked on are designed
to handle the long mean-free-path limit, and extensions to include
collisions can then interpolate smoothly between the two regimes with a
robust Pad\'e-type of approximation (see for example Eq. 51 in ``Landau
fluid models of collisionless magnetohydrodynamics,'' P.B. Snyder,
G.W. Hammett, and W. Dorland, Phys. Plasmas 4, 3974 (1997) or sections in
Stephen Smith's thesis or papers, or ``Transport theory in the
collisionless limit'', R.D. Hazeltine, Phys. Plasmas 5, 3282 (1998)). In
$k$ space, the crudest 1-moment Landau-fluid model is of the form
%
\[
{\partial n_k \over \partial t} = - {v_t^2 k^2 \over \nu + v_t |k|} n_k
\]
%
Comparing with \eqr{Burnettk-bad}, we see that this is better behaved, and
provides a smooth transition between the collision dominated result at
small $k$, and damping at a phase-mixing rate $\sim v_t |k]$ at high $k$.
Of course my work has focused on plasmas in the strong magnetic field
limit with a 4 or 6-moment method, while in a neutral gas the formulation
would be closer to Grad's 13 moment approach (modified to include
phase-mixing/Landau-damping types of terms). (Others who have worked on
such Landau fluid closures include Callen and Chang, Dorland, Beer, Waltz,
Smith, Snyder, Mattor and Parker, etc. and recent papers by Sugama et.al.).
These Landau-fluid approximations work well and are useful for some
problems, but may have inaccuracies for certain types of problems. To be
really accurate in a longish mean-free-path regime, one should either use
fully kinetic treatments, such as particle-in-cell methods or phase-space
continuum or ``Vlasov'' methods.
\newpage
\section{From the Landau Collision Operator to the Rosenbluth Potential
Form}
This mostly followed Krommes' Chapter 14 Sec. VI.
The Rosenbluth Potential form of the Collision operator has the advantage
that one can use the results of potential theory (from gravitation and
electrostatics) to greatly simplify the integrals if the species being
collided with is spherically symmetric in velocity space. In particular it
makes it easy to show that if one is colliding with a Maxwellian species,
the collision operator reduces to the form on p. 36 of the NRL Plasma
Formulary (the section on ``Fokker-Planck Equation''). This Maxwellian
limit makes clear the physics of a (directed velocity) slowing down term, a
pitch-angle scattering term, and an energy diffusion term, and is an easier
way to prove the formulas in the NRL for these various rates.
To summarize the resulting formulas:
\def\erf{\mathop{\rm erf}\nolimits}
\def\gradv{\nabla_{\bf v}}
The Rosenbluth potential form derived in class (which is slightly different
from the form in the Formulary, where the order of one of the derivatives
has been interchanged) can be written as:
\[
\left({\partial f_a \over \partial t} \right)_{\rm coll} = \sum_b
C(f_a,f_b) = -\sum_b {\partial \over \partial \vec v } \cdot {\bf J}^{a/b}
\]
(note that the NRL convention used here for the sign of $C$ is the reverse
of Krommes', and note that where the NRL uses $\alpha \backslash
\beta$ subscripts, I use $a/b$ subscripts, which is faster to type),
where
\[
{\bf J}^{a/b}= \nu_0^{a/b}v^3
\biggl[{m_a\over m_b}(\gradv h_b)f_a
-{1 \over 2}(\gradv\gradv g_b)\cdot\gradv f_a\biggr]
\]
where
\[
\nu_0^{a/b}={4 \pi e_a^2 e_b^2 \log\Lambda_{ab} \, n_b\over
m_a^2 \, v^3},
\]
(as defined in the NRL Plasma Formulary) and
$$\gradv^2 h_b= -4 \pi f_b,\qquad \gradv^2 g_b=2 h_b,\qquad
\int f_b\,d^3{\bf v}=1.$$
Note that in MKS units, $\nu_0^{a/b}$ is:
\[
\nu_0^{a/b}={e_a^2 e_b^2 \log\Lambda_{ab} \, n_b\over
4\pi \epsilon_0^2 m_a^2 v^3},
\]
If $f_b$ is spherically symmetric, $h_b$ and $g_b$ are
found by straightforward integration. For example, if $f_b$ is a Maxwellian
with temperature $T_b$ then
\[
{d h_b\over dv}= - {\psi(x^{a/b}) \over v^2 } =
- {1\over v^2}
{\nu_s^{a/b}\over (1+m_a/m_b)\nu_0^{a/b}}
\]
where $x^{a/b} = v_a^2/(2 T_b/m_b)$. (A minor point: This is the
definition of $x^{a/b}$ as given on p. 31 of the NRL formulary, and the $a$
subscript reminds us which velocity to use. In our context, or when used
in the operator on NRL p. 34 for collisions with Maxwellian species, $v_a =
v$, where $\vec v$ is the argument of $f_a$.) We can express $\psi(x)$ in
the NRL form or in a form using the standard error function
\[
\psi(x) = {2 \over \sqrt{\pi}} \int_0^x dt \, t^{1/2} e^{-t}
= \erf(\sqrt{x}) - \sqrt{x} {2 \over \sqrt{\pi}} e^{-x}
\]
where
\[
\erf(u)={2\over \sqrt\pi}\int_0^u \exp(-v^2)\,dv,\qquad
\]
As a useful reference (from notes by Karney), we collect here some of the
equations in terms of the error function (instead of the $\psi(x)$
function the NRL uses):
\bea-non
{d h_b\over dv}&=& - {1\over v^2}
\bigl(\erf(u)-u\erf'(u)\bigr)=- {1\over v^2}
{\nu_s^{a/b}\over (1+m_a/m_b)\nu_0^{a/b}} \\
{d g_b\over dv}&=&{1\over 2}
\biggl(\biggl(2-{1\over u^2}\biggr)\erf(u)+{\erf'(u)\over u}\biggr)
={1\over 2}{\nu_\perp^{a/b}\over\nu_0^{a/b}} \\
{d^2 g_b\over dv^2}&=&{1\over v}
\biggl({\erf(u)\over u^2}-{\erf'(u)\over u}\biggr)
={1\over v}{\nu_\parallel^{a/b}\over\nu_0^{a/b}} \\
\erf(u)&=&{2\over \sqrt\pi}\int_0^u \exp(-x^2)\,dx,\qquad
\erf'(u)={2\over \sqrt\pi}\exp(-u^2) \\
u&=&v/\sqrt{2T_b/m_b}
\eea-non
% \section*{Acknowledgements}
% \hspace{\parindent}
%
% This work was supported by U.S. Department of Energy contract
% No.~DE-AC02-76CH03073.
% \begin{thebibliography}{999}
%
% {\footnotesize
%
% \bibitem{Hammett90}
% G.~W.~Hammett and F.~W.~Perkins, Phys.~Rev.~Lett. {\bf 64}, 3019 (1990).
%
% } % end footnotesize
%
% \end{thebibliography}
%% ??
%% FIGURES AT THE END:::
% \newpage
%
% \mfs{0.5}{The evolution of a density
% perturbation from the exact kinetic theory, illustrating phase-mixing,
% and from 2, 3, and 4 moment fluid models (all without dissipation) which
% fail to reproduce phase-mixing.}
%% ??
%% FIGURES AT THE END:::
\end{document}