plasma_writeup.lyx

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\begin_body

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\tht}{\vartheta}
\end_inset


\begin_inset FormulaMacro
\newcommand{\ph}{\varphi}
\end_inset


\begin_inset FormulaMacro
\newcommand{\balpha}{\boldsymbol{\alpha}}
\end_inset


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\newcommand{\btheta}{\boldsymbol{\theta}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bJ}{\boldsymbol{J}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bGamma}{\boldsymbol{\Gamma}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\bOmega}{\boldsymbol{\Omega}}
\end_inset


\begin_inset FormulaMacro
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\end_inset


\begin_inset FormulaMacro
\newcommand{\t}[1]{\text{#1}}
\end_inset


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\newcommand{\m}{\text{m}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\v}[1]{\boldsymbol{#1}}
\end_inset


\end_layout

\begin_layout Standard
\begin_inset FormulaMacro
\renewcommand{\t}[1]{\mathbf{#1}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\tT}{\boldsymbol{\mathsf{T}}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\tI}{\boldsymbol{\mathsf{I}}}
\end_inset


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\newcommand{\tP}{\boldsymbol{\mathsf{P}}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\vv}{\boldsymbol{v}}
\end_inset


\end_layout

\begin_layout Title
Plasma Physics
\end_layout

\begin_layout Author
Christopher Albert
\end_layout

\begin_layout Date
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
today
\end_layout

\end_inset


\end_layout

\begin_layout Standard
This document is a write-up for the plasma physics course at TU Graz with
 the goal of becoming the new lecture notes.
 While derivations in the original lecture notes are kept short, they are
 listed more extensively here for better understanding.
 Before going to the actual problems, we summarize some general concepts
 that are required for treating them.
 Most of those concepts have a much wider application than plasma physics,
 so it's a good idea to be familiar with them.
\end_layout

\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents

\end_inset


\end_layout

\begin_layout Section*
Common Concepts
\end_layout

\begin_layout Subsection*

\series bold
Concept 1: Tensor calculus
\end_layout

\begin_layout Standard
First we summarize some rules of tensor calculus that are required for certain
 tasks within plasma physics.
 A summary of rules of vector and tensor calculus is available in the NRL
 plasma formulary.
 Some rules are
\end_layout

\begin_layout Standard

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\begin_inset Formula 
\begin{equation}
\v a\times(\v b\times\v c)=(\v c\times\v b)\times\v a=\v b(\v a\cdot\v c)-\v c(\v a\cdot\v b)
\end{equation}

\end_inset

The scalar product is not commutative for tensors:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\v a\cdot\nabla\v b=(\v a\cdot\nabla)\v b=\v a\cdot(\nabla\v b)\neq(\nabla\v b)\cdot\v a
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Cross-product with a curl:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\v a\times(\nabla\times\v b)=(\nabla\v b)\cdot\v a-\v a\cdot\nabla\v b\label{eq:eq3}
\end{equation}

\end_inset

Divergence of a dyad:
\begin_inset Formula 
\begin{equation}
\nabla\cdot(\v a\v b)=(\v a\cdot\nabla)\v b+(\nabla\cdot\v a)\v b\label{eq:diaddiv}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Those rules can be deduced from using index notation for the respective
 operations.
 For application in research and engineering it is important to note that
 tensor calculus can be formulated in a flexible and coordinate-independent
 way within the co-contravariant formalism.
 More detailed derivations on this very convenient way of treating curvilinear
 coordinates can be found in the first chapters of the book of Callen/d'haeselee
r 
\begin_inset CommandInset citation
LatexCommand citep
key "dhaeseleer1991"
literal "false"

\end_inset

.
 
\end_layout

\begin_layout Subsubsection*

\series bold
Taylor expansion of vector fields
\begin_inset CommandInset label
LatexCommand label
name "subsec:Taylor-expansion-of"

\end_inset


\end_layout

\begin_layout Standard
The second order expansion of a vector field 
\begin_inset Formula $\v A(\v r)$
\end_inset

 around a point 
\begin_inset Formula $\v R$
\end_inset

 is given by
\begin_inset Formula 
\begin{align}
\v A(\v R+\v{\rho}) & =\v A(\v R)+(\v{\rho}\cdot\nabla)\v A(\v R)+(\v{\rho}\v{\rho}:\nabla\nabla))\v A(\v R)+\mathcal{O}(\rho^{3}).
\end{align}

\end_inset

Here 
\begin_inset Formula $\v{\rho}\v{\rho}$
\end_inset

 and 
\begin_inset Formula $\nabla\nabla$
\end_inset

 are two dyades combined by a scalar product.
 where differentiation doesn't act on 
\begin_inset Formula $\v{\rho}$
\end_inset

.
 In index notation this means
\begin_inset Formula 
\begin{equation}
A_{i}(\v R+\v{\rho})=A_{i}(\v R)+\rho_{j}\partial_{j}A_{i}(\v R)+(\rho_{k}\rho_{l}\partial_{k}\partial_{l})A_{i}(\v R)+\mathcal{O}(\rho^{3}).
\end{equation}

\end_inset

The convention is that a sum is taken automatically over indexes appearing
 twice, and we skip the sum sign.
 If we consider the dyade
\begin_inset Formula 
\begin{equation}
(\nabla\v A)_{kl}=\partial_{k}A_{l}.
\end{equation}

\end_inset

We see that the first order term can be written without brackets as
\begin_inset Formula 
\begin{equation}
(\v{\rho}\cdot\nabla)\v A=\v{\rho}\cdot\nabla\v A.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
For the second order term we combine
\begin_inset Formula 
\begin{equation}
\rho_{k}\rho_{l}\partial_{k}\partial_{l}=(\rho_{k}\partial_{k})(\rho_{l}\partial_{l}).
\end{equation}

\end_inset

Thus it can also be written as a dyad itself via
\begin_inset Formula 
\begin{equation}
\v{\rho}\v{\rho}:\nabla\nabla=(\v{\rho}\cdot\nabla)(\v{\rho}\cdot\nabla),
\end{equation}

\end_inset

where derivatives don't act on 
\begin_inset Formula $\v{\rho}$
\end_inset

.
 As an example we can look at Cartesian coordinates 
\begin_inset Formula $(x^{1},x^{2})$
\end_inset

 in 2D and a vector component
\begin_inset Formula 
\begin{align*}
A_{i}(\v R+\v{\rho}) & =A_{i}(\v R)\\
 & +\rho^{1}\partial_{1}A_{i}(\v R)+\rho^{2}\partial_{2}A_{i}(\v R)\\
 & +(\rho^{1})^{2}\partial_{11}A_{i}(\v R)+2\rho^{1}\rho^{2}\partial_{12}A_{i}(\v R)+(\rho^{2})^{2}\partial_{22}A_{i}(\v R)\\
 & +\mathcal{O}(\rho^{3}).
\end{align*}

\end_inset

Here we used the shorthand notation 
\begin_inset Formula $\partial_{k}\partial_{l}\equiv\partial_{kl}$
\end_inset

.
 Since 
\begin_inset Formula $\rho^{1}\rho^{2}\partial_{12}=\rho^{2}\rho^{1}\partial_{21}$
\end_inset

 a coefficient 
\begin_inset Formula $2$
\end_inset

 appears in front of this term in the second order term of the Taylor expansion.
 Going to higher orders, such factors will be binomial coefficients in 2D
 and multinomial coefficients in 
\begin_inset Formula $N$
\end_inset

-D.
\end_layout

\begin_layout Subsection*

\series bold
Concept 2: Fluid mechanics
\end_layout

\begin_layout Standard
Let's look at the general form of fluid equations.
 The derivation relies on the intuitive picture what a 
\begin_inset Quotes eld
\end_inset

flux
\begin_inset Quotes erd
\end_inset

 is.
 A rigorous derivation of fluid equations can be performed from individual
 particles via kinetic theory.
\end_layout

\begin_layout Standard
We start with the conservation law for mass density 
\begin_inset Formula $\rho_{m}(\v r,t)$
\end_inset

 known as the continuity equation.
 In a fluid moving with velocity 
\begin_inset Formula $\v v(\v r,t)$
\end_inset

, mass is transported by the mass flux 
\begin_inset Formula 
\begin{equation}
\boldsymbol{\Gamma}_{m}(\v r,t)=\rho_{m}(\v r,t)\v v(\v r,t).
\end{equation}

\end_inset

The mass flux is a vector field parallel to 
\begin_inset Formula $\v v(\v r,t)$
\end_inset

 and of scales with the mass available to move in 
\begin_inset Formula $\v x$
\end_inset

 quantified by 
\begin_inset Formula $\rho_{m}(\v r,t)$
\end_inset

.
 Now we consider a volume 
\begin_inset Formula $\Omega$
\end_inset

 fixed to the lab frame.
\begin_inset Foot
status open

\begin_layout Plain Layout
This is the Eulerian picture, as opposed to the Lagrangian picture, where
 a volume is moved and distorted together with the fluid flow.
 
\end_layout

\end_inset

 We compute the total mass inside 
\begin_inset Formula $\Omega$
\end_inset

 via a volume integral over 
\begin_inset Formula $\rho_{m}$
\end_inset

.
 If the system's total mass should be conserved, a change in mass in the
 volume 
\begin_inset Formula $\Omega$
\end_inset

 can only be caused by an in-/outflux of mass via 
\begin_inset Formula $\v{\Gamma}_{m}$
\end_inset

 over its boundary surface 
\begin_inset Formula $\partial\Omega$
\end_inset

.
 Mathematically we formulate this as a relation between a partial time derivativ
e over the total mass and a surface integral over 
\begin_inset Formula $\v{\Gamma}_{m}$
\end_inset

 and obtain the law of
\series bold

\begin_inset Newline newline
\end_inset

Mass conservation (integral)
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\frac{\partial}{\partial t}\int_{\Omega}\rho_{m}(\v r,t)\d V+\int_{\partial\Omega}\boldsymbol{\Gamma}_{m}(\v r,t)\cdot\d\v S=0.
\end{equation}

\end_inset

Note that a positive flux 
\begin_inset Formula $\boldsymbol{\Gamma}_{m}$
\end_inset

 points outwards, such that it needs to be compensated by a reduction of
 integral mass via its partial time derivative.
 Now we can exchange derivative and integral, as 
\begin_inset Formula $\Omega$
\end_inset

 is fixed during time and use the divergence theorem to obtain
\begin_inset Formula 
\begin{equation}
\int_{\Omega}\frac{\partial\rho_{m}(\v r,t)}{\partial t}+\nabla\cdot\boldsymbol{\Gamma}_{m}(\v r,t)\d V=0.
\end{equation}

\end_inset

Since this relation is true independent on the choice of 
\begin_inset Formula $\Omega$
\end_inset

, it must hold for the integrand.
 Thus we obtain the law of
\series bold

\begin_inset VSpace defskip
\end_inset


\begin_inset Newline newline
\end_inset

Mass conservation (differential) / continuity equation
\series default

\begin_inset Formula 
\begin{equation}
\frac{\partial\rho_{m}(\v r,t)}{\partial t}+\nabla\cdot\boldsymbol{\Gamma}_{m}(\v r,t)=0.
\end{equation}

\end_inset

or explicitly
\begin_inset Formula 
\begin{equation}
\frac{\partial\rho_{m}(\v r,t)}{\partial t}+\nabla\cdot(\rho_{m}(\v r,t)\v v(\v r,t))=0.
\end{equation}

\end_inset

For a single species of particles of mass 
\begin_inset Formula $m$
\end_inset

 we can write 
\begin_inset Formula $\rho_{m}(\v r,t)=m\,n(\v r,t)$
\end_inset

 via the number density 
\begin_inset Formula $n(\v x,t)$
\end_inset

 and write the law of
\series bold

\begin_inset VSpace defskip
\end_inset


\begin_inset Newline newline
\end_inset

Number density conservation
\series default

\begin_inset Formula 
\begin{equation}
\frac{\partial n(\v r,t)}{\partial t}+\nabla\cdot(\v v(\v r,t)n(\v r,t))=0.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The second conserved quantity we are interested in is the momentum density
 
\begin_inset Formula $\boldsymbol{p}$
\end_inset

 in absence of external forces.
 Since momentum is a vector, its flux 
\begin_inset Formula $\boldsymbol{\Gamma}_{\v p}$
\end_inset

 must be a rank-2 tensor.
 The result is still the same as for mass or any other conservation law,
 yielding the
\series bold

\begin_inset VSpace defskip
\end_inset


\begin_inset Newline newline
\end_inset

Total momentum conservation
\series default
 
\series bold
law (differential)
\series default

\begin_inset Formula 
\begin{equation}
\frac{\partial\boldsymbol{p}(\v r,t)}{\partial t}+\nabla\cdot\boldsymbol{\Gamma}_{\v p}(\v r,t)=0.
\end{equation}

\end_inset

In case of external forces, they would appear as a 
\emph on
source
\emph default
 on the right-hand side as force densities 
\begin_inset Formula $\boldsymbol{f}$
\end_inset

.
 The overall momentum density consists of 
\emph on
kinematic fluid
\emph default
 
\begin_inset Formula $\v p^{\text{fl}}=\rho_{m}\v v$
\end_inset

 and 
\emph on
electromagnetic
\emph default
 momentum density 
\begin_inset Formula $\v p^{\text{EM}}=\boldsymbol{S}/c^{2}$
\end_inset

, where 
\begin_inset Formula $\boldsymbol{S}$
\end_inset

 is the Poynting vector.
 The momentum flux 
\begin_inset Formula $\boldsymbol{\Gamma}_{\v p}$
\end_inset

 consists of a fluid part
\begin_inset Foot
status open

\begin_layout Plain Layout
Here 
\begin_inset Formula $\t T=\boldsymbol{v}\v w$
\end_inset

 means the outer product 
\begin_inset Formula $\boldsymbol{v}\otimes\boldsymbol{w}$
\end_inset

, being a rank-2 tensor with Cartesian components 
\begin_inset Formula $T_{ij}=v_{i}w_{j}$
\end_inset

.
\end_layout

\end_inset

 
\begin_inset Formula $\boldsymbol{\Gamma}_{\v p}^{\text{fl}}=\v v\v p_{\text{fl}}$
\end_inset

, but is also influenced by 
\emph on
thermodynamic
\emph default
 effects, namely the width of the distribution around the average velocity
 
\begin_inset Formula $\boldsymbol{v}$
\end_inset

.
 We separate out 
\emph on
electrodynamic
\emph default
 and 
\emph on
thermodynamic
\emph default
 effects in the total momentum conservation law, such that they appear as
 forces on the right-hand side, leading to the
\series bold

\begin_inset VSpace defskip
\end_inset


\begin_inset Newline newline
\end_inset

Kinematic fluid momentum law
\series default

\begin_inset Formula 
\begin{equation}
\frac{\partial\v p^{\mathrm{fl}}(\v r,t)}{\partial t}+\nabla\cdot(\v v(\v r,t)\v p^{\mathrm{fl}}(\v r,t))=-\nabla p(\v r,t)+q\left(\v E(\v r,t)+\v v(\v r,t)\times\v B(\v r,t)\right).
\end{equation}

\end_inset

In Cartesian coordinates the divergence of a rank-2 tensor 
\begin_inset Formula $\t T$
\end_inset

 is a column vector whose components are the divergence of tensor columns,
 so
\begin_inset Formula 
\begin{equation}
\nabla\cdot\t T=\left(\begin{array}{c}
\nabla\cdot\boldsymbol{T}_{\cdot1}\\
\nabla\cdot\boldsymbol{T}_{\cdot2}\\
\nabla\cdot\boldsymbol{T}_{\cdot3}
\end{array}\right)=\left(\begin{array}{c}
\frac{\partial}{\partial x^{1}}T_{11}+\frac{\partial}{\partial x^{2}}T_{21}+\frac{\partial}{\partial x^{3}}T_{31}\\
\frac{\partial}{\partial x^{1}}T_{12}+\frac{\partial}{\partial x^{2}}T_{22}+\frac{\partial}{\partial x^{3}}T_{32}\\
\frac{\partial}{\partial x^{1}}T_{13}+\frac{\partial}{\partial x^{2}}T_{23}+\frac{\partial}{\partial x^{3}}T_{33}
\end{array}\right).
\end{equation}

\end_inset

Now we can further modify this using mass continuity by using a rule from
 tensor calculus to write the flux vector 
\begin_inset Formula $\nabla\cdot\boldsymbol{\Gamma}_{\v p}^{\text{fl}}$
\end_inset

 as
\begin_inset Formula 
\begin{align}
\nabla\cdot\boldsymbol{\Gamma}_{\v p}^{\mathrm{fl}}=\nabla\cdot(\v v\v p^{\mathrm{fl}}) & =\v p^{\mathrm{fl}}(\nabla\cdot\v v)+(\v v\cdot\nabla)\v p^{\mathrm{fl}}.
\end{align}

\end_inset

Here, 
\begin_inset Formula $\boldsymbol{v}\cdot\nabla$
\end_inset

 performs a convective derivative, i.e.
 
\begin_inset Formula 
\begin{equation}
(\v v\cdot\nabla)\boldsymbol{w}=v_{1}\frac{\partial}{\partial x^{1}}\boldsymbol{w}+v_{2}\frac{\partial}{\partial x^{2}}\boldsymbol{w}+v_{3}\frac{\partial}{\partial x^{3}}\boldsymbol{w}.
\end{equation}

\end_inset

Now let's look back and see that 
\emph on
incidentally 
\emph default
(well, not entirely) the 
\emph on
fluid momentum density
\emph default
 
\begin_inset Formula $\v p^{\mathrm{fl}}=\rho_{m}\v v=\boldsymbol{\Gamma}_{m}$
\end_inset

 matches the 
\emph on
mass flux
\emph default
 defined before.
 
\end_layout

\begin_layout Section
Basics
\end_layout

\begin_layout Subsection*
1.1 Criteria for a plasma
\end_layout

\begin_layout Subsection*

\series bold
Problem 1: Work needed to introduce charge density fluctuation
\end_layout

\begin_layout Standard
see lecture notes
\end_layout

\begin_layout Subsection*

\series bold
Problem 2: Debye shielding - the easy way
\end_layout

\begin_layout Standard
see lecture notes
\end_layout

\begin_layout Subsection*

\series bold
Problem 3: Debye shielding - the hard way
\end_layout

\begin_layout Standard
see lecture notes
\end_layout

\begin_layout Standard
Helmholtz equation
\begin_inset Formula 
\begin{equation}
\Delta\Phi(\v x)-\frac{1}{\lambda^{2}}\Phi(\v x)=-\frac{Q}{\varepsilon_{0}}\delta(\v x).
\end{equation}

\end_inset

Linear problem and infinite or periodic space: Use spatial Fourier transform.
 Be careful about normalization with 
\begin_inset Formula $1/\sqrt{2\pi}$
\end_inset

 (symmetric, unitary) or 
\begin_inset Formula $1/(2\pi)$
\end_inset

 (non-unitary) on one of either transform or inverse transform.
\end_layout

\begin_layout Standard
1-dimensional Fourier transform of 
\begin_inset Formula $\delta(x)$
\end_inset

: 
\begin_inset Formula 
\begin{align}
\mathcal{F}\delta(x) & =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\d x\,\delta(x)e^{-ikx}\nonumber \\
 & =\frac{e^{-ik0}}{\sqrt{2\pi}}=\frac{1}{\sqrt{2\pi}}.
\end{align}

\end_inset


\begin_inset Formula $N$
\end_inset

-dimensional Fourier transform of 
\begin_inset Formula $\delta(\v x)$
\end_inset

:
\begin_inset Formula 
\begin{align}
\mathcal{F}_{N}\delta(\v x) & =\frac{1}{(2\pi)^{N/2}}\int_{-\infty}^{\infty}\d^{N}x\,\delta(\v x)e^{-i\v k\cdot\v x}\nonumber \\
 & =\frac{e^{-i\v k\cdot\v 0}}{(2\pi)^{N/2}}=\frac{1}{(2\pi)^{N/2}}.
\end{align}

\end_inset

So in our 3D equation:
\begin_inset Formula 
\begin{equation}
-k^{2}\Phi(\v k)-\frac{1}{\lambda^{2}}\Phi(\v k)=-\frac{Q}{(2\pi)^{3/2}\varepsilon_{0}}.
\end{equation}

\end_inset

Solution in 
\begin_inset Formula $\v k$
\end_inset

-space:
\begin_inset Formula 
\begin{equation}
\Phi(\v k)=\frac{Q}{(2\pi)^{3/2}\varepsilon_{0}(k^{2}+1/\lambda_{D}^{\,2})}.
\end{equation}

\end_inset

Solution in real space:
\begin_inset Formula 
\begin{align}
\Phi(\v x)=\mathcal{F}_{3}^{\,-1}\Phi(\v k) & =\frac{1}{(2\pi)^{3/2}}\int_{-\infty}^{\infty}\d^{3}k\,\Phi(\v k)\\
 & =\frac{1}{(2\pi)^{3/2}}\int_{-\infty}^{\infty}\d^{3}k\,\frac{Q}{(2\pi)^{3/2}\varepsilon_{0}(k^{2}+1/\lambda_{D}^{\,2})}e^{i\v k\cdot\v x}.
\end{align}

\end_inset

How do we represent solve this? Represent scalar product by
\begin_inset Formula 
\begin{equation}
\v k\cdot\v x=|\v k|\,|\v x|\cos\tht_{(k)}=kr\cos\tht_{kx}
\end{equation}

\end_inset

where 
\begin_inset Formula $k=|\v k|$
\end_inset

, 
\begin_inset Formula $r=|\v x|$
\end_inset

 and 
\begin_inset Formula $\tht_{kx}$
\end_inset

 is the angle between vectors 
\begin_inset Formula $\v k$
\end_inset

 and 
\begin_inset Formula $\v x$
\end_inset

.
 For the integral we first rotate the coordinate system of 
\begin_inset Formula $\v k$
\end_inset

-space such that the 
\begin_inset Formula $z$
\end_inset

-axis coincides with the vector 
\begin_inset Formula $\v x$
\end_inset

.
 The advantage of this coordinate frame is that
\begin_inset Formula 
\begin{equation}
k_{z}=\v k\cdot\frac{\v x}{|\v x|}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Now we switch from 
\begin_inset Formula $(k_{x},k_{y},k_{z})$
\end_inset

 to spherical coordinates 
\begin_inset Formula $(k,\tht_{(k)},\ph_{(k)})$
\end_inset

 where
\begin_inset Formula 
\begin{align}
k & =|\v k|,\\
k_{z} & =k\cos\tht_{(k)}.
\end{align}

\end_inset

We observe that the a
\begin_inset Formula 
\begin{equation}
\v k\cdot\v x=kr\cos\tht_{(k)}=rk_{z}.
\end{equation}

\end_inset

With this trick, we can thus represent the term appearing in the complex
 exponent as
\begin_inset Formula 
\begin{equation}
e^{i\v k\cdot\v x}=e^{irk_{z}}=e^{ikr\cos\tht_{(k)}}.
\end{equation}

\end_inset

Now back to our original problem.
 We compute
\begin_inset Formula 
\begin{align}
\Phi(\v x) & =\frac{Q}{(2\pi)^{3}\varepsilon_{0}}\int_{0}^{\infty}\d k\int_{0}^{\pi}\d\tht_{(k)}\int_{0}^{2\pi}\d\ph_{(k)}\,k^{2}\sin\tht\frac{k^{2}}{k^{2}+1/\lambda_{D}^{\,2}}e^{ikr\cos\tht_{(k)}}.
\end{align}

\end_inset

The next trick is to substitute
\begin_inset Formula 
\begin{align}
u= & \cos\tht_{(k)},\\
\d u= & -\sin\tht_{(k)}\d\tht_{(k)}.
\end{align}

\end_inset

We cancel the negative sign in 
\begin_inset Formula $\d u$
\end_inset

 by swapping the integration boundaries, which become 
\begin_inset Formula $(-1,1)$
\end_inset

 for 
\begin_inset Formula $u$
\end_inset

 where 
\begin_inset Formula $\tht_{(k)}=(\pi,0)$
\end_inset

.
 As there are no dependencies on 
\begin_inset Formula $\varphi_{(k)}$
\end_inset

 integration just adds a factor of 
\begin_inset Formula $2\pi$
\end_inset

.
\begin_inset Formula 
\begin{align}
\Phi(\v x) & =\frac{Q}{(2\pi)^{2}\varepsilon_{0}}\int_{0}^{\infty}\d k\frac{k^{2}}{k^{2}+1/\lambda_{D}^{\,2}}\int_{-1}^{1}\d ue^{ikru}\\
 & =\frac{Q}{(2\pi)^{2}\varepsilon_{0}}\int_{0}^{\infty}\d k\frac{k^{2}}{k^{2}+1/\lambda_{D}^{\,2}}\frac{1}{ikr}(e^{ikr}-e^{-ikr})
\end{align}

\end_inset

The next trick is to replace the integral over the second term 
\begin_inset Formula $e^{-ikr}$
\end_inset

 by setting 
\begin_inset Formula $k\rightarrow-k$
\end_inset

, and include it by extending the remaining integral in the range 
\begin_inset Formula $(-\infty,0)$
\end_inset

,
\begin_inset Formula 
\begin{equation}
\Phi(\v x)=\frac{Q}{ir(2\pi)^{2}\varepsilon_{0}}\int_{-\infty}^{\infty}\d k\frac{k}{k^{2}+1/\lambda_{D}^{\,2}}e^{ikr}.
\end{equation}

\end_inset

We see that the integral is now an inverse 1D Fourier transform of an expression
 of the form 
\begin_inset Formula $k/(k^{2}+a^{2})$
\end_inset

 that we can look up to be
\begin_inset Formula 
\begin{equation}
\mathcal{F}^{(-1)}\frac{a}{k^{2}+a^{2}}=\sqrt{\frac{\pi}{2}}e^{-a|x|},
\end{equation}

\end_inset

so taking one derivative
\begin_inset Formula 
\begin{equation}
\mathcal{F}^{(-1)}\frac{ik}{k^{2}+a^{2}}=\sqrt{\frac{\pi}{2}}\frac{1}{a}\frac{\d}{\d x}e^{-a|x|}=-\sqrt{\frac{\pi}{2}}e^{-a|x|}.
\end{equation}

\end_inset

Finally, with 
\begin_inset Formula $a=1/\lambda_{D}$
\end_inset

, and 
\begin_inset Formula $x=r>0$
\end_inset

 we obtain 
\begin_inset Formula 
\begin{align}
\Phi(\v x) & =\frac{Q}{ir(2\pi)^{2}\varepsilon_{0}}\frac{\sqrt{2\pi}}{i}\mathcal{F}^{(-1)}\frac{ik}{k^{2}+1/\lambda_{D}^{\,2}}\nonumber \\
 & =\frac{Q}{4\pi\varepsilon_{0}r}e^{-r/\lambda_{D}}.
\end{align}

\end_inset


\end_layout

\begin_layout Itemize
Summary: 
\end_layout

\begin_deeper
\begin_layout Itemize
We computed 
\emph on
potential 
\emph default
of 
\emph on
point charge
\emph default
 inside 
\emph on
electron fluid
\emph default
 with a certain 
\emph on
temperature 
\emph default
where electrons are allowed to 
\emph on
move
\emph default
 and thus 
\emph on
shield
\emph default
 the potential.
\end_layout

\begin_layout Itemize
First we used vortex-free 
\emph on
momentum law
\emph default
 (Euler/Navier-Stokes) to compute 
\emph on
local
\emph default
 
\emph on
Boltzmann law
\emph default
 for 
\emph on
electron density
\emph default
 according to 
\begin_inset Formula $V$
\end_inset

, 
\begin_inset Formula $\Phi$
\end_inset

 and 
\begin_inset Formula $T$
\end_inset

.
\end_layout

\begin_layout Itemize
Then we inserted convection-free (
\begin_inset Formula $V=0$
\end_inset

) case into 
\emph on
Poisson equation
\series bold
 
\series default
\emph default
and 
\emph on
linearized
\emph default
 to obtain linear 
\emph on
Helmholtz equation
\emph default
, where 
\emph on
Debye length 
\emph default

\begin_inset Formula $\lambda_{D}$
\end_inset

 appeared.
\end_layout

\begin_layout Itemize
We solved the resulting linear equation via a spatial 
\emph on
Fourier transform
\emph default
 and a few tricks.
\end_layout

\begin_layout Itemize
The result is the usual potential of a point charge weighted by an 
\emph on
exponential decaying
\emph default
 term on the scale of 
\begin_inset Formula $\lambda_{D}$
\end_inset

.
\end_layout

\begin_layout Itemize
The interpretation is, that outside of 
\begin_inset Formula $\lambda_{D}$
\end_inset

, perturbations in charge density become 
\emph on
invisible
\emph default
 
\begin_inset Formula $\rightarrow$
\end_inset

 
\emph on
Debye shielding
\emph default
.
\end_layout

\end_deeper
\begin_layout Itemize
Remember: Fourier 
\emph on
transform
\emph default
 works only in an infinite domain 
\begin_inset Formula $\mathbb{R}^{N}$
\end_inset

 where no boundary conditions are imposed, since we need to take integrals
 up to infinity.
 Fourier 
\emph on
series
\emph default
 are similar and work with periodic boundary conditions (lattice) or topology
 (cylinder, torus, sphere).
 Both methods work only well for 
\emph on
linear
\emph default
 equations 
\begin_inset Formula $\rightarrow$
\end_inset

 no coupling between different Fourier harmonics.
\end_layout

\begin_layout Itemize
Remark: Here we could see that notation introduces quite some ambiguity.
 We often identify the coordinates of a point 
\begin_inset Formula $\v x$
\end_inset

 with the coordinate system itself.
 If we have another point 
\begin_inset Formula $\v k$
\end_inset

 and switch to spherical coordinates 
\begin_inset Formula $(r,\tht,\ph)$
\end_inset

 we cannot distinguish 
\begin_inset Formula $r$
\end_inset

 from 
\begin_inset Formula $|\v x|$
\end_inset

, even if it could be the radius 
\begin_inset Formula $|\v y|$
\end_inset

 of another point 
\begin_inset Formula $\v y$
\end_inset

.
 To resolve this, we gave different names to our coordinates 
\begin_inset Formula $(k,\tht_{(k)},\ph_{(k)})$
\end_inset

 in 
\begin_inset Formula $k$
\end_inset

-space.
 We could also have introduced the notation 
\begin_inset Formula $r_{\v x}$
\end_inset

 and 
\begin_inset Formula $r_{\v k}$
\end_inset

, which is non-standard and maybe even more confusing.
\end_layout

\begin_layout Subsection*

\series bold
Problem 4: Electron plasma oscillations - the easier way
\end_layout

\begin_layout Standard
see lecture notes
\end_layout

\begin_layout Subsection*

\series bold
Problem 5: Electron plasma oscillations - the harder way
\end_layout

\begin_layout Standard
We look at oscillations/waves in an unmagnetized plasma due to electrostatic
 forces.
\end_layout

\begin_layout Standard
Approximations:
\end_layout

\begin_layout Enumerate
Ions are resting compared to fast electron dynamics.
\end_layout

\begin_layout Enumerate
Electrons modeled as fluid.
\end_layout

\begin_layout Enumerate
Electrostatic forces are stronger than thermodynamics forces (
\begin_inset Quotes eld
\end_inset

cold plasma
\begin_inset Quotes erd
\end_inset

).
\end_layout

\begin_layout Enumerate
Linear expansion around equilibrium with subsonic flow and quasineutrality.
\end_layout

\begin_layout Standard
Take mass continuity and momentum equation for electron fluid with particles
 of mass 
\begin_inset Formula $m_{e}$
\end_inset

 and charge 
\begin_inset Formula $q_{e}=-e$
\end_inset

, as well as Poisson equation for electric potential and constant and static
 ion density 
\begin_inset Formula $n_{i}$
\end_inset

,
\begin_inset Formula 
\begin{align}
m_{e}\frac{\partial n_{e}(\v x,t)}{\partial t}+\nabla\cdot\v p_{\text{fl}}(\v x,t) & =0,\label{eq:conti}\\
\frac{\partial\v p_{\text{fl}}(\v x,t)}{\partial t}+m_{e}\nabla\cdot(\v v(\v x,t)\v p_{\text{fl}}(\v x,t)) & =-\nabla p(\v x,t)+e\nabla\Phi(\v x,t),\\
\Delta\Phi(\v x,t) & =-\frac{e}{\varepsilon_{0}}(n_{i}-n_{e}(\v x,t)).\label{eq:poiss}
\end{align}

\end_inset

Here we could pull out 
\begin_inset Formula $m_{e}$
\end_inset

 as a constant, and replaced mass flux 
\begin_inset Formula $\v{\Gamma}_{m}$
\end_inset

 by the identical kinematic momentum density 
\begin_inset Formula $\v p^{fl}$
\end_inset

.
 In order to combine all three equations we now take the time derivative
 of the continuity equation, and the divergence of the momentum equation
 to obtain
\begin_inset Formula 
\begin{align}
m_{e}\frac{\partial^{2}n_{e}(\v x,t)}{\partial t^{2}}+\frac{\partial}{\partial t}\nabla\cdot\v p_{\text{fl}}(\v x,t) & =0,\label{eq:conti-1}\\
\nabla\cdot\frac{\partial\v p_{\text{fl}}(\v x,t)}{\partial t}+\nabla\cdot(\nabla\cdot(\v v(\v x,t)\v p_{\text{fl}}(\v x,t))) & =-\Delta p(\v x,t)+en_{e}(\v x,t)\Delta\Phi(\v x,t),\\
\Delta\Phi(\v x,t) & =-\frac{e}{\varepsilon_{0}}(n_{i}-n_{e}(\v x,t)).\label{eq:poiss-1}
\end{align}

\end_inset

Exchanging divergence and time derivative in front of 
\begin_inset Formula $\v p^{fl}$
\end_inset

 we can eliminate the respective terms by subtracting the momentum equation
 from the continuity equation.
 The Laplacian 
\begin_inset Formula $\Delta\Phi$
\end_inset

 is substituted from Poisson's equation to yield a single equation
\begin_inset Formula 
\begin{align}
\frac{\partial^{2}n_{e}(\v x,t)}{\partial t^{2}}+\frac{e^{2}n_{e}(\v x,t)}{m_{e}\varepsilon_{0}}(n_{e}(\v x,t)-n_{i}(\v x))-\nabla\cdot(\nabla\cdot(\v v(\v x,t)\v v(\v x,t))) & =\frac{\Delta p(\v x,t)}{m_{e}}.
\end{align}

\end_inset

Now we start with a stationary equilibrium without time derivatives, and
 flow velocity small enough (
\begin_inset Quotes eld
\end_inset

subsonic
\begin_inset Quotes erd
\end_inset

) such that the term 
\begin_inset Formula $\nabla\cdot(\nabla\cdot(\v v(\v x,t)\v v(\v x,t)))$
\end_inset

 can be neglected.
 The equilibrium should fulfill quasineutrality,
\begin_inset Formula 
\begin{equation}
n_{0}(\v x)=n_{e}^{0}(\v x)=n_{i}^{0}(\v x),
\end{equation}

\end_inset

so 
\begin_inset Formula $\Delta p^{0}=0$
\end_inset

 in the subsonic approximation.
\end_layout

\begin_layout Standard
Now we introduce a perturbation in 
\begin_inset Formula $n_{e}(\v x,t)=n_{e}^{0}(\v x)+\delta n_{e}(\v x,t)$
\end_inset

 and 
\begin_inset Formula $\v v=\v v^{0}+\delta\v v(\v x,t)$
\end_inset

, but neglect changes in 
\begin_inset Formula $p$
\end_inset

 (
\begin_inset Quotes eld
\end_inset

cold plasma
\begin_inset Quotes erd
\end_inset

).
 Keeping only first order terms (remember, 
\begin_inset Formula $n_{e}^{0}-n_{i}^{0}=0$
\end_inset

), we obtain
\begin_inset Formula 
\begin{align}
\frac{\partial^{2}\delta n_{e}(\v x,t)}{\partial t^{2}}+\frac{e^{2}n_{0}(\v x)}{m_{e}\varepsilon_{0}}\delta n_{e}(\v x,t) & =0.
\end{align}

\end_inset

Here the electron plasma frequency appears locally at position 
\begin_inset Formula $\v x$
\end_inset

 as
\begin_inset Formula 
\begin{equation}
\omega_{\text{pe}}^{\,2}(\v x)=\frac{e^{2}n_{0}(\v x)}{m_{e}\varepsilon_{0}}.
\end{equation}

\end_inset


\end_layout

\begin_layout Section
Collisions
\end_layout

\begin_layout Subsection*
2.1 Binary Coulomb Collisions
\end_layout

\begin_layout Standard
see lecture notes
\end_layout

\begin_layout Section
Charged Particle Dynamics
\end_layout

\begin_layout Subsection*
3.2 Motion in static and homogeneous fields
\end_layout

\begin_layout Standard
Our 
\series bold
goal
\series default
 is to solve
\begin_inset Formula 
\begin{align}
m\ddot{\v r}(t) & =\frac{e}{c}\dot{\v r}(t)\times\v B+e\v E.\label{eq:Lorentz}
\end{align}

\end_inset

or written in terms of a differential operator describing Lorentz force:
\begin_inset Formula 
\begin{equation}
\hat{L}\v r\equiv\left(\frac{\d^{2}}{\d t^{2}}+\left(\text{\frac{e\v B}{mc}\frac{\d}{\d t}}\right)\times\right)\v r(t)=\frac{e}{m}\v E\label{eq:LorentzOperator}
\end{equation}

\end_inset

and identify
\emph on
 gyromotion
\emph default
 circling perpendicular around 
\begin_inset Formula $\v B$
\end_inset

 with 
\begin_inset Formula $v_{\perp}$
\end_inset

,
\end_layout

\begin_layout Standard

\emph on
parallel 
\emph default
motion along 
\begin_inset Formula $\v B$
\end_inset

 with 
\begin_inset Formula $v_{\parallel}$
\end_inset

 and
\end_layout

\begin_layout Standard
average 
\emph on
drift
\emph default
 motion perpendicular to 
\begin_inset Formula $\v B$
\end_inset

 with 
\begin_inset Formula $\v v_{D}$
\end_inset

.
\begin_inset Newline newline
\end_inset


\end_layout

\begin_layout Standard
Our 
\series bold
strategy
\series default
 is to separate out gyromotion in 
\begin_inset Formula $\v B$
\end_inset

-field alone and solve its equations first.
 Then we could solve parallel and drift motion with the constraint that
 
\begin_inset Formula $\v v_{D}$
\end_inset

 remains constant.
 In the combined solution we verified there are 6 free parameters that can
 reproduce any initial conditions.
\begin_inset Newline newline
\end_inset


\begin_inset space ~
\end_inset


\end_layout

\begin_layout Standard
The
\series bold
 assumption
\emph on
 
\series default
\emph default
is that we considere only static and homogeneous fields.
 Under this simplification the solution describes particle motion exactly,
 as 
\begin_inset Formula $\hat{L}$
\end_inset

 in Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:LorentzOperator"

\end_inset

 is 
\emph on
linear
\emph default
 in 
\series bold

\begin_inset Formula $\v r(t)$
\end_inset


\series default
, so the overall solution can be described as a linear superposition of
 two solutions.
\end_layout

\begin_layout Standard
Splitting Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Lorentz"

\end_inset

 into a sum of two solutions 
\begin_inset Formula $\v R+\v{\rho}$
\end_inset

 with 
\begin_inset Formula $\v r=\v R+\v{\rho}$
\end_inset

 yields 
\begin_inset Formula 
\begin{align}
m(\ddot{\v R}+\ddot{\v{\rho}}) & =e\v E+\frac{e}{c}(\dot{\v R}+\dot{\v{\rho}})\times\v B.\label{eq:split}
\end{align}

\end_inset

Since we introduced 3 more variables, we can also introduce 3 more equations,
 as long as they are not in conflict with Eq.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:split"

\end_inset

.
 We choose
\begin_inset Formula 
\begin{align}
m\ddot{\v{\rho}} & =\frac{e}{c}\dot{\v{\rho}}\times\v B,\\
m\ddot{\v R} & =e\v E+\frac{e}{c}\dot{\v R}\times\v B.
\end{align}

\end_inset

This way, the gyromotion in 
\begin_inset Formula $\v{\rho}$
\end_inset

 is only affected by the magnetic, but not the electric field.
\end_layout

\begin_layout Subsubsection*
Solution for gyration
\end_layout

\begin_layout Standard
First we solve 
\begin_inset Formula $m\ddot{\v{\rho}}=\frac{e}{c}\dot{\v{\rho}}\times\v B$
\end_inset

.
 We choose our coordinate system such that field 
\begin_inset Formula $\v B=B\v b=B\v e_{1}\times\v e_{2}$
\end_inset

.
 (or: 
\begin_inset Formula $\v e_{3}=\v b$
\end_inset

) Then
\begin_inset Formula 
\begin{align}
m\ddot{\v{\rho}} & =\frac{e}{c}\dot{\v{\rho}}\times(B\v b)=\frac{eB}{c}(\dot{\rho}_{1}\v e_{1}+\dot{\rho}_{2}\v e_{2})\times\v e_{3}\nonumber \\
 & =\frac{eB}{c}(-\dot{\rho}_{1}\v e_{2}+\dot{\rho}_{2}\v e_{1}).
\end{align}

\end_inset

We combine this to
\begin_inset Formula 
\begin{align}
\ddot{\rho}_{1} & =\frac{eB}{mc}\dot{\rho}_{2}=\omega_{c}\dot{\rho}_{2},\\
\ddot{\rho}_{2} & =-\frac{eB}{mc}\dot{\rho}_{1}=-\omega_{c}\dot{\rho}_{1}.
\end{align}

\end_inset

Taking another time derivative and denoting 
\begin_inset Formula $v_{1}\equiv\dot{\rho}_{1}$
\end_inset

, 
\begin_inset Formula $v_{2}\equiv\dot{\rho}_{2}$
\end_inset

 we obtain
\begin_inset Formula 
\begin{equation}
\ddot{v}_{1}=\omega_{c}\dot{v}_{2}=-\omega_{c}^{\,2}v_{1}.
\end{equation}

\end_inset

In this second order equation we have six free constants.
 We choose one to be the 
\series bold
perpendicular velocity
\series default
 
\begin_inset Formula $v_{\perp}$
\end_inset

, which is the amplitude of the gyration velocity and another one as a phase-shi
ft 
\begin_inset Formula $\phi_{0}$
\end_inset

.
 Then we obtain the solution of position, velocity, and acceleration
\begin_inset Formula 
\begin{align}
\v{\rho} & =\frac{v_{\perp}}{\omega_{c}}(\sin(\omega_{c}t+\phi_{0})\,\v e_{1}+\cos(\omega_{c}t+\phi_{0})\,\v e_{2}),\\
\dot{\v{\rho}} & =v_{\perp}(\cos(\omega_{c}t+\phi_{0})\,\v e_{1}-\sin(\omega_{c}t+\phi_{0})\,\v e_{2}),\\
\ddot{\v{\rho}} & =-v_{\perp}\omega_{c}(\sin(\omega_{c}t+\phi_{0})\,\v e_{1}+\cos(\omega_{c}t+\phi_{0})\,\v e_{2}).
\end{align}

\end_inset

From here we define the 
\series bold
gyroradius
\series default
 
\begin_inset Formula $\rho_{c}=v_{\perp}/\omega_{c}$
\end_inset

 and the 
\series bold
gyrophase
\series default
 
\begin_inset Formula $\phi=\omega_{c}t+\phi_{0}$
\end_inset

.
\begin_inset Newline newline
\end_inset


\begin_inset Newline newline
\end_inset

Additional four integration constants are a translational shift by 
\begin_inset Formula $\v{\rho}_{0}$
\end_inset

 plus a constant parallel velocity 
\begin_inset Formula $\dot{\v{\rho}}_{0}\cdot\v b$
\end_inset

 which we both set to zero! This means that we didn't include any motion
 parallel to the magnetic field, or drift across the field intro 
\begin_inset Formula $\v{\rho}$
\end_inset

 and need to account for them in 
\begin_inset Formula $\v R$
\end_inset

.
\begin_inset Newline newline
\end_inset


\begin_inset Newline newline
\end_inset

(Have 
\begin_inset Formula $6\times2$
\end_inset

 integration constants in 
\begin_inset Formula $\v{\rho},\v R$
\end_inset

 but need only 
\begin_inset Formula $6$
\end_inset

 of them for overall problem in 
\begin_inset Formula $\v r$
\end_inset

).
\end_layout

\begin_layout Subsubsection*
Solution for parallel and drift motion
\end_layout

\begin_layout Standard
Now we split
\begin_inset Formula 
\begin{equation}
\dot{\v R}=v_{\parallel}\v b+\v v_{D},\qquad\v v_{D}\cdot\v b=0.
\end{equation}

\end_inset

We need to solve
\begin_inset Formula 
\begin{equation}
m\ddot{\v R}=\dot{v}_{\parallel}\v b+\dot{\v v}_{D}=e\v E+\frac{e}{c}\dot{\v R}\times\v B.
\end{equation}

\end_inset

For parallel motion, obtain immediately
\begin_inset Formula 
\begin{equation}
m\dot{v}_{\parallel}=eE_{\parallel}\Rightarrow v_{\parallel}(t)=v_{\parallel0}+eE_{\parallel}t.
\end{equation}

\end_inset

For perpendicular drift use
\begin_inset Formula 
\begin{equation}
m\dot{\v v}_{D}=e\v E_{\perp}+\frac{e}{c}\v v_{D}\times\v B
\end{equation}

\end_inset

take 
\begin_inset Formula $\times\v B$
\end_inset

 and use bac-cab rule (NRL)
\begin_inset Formula 
\begin{equation}
m\dot{\v v}_{D}\times\v B=e\v E\times\v B-\frac{e}{c}B^{2}v_{D}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Consistent solution is possible if 
\begin_inset Formula $\dot{\v v}_{D}\times\v B$
\end_inset

 remains zero.
 This way we do not include any gyration in the perpendicular motion.
 Solution for this drift motion (
\series bold
ExB drift
\series default
):
\begin_inset Formula 
\begin{equation}
\v v_{D}=c\frac{\v E\times\v B}{B^{2}}.
\end{equation}

\end_inset


\emph on
...
 independent of charge and mass! 
\emph default
This also means setting initial condition 
\begin_inset Formula $v_{D0}=c\frac{\v E\times\v B}{B^{2}}$
\end_inset

 which fixes another integration constant.
 
\end_layout

\begin_layout Standard
The solution for parallel and drift motion together is thus
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\v R(t)=\v R_{0}+c\frac{\v E\times\v B}{B^{2}}t+\left(v_{\parallel0}t+\frac{eE_{\parallel}}{2}t^{2}\right)\v b.
\end{equation}

\end_inset


\end_layout

\begin_layout Subsubsection*
Combined solution
\end_layout

\begin_layout Standard
The combined exact solution for particle motion in static, homogeneous 
\begin_inset Formula $\v E$
\end_inset

 and 
\begin_inset Formula $\v B$
\end_inset

 is
\begin_inset Formula 
\begin{align}
\v r(t) & =\v R(t)+\v{\rho}(t)\nonumber \\
 & =\v R_{0}+c\frac{\v E\times\v B}{B^{2}}t+\left(v_{\parallel0}t+\frac{eE_{\parallel}}{2}t^{2}\right)\v b+\frac{v_{\perp}}{\omega_{c}}\left(\sin(\omega_{c}t+\phi_{0})\,\v e_{1}+\cos(\omega_{c}t+\phi_{0})\,\v e_{2}\right).
\end{align}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset space ~
\end_inset


\end_layout

\begin_layout Standard
Let's count integration constants: 
\end_layout

\begin_layout Itemize
1,2,3: 
\begin_inset Formula $\v R_{0}$
\end_inset

 contains 3 constants of initial position of Larmor circle center (
\series bold
gyrocenter
\series default
).
\end_layout

\begin_layout Itemize
4: Constant 
\begin_inset Formula $v_{\parallel0}$
\end_inset

 sets initial gyrocenter parallel velocity.
\end_layout

\begin_layout Itemize
5,6: Constants 
\begin_inset Formula $v_{\perp}$
\end_inset

 and 
\begin_inset Formula $\phi_{0}$
\end_inset

 fix the gyromotion (
\begin_inset Formula $v_{\perp}$
\end_inset

 can optionally be replaced by 
\begin_inset Formula $\rho_{c}$
\end_inset

).
\begin_inset Newline newline
\end_inset


\end_layout

\begin_layout Standard
Thus we could find a consistent solution that separates 
\emph on
gyromotion
\emph default
 from 
\emph on
parallel
\emph default
 and 
\emph on
drift
\emph default
 motion in 
\emph on
static
\emph default
 and 
\emph on
homogeneous
\emph default
 fields.
 In more complicated fields we will rely on methods of averaging that work
 as long as gyromotion i.e.
 
\begin_inset Formula $\omega_{c}$
\end_inset

 is fast, and 
\begin_inset Formula $\rho_{c}$
\end_inset

 is small, compared to field scales.
\end_layout

\begin_layout Subsection*
3.4 Motion in weakly inhomogeneous fields
\begin_inset CommandInset label
LatexCommand label
name "subsec:Inhomo"

\end_inset


\end_layout

\begin_layout Standard
Now the 
\series bold
goal
\series default
 is to solve
\begin_inset Formula 
\begin{align}
m\ddot{\v r} & =\frac{e}{c}\dot{\v r}\times\v B(\v r)+e\v E(\v r).
\end{align}

\end_inset

or written in terms of a 
\emph on
non-linear
\emph default
 Lorentz operator:
\begin_inset Formula 
\begin{equation}
\hat{L}\v r\equiv\left(\frac{\d^{2}}{\d t^{2}}+\frac{e\v B(\v r)}{mc}\frac{\d}{\d t}\times\right)\v r=\frac{e}{m}\v E(\v r).
\end{equation}

\end_inset


\end_layout

\begin_layout Standard

\series bold
Strategy:
\series default
 Instead of an exact splitting as before, separate into 
\emph on
fast
\emph default
 gyromotion for 
\begin_inset Formula $\v{\rho}$
\end_inset

 and 
\emph on
gyro-averaged
\emph default
 motion in 
\begin_inset Formula $\v R$
\end_inset

.
 The latter means to take time averages over
\emph on
 gyroperiod
\emph default
 
\begin_inset Formula $\tau_{c}=2\pi/\omega_{c}$
\end_inset

 and spatial averages within 
\emph on
gyroradius 
\begin_inset Formula $\rho_{c}=v_{\perp}/\omega_{c}$
\end_inset

.
 
\emph default
Perform a series expansion of fields around average 
\begin_inset Formula $\v R$
\end_inset

 in direction of 
\begin_inset Formula $\v{\rho}$
\end_inset

.
\begin_inset Newline newline
\end_inset


\begin_inset space ~
\end_inset


\end_layout

\begin_layout Standard
We
\series bold
 assume
\emph on
 
\series default
\emph default
that relative variations of 
\begin_inset Formula $\v E$
\end_inset

 and 
\begin_inset Formula $\v B$
\end_inset

 in time and space are small over 
\begin_inset Formula $\tau_{c}$
\end_inset

 and within 
\begin_inset Formula $\rho_{c}$
\end_inset

.
 This means that the plasma must be 
\emph on
strongly magnetized 
\emph default
to get reasonably large 
\begin_inset Formula $\omega_{c}=eB/(mc)$
\end_inset

.
 Due to the expansion the solution is only 
\series bold
approximate
\series default
\emph on
.
\end_layout

\begin_layout Subsubsection*
Calculation
\end_layout

\begin_layout Standard
A detailed calculation can be found in the lecture notes, chapter 3.4.
 For details how to do an expansion of tensor fields, see p.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand pageref
reference "subsec:Taylor-expansion-of"

\end_inset

.
\end_layout

\begin_layout Subsubsection*
Result
\end_layout

\begin_layout Standard
The orbital motion is split into the sum of the gyrocenter position 
\begin_inset Formula $\v R$
\end_inset

 and the Larmor gyration vector 
\begin_inset Formula $\v{\rho}$
\end_inset

, where
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\v{\rho} & =\frac{v_{\perp}}{\omega_{c}(\v R)}(\sin(\phi(\v R,t))\,\v e_{1}(\v R)+\cos(\phi(\v R,t))\,\v e_{2}(\v R)),\\
m\ddot{\v R} & =e\left(1+\frac{\rho_{c}^{2}}{4}\nabla_{\perp}^{\,2}\right)\v E(\v R)+\frac{e}{c}\dot{\v R}\times\v B(\v R)-\mu\nabla B(\v R).\label{eq:R}
\end{align}

\end_inset

Fields are evaluated in the gyrocenter position 
\begin_inset Formula $\v R$
\end_inset

 that can be computed ignoring 
\begin_inset Formula $\v{\rho}$
\end_inset

.
 The magnetic moment 
\begin_inset Formula $\mu$
\end_inset

 is a conserved quantity and related to 
\begin_inset Formula $v_{\perp}$
\end_inset

 via
\begin_inset Formula 
\begin{equation}
\mu=\frac{mv_{\perp}^{\,2}}{2B(\v R)}=\frac{ev_{\perp}}{2\pi\rho_{c}}\pi\rho_{c}^{\,2}.\label{eq:magmoment}
\end{equation}

\end_inset

This is the ring current over one gyration period and 
\begin_inset Formula $\mu=I\cdot A$
\end_inset

 over the gyration circle with surface area 
\begin_inset Formula $A=\pi\rho_{c}^{2}$
\end_inset

.
 If 
\begin_inset Formula $E_{\parallel}=\v b\cdot\v E$
\end_inset

 then 
\begin_inset Formula $\mu$
\end_inset

 is approximately conserved – an 
\series bold
adiabatic invariant
\series default
.
 
\begin_inset Formula 
\begin{align}
\frac{\d\mu}{\d t} & =\frac{1}{B}\frac{\d}{\d t}\left(\frac{mv_{\perp}^{\,2}}{2}\right)-\frac{mv_{\perp}^{\,2}}{2B^{2}}\dot{\v R}\cdot\nabla B\nonumber \\
 & \approx\frac{1}{B}\frac{\d}{\d t}\left(\frac{mv_{\perp}^{\,2}}{2}\right)-\frac{\mu}{B}v_{\parallel}\v b\cdot\nabla B.
\end{align}

\end_inset

Compare this with total energy conservation
\begin_inset Formula 
\begin{align}
0 & =\frac{\d}{\d t}\left(\frac{mv_{\parallel}^{\,2}}{2}+\frac{mv_{\perp}^{\,2}}{2}+e\Phi\right)\nonumber \\
 & \approx\frac{\d}{\d t}\left(\frac{mv_{\perp}^{\,2}}{2}\right)-v_{\parallel}\mu\v b\cdot\nabla B.
\end{align}

\end_inset

Here 
\begin_inset Formula $v_{\parallel}\v b\cdot\nabla\Phi$
\end_inset

 has vanished due to 
\begin_inset Formula $E_{\parallel}=0$
\end_inset

 and 
\begin_inset Formula $\dot{v}_{\parallel}$
\end_inset

 has been replaced by its expression from the equations of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:R"

\end_inset

 with 
\begin_inset Formula $\dot{\v R}\approx v_{\parallel}\v b$
\end_inset

.
 Together we see that the total time derivative of 
\begin_inset Formula $\mu$
\end_inset

 vanishes in this first order calculation.
 A subtle point that should concern us is the actual nature of 
\begin_inset Formula $v_{\perp}$
\end_inset

 and 
\begin_inset Formula $\phi$
\end_inset

.
 If we assume 
\begin_inset Formula $\mu$
\end_inset

 to be exactly conserved, Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:magmoment"

\end_inset

 becomes a definition for 
\begin_inset Formula $v_{\perp}=v_{\perp}(\v R)$
\end_inset

 rather than a quantity resulting from it.
 While the motion of the gyrocenter 
\begin_inset Formula $\v R$
\end_inset

 can be computed from Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:R"

\end_inset

 without referring to the gyromotion, the definition what 
\begin_inset Formula $\v R$
\end_inset

 actually describes is still somewhat dubious.
 For this reason we will perform a more 
\begin_inset Quotes eld
\end_inset

watertight
\begin_inset Quotes erd
\end_inset

 treatment of the problem using Lagrangian mechanics.
\end_layout

\begin_layout Section
Guiding-Center Dynamics
\end_layout

\begin_layout Standard
The modern way to treat guiding-center motion via a phase-space Lagrangian
 (Goldstein 8-5 
\begin_inset Quotes eld
\end_inset

Derivation of Hamilton's equations from a variational principle
\begin_inset Quotes erd
\end_inset

) has been introduced by Littlejohn
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citep
key "littlejohn1983"
literal "false"

\end_inset

 in the 1980s.
 Besides that there exists a review article on guiding-center motion by
 Cary and Brizard 
\begin_inset CommandInset citation
LatexCommand citep
key "cary2009"
literal "false"

\end_inset

 containing background, main derivation steps and further literature references.
 A detailed derivation of the guiding-center Lagrangian can also be found
 in Lectures 3-4 of 2017 and https://itp.tugraz.at/~ert/assets/internal/plasma/201
7/A1_Guiding_Center_Lagrangian.pdf .
 More information on asymptotic expansions with a small parameter 
\begin_inset Formula $\varepsilon$
\end_inset

 can be found in the lecture of Stefan Possanner at TUM.
 Here we treat the case where fields are not time-dependent.
\end_layout

\begin_layout Standard
We transform from the 
\series bold
particle
\series default
 coordinates 
\begin_inset Formula $\v z_{P}=(\v r,\v v)$
\end_inset

 in phase-space to new 
\series bold
guiding-center
\series default
 coordinates 
\begin_inset Formula $\v z=(\v R,\phi,v_{\parallel},v_{\perp})$
\end_inset

 related to particle position 
\begin_inset Formula $\v r$
\end_inset

 and velocity 
\begin_inset Formula $\v v$
\end_inset

 via
\begin_inset Formula 
\begin{align}
\v r(\v z) & =\v R+\v{\rho}(\v R,\phi,v_{\perp}),\text{ where }\v{\rho}(\v R,\phi,v_{\perp})\equiv\frac{mcv_{\perp}}{eB(\v R)}\hat{\v{\rho}}(\v R,\phi)\equiv\rho_{c}(v_{\perp},\v R)\hat{\v{\rho}}(\v R,\phi),\label{eq:rz}\\
\v v(\v z) & =v_{\parallel}\hat{\v b}(\v R)+v_{\perp}\hat{\v n}(\v R,\phi).\label{eq:vz}
\end{align}

\end_inset

Here 
\begin_inset Formula $\hat{\v b}(\v R)=\v B(\v R)/B(\v R)$
\end_inset

 and forms a tripod with some pair of 
\series bold
fixed 
\series default
but possibly space-dependent unit vectors 
\begin_inset Formula $\hat{\v e}_{1}(\v R)$
\end_inset

 and 
\begin_inset Formula $\hat{\v e}_{2}(\v R)$
\end_inset

, and definitions for 
\series bold
dynamic
\series default
 (gyrating) unit vectors 
\begin_inset Formula $\hat{\v{\rho}}$
\end_inset

 and 
\begin_inset Formula $\hat{\v n}$
\end_inset

 are
\begin_inset Formula 
\begin{align}
\hat{\v{\rho}}(\v R,\phi) & =\cos(\phi)\,\hat{\v e}_{1}(\v R)-\sin(\phi)\,\hat{\v e}_{2}(\v R)=-\frac{\partial\hat{\v n}(\v R,\phi)}{\partial\phi},\\
\hat{\v n}(\v R,\phi) & =\hat{\v{\rho}}(\v R,\phi)\times\hat{\v b}(\v R)=-\sin(\phi)\,\hat{\v e}_{1}(\v R)-\cos(\phi)\,\hat{\v e}_{2}(\v R)=\frac{\partial\hat{\v{\rho}}(\v R,\phi)}{\partial\phi}.
\end{align}

\end_inset

The transformation of Eqs.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:rz"
plural "false"
caps "false"
noprefix "false"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:vz"

\end_inset

) is still exact! We call 
\begin_inset Formula $\v R$
\end_inset

 the guiding-center position.
 Both, 
\begin_inset Formula $(\hat{\v e}_{1},\hat{\v e}_{2},\hat{\v b})$
\end_inset

 and 
\begin_inset Formula $(\hat{\v n},\hat{\v{\rho}},\hat{\v b})$
\end_inset

 form an orthonormal basis, where 
\begin_inset Formula $\hat{\v e}_{1},\hat{\v e}_{2}$
\end_inset

 depend only on 
\begin_inset Formula $\boldsymbol{R}$
\end_inset

, but 
\begin_inset Formula $\hat{\v n},\hat{\v{\rho}}$
\end_inset

 also on 
\begin_inset Formula $\phi$
\end_inset

.
 Variables have suspicious names 
\begin_inset Formula $\phi$
\end_inset

, 
\begin_inset Formula $v_{\parallel}$
\end_inset

 and 
\begin_inset Formula $v_{\perp}$
\end_inset

.
 Indeed, for the case of a homogeneous 
\begin_inset Formula $\v B$
\end_inset

-field those variables are equal to gyrophase, parallel and perpendicular
 velocity of the particle.
 If the field becomes weakly inhomogeneous, they still approximate the particle
 values averaged over one gyro-period.
 Similarly, 
\begin_inset Formula $\v R$
\end_inset

 describes the center of the Larmor circle only in the homogeneous limiting
 case, and remains close to it in a field with sufficiently weak gradients.
 To sum up one could say that 
\begin_inset Formula $(\phi,v_{\parallel},v_{\perp})$
\end_inset

 are values we would get if we pretend that 
\begin_inset Formula $\v B$
\end_inset

 is evaluated in the guiding-center position 
\begin_inset Formula $\v R$
\end_inset

 rather than the particle position 
\begin_inset Formula $\v R+\v{\rho}$
\end_inset

.
 In contrast to the treatment in section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Inhomo"

\end_inset

, all variables in 
\begin_inset Formula $\v z=(\v R,\phi,v_{\parallel},v_{\perp})$
\end_inset

 have a well-defined relation to the particle position 
\begin_inset Formula $\v r$
\end_inset

 and velocity 
\begin_inset Formula $\v v$
\end_inset

 if Eqs.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:rz"
plural "false"
caps "false"
noprefix "false"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:vz"

\end_inset

) are solved implicitly in 
\begin_inset Formula $\v z$
\end_inset

.
 
\end_layout

\begin_layout Standard
The phase-space Lagrangian of a particle in an electromagnetic field in
 terms of canonical 
\begin_inset Formula $(\v r,\v p$
\end_inset

) is given by
\begin_inset Formula 
\begin{equation}
L(\v r,\dot{\v r},\v p,\cancel{\dot{\v p}})=\v p\cdot\dot{\v r}-\left(\frac{(\v p-\frac{e}{c}\v A(\v r))^{2}}{2m}+e\Phi(\v r)\right).\label{eq:Lag1-1}
\end{equation}

\end_inset

Using non-canonical variables 
\begin_inset Formula $\v v=\frac{\v p}{m}-\frac{e}{mc}\v A$
\end_inset

 instead of the momenta 
\begin_inset Formula $\v p$
\end_inset

 the particle phase-space Lagrangian is
\begin_inset Formula 
\begin{equation}
L(\v r,\dot{\v r},\v v,\cancel{\dot{\v v}})=\left(m\v v+\frac{e}{c}\v A(\v r)\right)\cdot\dot{\v r}-\left(\frac{m\v v^{2}}{2}+e\Phi(\v r)\right).\label{eq:Lag1-1-1}
\end{equation}

\end_inset

In terms of 
\begin_inset Formula $\v z=(\v R,\phi,v_{\parallel},v_{\perp})$
\end_inset

 we have
\begin_inset Formula 
\begin{equation}
L(\v R,\dot{\v R},\phi,\dot{\phi},v_{\parallel},v_{\perp})=\left(mv_{\parallel}\hat{\v b}(\v R)+mv_{\perp}\hat{\v n}(\v R,\phi)+\frac{e}{c}\v A(\v R+\v{\rho})\right)\cdot(\dot{\v R}+\dot{\v{\rho}})-\left(\frac{m(v_{\parallel}^{\,2}+v_{\perp}^{\,2})}{2}+e\Phi(\v R+\v{\rho})\right),\label{eq:Lag1-1-1-1}
\end{equation}

\end_inset

with gyration vector 
\begin_inset Formula $\v{\rho}=\v{\rho}(\v R,\phi,v_{\perp})=\v r-\v R$
\end_inset

 from Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:rz"

\end_inset

.
 This Lagrangian is still exact, as are transformations to guiding-center
 variables.
 We will now make approximations to the Lagrangian, affecting the 
\emph on
dynamics
\emph default
 of motion in guiding-center variables.
 As in the previous chapter we use a Taylor expansion which we break at
 the first order.
 For higher order computations, Lie transform methods are better suited
 (see Littlejohn, 1983).
 Here the small parameter compared to typical length scales is the gyroradius
\begin_inset Formula 
\begin{equation}
\rho_{c}=|\v{\rho}|=\frac{mcv_{\perp}}{eB(\v R)}\sim\varepsilon.
\end{equation}

\end_inset

Furthermore we define the cyclotron frequency at the guiding-center position
\begin_inset Formula 
\begin{equation}
\omega_{c}=\frac{eB(\v R)}{mc}\sim\varepsilon^{-1},
\end{equation}

\end_inset

as a large parameter compared to typical time scales.
 This corresponds to the physical conditions where the guiding-center approximat
ion is applicable, leaves the perpendicular velocity 
\begin_inset Formula $v_{\perp}$
\end_inset

 being a quantity of interest with order 
\begin_inset Formula $\varepsilon^{0}=1$
\end_inset

 and thus prevents it from being swallowed by a perturbation expansion.
 The parameter 
\begin_inset Formula $\varepsilon$
\end_inset

 is called an ordering parameter that allows us to collect terms of similar
 magnitude.
 Vector and scalar fields are expanded in the following way,
\begin_inset Formula 
\begin{align}
\v A(\v r) & =\v A(\v R)+\varepsilon\v{\rho}\cdot\nabla\v A(\v R)+\mathcal{O}(\varepsilon^{2}),\\
\Phi(\v r) & =\Phi(\v R)+\varepsilon\v{\rho}\cdot\nabla\Phi(\v R)+\mathcal{O}(\varepsilon^{2}),
\end{align}

\end_inset

where 
\begin_inset Formula $\varepsilon$
\end_inset

 acts as a marker for intermediate calculations and is set to one only in
 the final result after neglecting all terms of order 
\begin_inset Formula $\varepsilon$
\end_inset

 or higher.
 We mark each small quantity with an 
\begin_inset Formula $\varepsilon$
\end_inset

 in Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Lag1-1-1-1"

\end_inset

 to obtain
\begin_inset Formula 
\begin{equation}
L(\v R,\dot{\v R},\phi,\dot{\phi},v_{\parallel},v_{\perp})=\left(mv_{\parallel}\hat{\v b}(\v R)+mv_{\perp}\hat{\v n}(\v R,\phi)+\varepsilon^{-1}\frac{e}{c}\v A(\v R+\varepsilon\v{\rho})\right)\cdot(\dot{\v R}+\dot{\v{\rho}})-\left(\frac{m(v_{\parallel}^{\,2}+v_{\perp}^{\,2})}{2}+e\Phi(\v R+\varepsilon\v{\rho})\right).\label{eq:Lag1-1-1-1-1}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
We cannot simply write 
\begin_inset Formula $\varepsilon$
\end_inset

 in front of 
\begin_inset Formula $\dot{\boldsymbol{\rho}}$
\end_inset

 just because the magnitude of 
\begin_inset Formula $\v{\rho}$
\end_inset

 is small, since gyration is fast.
 This becomes apparent by using the chain rule,
\begin_inset Formula 
\begin{align}
\dot{\v{\rho}} & =\frac{\d}{\d t}\v{\rho}(\v R,\phi,v_{\perp})=\dot{\v R}\cdot\nabla\v{\rho}+\dot{\phi}\frac{\partial\v{\rho}}{\partial\phi}+\dot{v}_{\perp}\frac{\partial\v{\rho}}{\partial v_{\perp}}\nonumber \\
 & =\varepsilon\dot{\v R}\cdot\nabla\v{\rho}+\varepsilon^{-1}\dot{\phi}\varepsilon\rho_{c}\hat{\v n}+\varepsilon\frac{\dot{v}_{\perp}}{v_{\perp}}\v{\rho}=\dot{\phi}\rho_{c}\hat{\v n}+\varepsilon(\dot{\v R}\cdot\nabla\v{\rho}+\frac{\dot{v}_{\perp}}{v_{\perp}}\v{\rho}).
\end{align}

\end_inset

Here in the first term the fast gyration 
\begin_inset Formula $\dot{\phi}$
\end_inset

 and the small 
\begin_inset Formula $\rho_{c}$
\end_inset

 cancel to order 
\begin_inset Formula $\varepsilon^{0}$
\end_inset

, and the last term should be even smaller than order 
\begin_inset Formula $\varepsilon$
\end_inset

 as long as 
\begin_inset Formula $v_{\perp}$
\end_inset

 doesn't change too fast due to strongly inhomogeneous fields.
 Now we look at the first term in
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Lag1-1-1-1-1"

\end_inset

 up to order 
\begin_inset Formula $\varepsilon^{0}$
\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\left(mv_{\parallel}\hat{\v b}(\v R)+mv_{\perp}\hat{\v n}(\v R,\phi)+\varepsilon^{-1}\frac{e}{c}\v A(\v R+\v{\rho})\right)\cdot(\dot{\v R}+\dot{\v{\rho}}) & =\left(mv_{\parallel}\hat{\v b}+mv_{\perp}\hat{\v n}+\varepsilon^{-1}\frac{e}{c}\v A+\frac{e}{c}\v{\rho}\cdot\nabla\v A\right)\cdot\dot{\v R}\nonumber \\
 & +mv_{\perp}\dot{\phi}\rho_{c}+\varepsilon^{-1}\frac{e}{c}\v A\cdot\dot{\v{\rho}}+\frac{e}{c}(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v{\rho}}+\mathcal{O}(\varepsilon).
\end{align}

\end_inset

Here the term with 
\begin_inset Formula $\hat{\v b}\cdot\hat{\v n}$
\end_inset

 vanished and we took a scalar product 
\begin_inset Formula $\hat{\v n}\cdot\hat{\v n}=1$
\end_inset

 to obtain the first term in the second line.
 Now we use the fact that we can add/subtract an arbitrary total time derivative
 to the Lagrangian without changing its dynamics.
 We will use this trick two times in a row.
\end_layout

\begin_layout Standard
First we take
\begin_inset Formula 
\begin{align}
\frac{\d}{\d t}(\v{\rho}\cdot\v A(\v R)) & =\dot{\v{\rho}}\cdot\v A+\v{\rho}\cdot\dot{\v A}\nonumber \\
 & =\dot{\v{\rho}}\cdot\v A+\v{\rho}\cdot(\dot{\v R}\cdot\nabla\v A).\label{eq:dt}
\end{align}

\end_inset

This we can combine with the term 
\begin_inset Formula $(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v R}=\v{\rho}\cdot((\nabla\v A)\cdot\dot{\v R})$
\end_inset

 via tensor algebra rule 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:eq3"

\end_inset

,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
(\nabla\v A)\cdot\dot{\v R}-\dot{\v R}\cdot\nabla\v A=\dot{\v R}\times(\nabla\times\v A)=\dot{\v R}\times\boldsymbol{B}.\label{eq:eq3-1}
\end{equation}

\end_inset

So 
\begin_inset Formula 
\begin{align}
(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v R}-\v{\rho}\cdot(\dot{\v R}\cdot\nabla\v A) & =\v{\rho}\cdot(\dot{\v R}\times\v B).
\end{align}

\end_inset

and together with
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:dt"

\end_inset

 we obtain
\begin_inset Formula 
\begin{equation}
(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v R}=\v{\rho}\cdot(\dot{\v R}\times\v B+\dot{\v R}\cdot\nabla\v A)=\v{\rho}\cdot(\dot{\v R}\times\v B)+\frac{\d}{\d t}(\v{\rho}\cdot\v A)-\dot{\v{\rho}}\cdot\v A.
\end{equation}

\end_inset

Inserting everything into 
\begin_inset Formula $L$
\end_inset

 cancels the term 
\begin_inset Formula $\v A\cdot\dot{\v{\rho}}$
\end_inset

 and we obtain
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
L-\frac{e}{c}\frac{\d}{\d t}(\v{\rho}\cdot\v A)+\mathcal{O}(\varepsilon) & =\left(mv_{\parallel}\hat{\v b}+mv_{\perp}\hat{\v n}+\varepsilon^{-1}\frac{e}{c}\v A\right)\cdot\dot{\v R}\nonumber \\
 & +\frac{e}{c}\v{\rho}\cdot(\dot{\v R}\times\v B)+mv_{\perp}\dot{\phi}\rho_{c}+\frac{e}{c}(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v{\rho}}\nonumber \\
 & -\left(\frac{m(v_{\parallel}^{\,2}+v_{\perp}^{\,2})}{2}+e\Phi\right),
\end{align}

\end_inset

where 
\begin_inset Formula $\v A,\v B,$
\end_inset

 and 
\begin_inset Formula $\Phi$
\end_inset

 are now evaluated in the guiding-center position 
\begin_inset Formula $\v R$
\end_inset

 instead of the particle position 
\begin_inset Formula $\v r$
\end_inset

.
 Two terms cancel since by rotating the triple-product and extracting basis
 vectors,
\begin_inset Formula 
\begin{equation}
\frac{e}{c}\v{\rho}\cdot(\dot{\v R}\times\v B)=\frac{e}{c}\rho_{c}B\dot{\boldsymbol{R}}\cdot(\hat{\v b}\times\hat{\v{\rho}})=-mv_{\perp}\dot{\boldsymbol{R}}\cdot\hat{\v n}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
To eliminate the term 
\begin_inset Formula $(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v{\rho}}$
\end_inset

 we use another total time derivative (see also review paper of Cary and
 Brizard, 2009),
\begin_inset Formula 
\begin{align}
\frac{\d}{\d t}\left((\v{\rho}\cdot\nabla\v A)\cdot\v{\rho}\right) & =\left((\dot{\v{\rho}}\cdot\nabla\v A)\cdot\v{\rho}+(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v{\rho}}\right)+\varepsilon\v{\rho}\cdot\left(\frac{\d\nabla\v A}{\d t}\right)\cdot\v{\rho}.
\end{align}

\end_inset

Despite being a total time derivative one could also argue that this term
 is of order 
\begin_inset Formula $\varepsilon$
\end_inset

 for the leading order approximation.
 Since it contains terms of higher order due to the time derivative and
 fast gyration one should be careful with this.
 Anyway, we can use the expression for the term
\begin_inset Formula 
\begin{align}
(\v{\rho}\times\dot{\v{\rho}})\cdot\v B & =(\v{\rho}\times\dot{\v{\rho}})\cdot(\nabla\times\v A)\nonumber \\
 & =(\varepsilon_{ijk}\rho_{j}\dot{\rho}_{k})(\varepsilon_{imn}\partial_{m}A_{n})\nonumber \\
 & =(\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km})(\rho_{j}\dot{\rho}_{k}\partial_{m}A_{n})\nonumber \\
 & =(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v{\rho}}-(\dot{\v{\rho}}\cdot\nabla\v A)\cdot\v{\rho}.
\end{align}

\end_inset

Further,
\begin_inset Formula 
\begin{equation}
(\v{\rho}\times\dot{\v{\rho}})\cdot\v B=\left(\rho_{c}\hat{\v{\rho}}\times\dot{\phi}\rho_{c}\hat{\v n}\right)\cdot\boldsymbol{B}=-\dot{\phi}\rho_{c}^{\,2}B=-\frac{mcv_{\perp}}{e}\rho_{c}\dot{\phi}.
\end{equation}

\end_inset

Thus,
\begin_inset Formula 
\begin{align}
(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v{\rho}} & =\frac{1}{2}\underbrace{\left((\dot{\v{\rho}}\cdot\nabla\v A)\cdot\v{\rho}+(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v{\rho}}\right)}_{\frac{1}{2}\frac{\d}{\d t}((\v{\rho}\cdot\nabla\v A)\cdot\v{\rho})+\mathcal{O}(\varepsilon)}+\frac{1}{2}\underbrace{\left((\v{\rho}\cdot\nabla\v A)\cdot\dot{\v{\rho}}-(\dot{\v{\rho}}\cdot\nabla\v A)\cdot\v{\rho}\right)}_{-\frac{mcv_{\perp}}{e}\rho_{c}\dot{\phi}}.
\end{align}

\end_inset

and we can replace 
\begin_inset Formula $(\v{\rho}\cdot\nabla\v A)\cdot\dot{\v{\rho}}$
\end_inset

 in the Lagrangian to obtain to order 
\begin_inset Formula $\varepsilon$
\end_inset

 (formally setting 
\begin_inset Formula $\varepsilon=1$
\end_inset

) 
\begin_inset Formula 
\begin{align}
L-\frac{e}{c}\frac{\d}{\d t}\left(\v{\rho}\cdot\v A-\frac{1}{2}\v{\rho}\cdot\nabla\v A\cdot\v{\rho}\right)+\mathcal{O}(\varepsilon) & =\left(mv_{\parallel}\hat{\v b}+\frac{e}{c}\v A\right)\cdot\dot{\v R}\nonumber \\
 & +mv_{\perp}\dot{\phi}\rho_{c}+\frac{mv_{\perp}}{2}\rho_{c}\dot{\phi}\nonumber \\
 & -\left(\frac{m(v_{\parallel}^{\,2}+v_{\perp}^{\,2})}{2}+e\Phi\right).
\end{align}

\end_inset

In the bracket under the total time derivative we can see that there is
 some system to the way the gauge transform in phase-space is done.
 More generally the technique is known as the 
\series bold
Lie-transform method
\series default
.
 To leading order in 
\begin_inset Formula $\varepsilon$
\end_inset

 the 
\series bold
guiding-center Lagrangian
\series default
 follows as
\begin_inset Formula 
\[
L_{\mathrm{gc}}(\v R,\dot{\v R},\cancel{\phi},\dot{\phi},v_{\parallel},v_{\perp})=\left(mv_{\parallel}\hat{\v b}(\v R)+\frac{e}{c}\v A(\v R)\right)\cdot\dot{\v R}+\frac{m^{2}cv_{\perp}^{\,2}}{2eB(\v R)}\dot{\phi}-\left(\frac{m(v_{\parallel}^{\,2}+v_{\perp}^{\,2})}{2}+e\Phi(\v R)\right).
\]

\end_inset

Here all fields are evaluated in 
\begin_inset Formula $\v R$
\end_inset

 and dependencies on 
\begin_inset Formula $\phi$
\end_inset

 have vanished.
 Thus we can identify a conserved generalized momentum
\begin_inset Formula 
\begin{equation}
\frac{\partial L_{\mathrm{gc}}}{\partial\dot{\phi}}=\frac{m^{2}cv_{\perp}^{\,2}}{2eB(\v R)}\equiv J_{\perp}=\frac{mc}{e}\mu,
\end{equation}

\end_inset

known as the perpendicular invariant with 
\begin_inset Formula $\dot{J}_{\perp}=0$
\end_inset

 along the trajectory.
 The perpendicular invariant is proportional to the magnetic moment 
\begin_inset Formula $\mu$
\end_inset

 of the guiding-center orbit, which is thus also conserved.
 Again, 
\begin_inset Formula $\mu$
\end_inset

 is not the exact magnetic moment of the particle, but rather an approximation
 up to order 
\begin_inset Formula $\varepsilon$
\end_inset

.
 If we replace 
\begin_inset Formula $v_{\perp}$
\end_inset

 by the variable 
\begin_inset Formula $\mu$
\end_inset

 we can write
\begin_inset Formula 
\begin{equation}
L_{\mathrm{gc}}(\v R,\dot{\v R},\dot{\phi},v_{\parallel},\mu)=\left(mv_{\parallel}\hat{\v b}(\v R)+\frac{e}{c}\v A(\v R)\right)\cdot\dot{\v R}+\frac{mc}{e}\mu\dot{\phi}-\left(\frac{mv_{\parallel}^{\,2}}{2}+\mu B(\v R)+e\Phi(\v R)\right).
\end{equation}

\end_inset

The Hamiltonian is the last part in brackets,
\begin_inset Formula 
\begin{equation}
H=\frac{mv_{\parallel}^{\,2}}{2}+\mu B(\v R)+e\Phi(\v R),
\end{equation}

\end_inset

and is equal to the total energy.
 For equations of motion, it is of limited use, since we use non-canonical
 variables.
 We rather need Euler-Lagrange equations of the guiding-center Lagrangian
 that follow as
\begin_inset Formula 
\begin{align}
\dot{\v R} & =\frac{v_{\parallel}\v B^{\star}(\v R,v_{\parallel})+c\v E^{\star}(\v R,\mu)\times\hat{\v b}(\v R)}{\hat{\v b}(\v R)\cdot\v B^{\star}(\v R,v_{\parallel})},\quad\dot{v}_{\parallel}=\frac{e}{m}\frac{\v E^{\star}(\v R,\mu)\cdot\v B^{\star}(\v R,v_{\parallel})}{\hat{\v b}(\v R)\cdot\v B^{\star}(\v R,v_{\parallel})},\label{eq:rdot-1}\\
\dot{\phi} & =-\omega_{c}(\v R)\equiv\frac{e\v B(\v R)}{mc},\quad\dot{\mu}=0.\label{eq:phidot}
\end{align}

\end_inset

where 
\begin_inset Formula $\hat{\v b}(\v R)=\v B(\v R)/B(\v R)$
\end_inset

 and
\begin_inset Formula 
\begin{align}
\v E^{\star}(\v R,\mu) & =-\frac{1}{e}\nabla H(\v R,v_{\parallel},\mu)=-\frac{\mu}{e}\nabla B(\v R)-\nabla\Phi(\v R),\\
\v B^{\star}(\v R,v_{\parallel}) & =\nabla\times\v A^{\star}(\v R,v_{\parallel}),\\
\v A^{\star}(\v R,v_{\parallel}) & =\v A(\v R)+\frac{mc}{e}v_{\parallel}\hat{\v b}(\v R).
\end{align}

\end_inset


\end_layout

\begin_layout Standard

\series bold
Derivation of equations of motion
\end_layout

\begin_layout Standard
From now on we will call the guiding-center Lagrangian 
\begin_inset Formula $L_{\mathrm{gc}}$
\end_inset

 just 
\begin_inset Formula $L$
\end_inset

 to reduce notational clutter.
 Take
\begin_inset Formula 
\begin{equation}
L(\v R,v_{\parallel},\mu,\dot{\v R},\dot{\phi})=\left(mv_{\parallel}\hat{\v b}(\v R)+\frac{e}{c}\v A(\v R)\right)\cdot\dot{\v R}+\frac{mc}{e}\mu\dot{\phi}-\left(\frac{mv_{\parallel}^{\,2}}{2}+\mu B(\v R)+e\Phi(\v R)\right).
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Derivatives are
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align}
\frac{\partial L}{\partial\v R} & =mv_{\parallel}(\nabla\hat{\v b})\cdot\dot{\v R}+\frac{e}{c}(\nabla\v A)\cdot\dot{\v R}-\left(\mu\nabla B+e\nabla\Phi\right)\\
\frac{\partial L}{\partial\phi} & =0\\
\frac{\partial L}{\partial v_{\parallel}} & =m\hat{\v b}\cdot\dot{\v R}-mv_{\parallel}\label{eq:vpar}\\
\frac{\partial L}{\partial\mu} & =\frac{mc}{e}\dot{\phi}-B\label{eq:mu}
\end{align}

\end_inset

Generalized velocities:
\begin_inset Formula 
\begin{align}
\frac{\d}{\d t}\frac{\partial L}{\partial\dot{\v R}} & =\frac{\d}{\d t}(mv_{\parallel}\hat{\v b}(\v R(t))+\frac{e}{c}\v A(\v R(t)))=m\dot{v}_{\parallel}\hat{\v b}+mv_{\parallel}\dot{\v R}\cdot\nabla\hat{\v b}+\frac{e}{c}\dot{\v R}\cdot\nabla\v A.\\
\frac{\d}{\d t}\frac{\partial L}{\partial\dot{\phi}} & =\frac{\d}{\d t}(\frac{mc}{e}\mu)=\frac{mc}{e}\dot{\mu}
\end{align}

\end_inset

Euler-Lagrange euqations in 
\begin_inset Formula $\v R$
\end_inset

 are
\begin_inset Formula 
\begin{align}
0=\frac{\d}{\d t}\frac{\partial L}{\partial\dot{\v R}}-\frac{\partial L}{\partial\v R} & =m\dot{v}_{\parallel}\hat{\v b}+mv_{\parallel}(\nabla\hat{\v b})\cdot\dot{\v R}+\frac{e}{c}(\nabla\v A)\cdot\dot{\v R}\nonumber \\
 & -\mu\nabla B-e\nabla\Phi-mv_{\parallel}\dot{\v R}\cdot\nabla\hat{\v b}-\frac{e}{c}\dot{\v R}\cdot\nabla\v A\nonumber \\
 & =m\dot{v}_{\parallel}\hat{\v b}+mv_{\parallel}\dot{\v R}\times(\nabla\times\hat{\v b})+\frac{e}{c}\dot{\v R}\times(\nabla\times\v A)\nonumber \\
 & -\mu\nabla B+e\v E-mv_{\parallel}\dot{\v R}\cdot\nabla\hat{\v b}\nonumber \\
 & =m\dot{v}_{\parallel}\hat{\v b}+\frac{e}{c}\dot{\v R}\times(\nabla\times\v A^{\star})-\mu\nabla B+e\v E-mv_{\parallel}\dot{\v R}\cdot\nabla\hat{\v b}\nonumber \\
 & =m\dot{v}_{\parallel}\hat{\v b}+\frac{e}{c}\dot{\v R}\times\v B^{\star}-\mu\nabla B+e\v E-mv_{\parallel}\dot{\v R}\cdot\nabla\hat{\v b}.
\end{align}

\end_inset

Here we have used 
\begin_inset Formula $\v a\times(\nabla\times\v b)=(\nabla\v a)\cdot\v b-\v b\cdot\nabla\v a$
\end_inset

 from 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:eq3"

\end_inset

 and defined
\begin_inset Formula 
\begin{align}
\v A^{\star}(\text{\v R},v_{\parallel}) & =\v A(\text{\v R})+\frac{mc}{e}v_{\parallel}\hat{\v b}(\text{\v R}),\\
\v B^{\star}(\text{\v R},v_{\parallel}) & =\nabla\times\v A^{\star}(\text{\v R},v_{\parallel})
\end{align}

\end_inset

as a modified vector potential and magnetic field.
 The latter depend on the parallel velocity 
\begin_inset Formula $v_{\parallel}$
\end_inset

 that can take both, positive or negative sign and vary during the orbit.
 The Euler-Lagrange equation in the ignorable 
\begin_inset Formula $\phi$
\end_inset

 yield conservation of magnetic moment 
\begin_inset Formula $\dot{\mu}=0$
\end_inset

.
 The remaining equations have no generalized velocity terms in 
\begin_inset Formula $L_{\text{gc}}$
\end_inset

 being a phase-space Lagrangian.
 Thus we can set terms (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:vpar"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:mu"

\end_inset

) to zero right away.
 To sum up our equations of motion become
\begin_inset Formula 
\begin{align}
m\dot{v}_{\parallel}\hat{\v b}(\v R)+\frac{e}{c}\dot{\v R}\times\v B^{\star}(\text{\v R},v_{\parallel}) & =\mu\nabla B(\v R)-e\v E(\v R),\label{eq:elg1}\\
\hat{\v b}(\v R)\cdot\dot{\v R} & =v_{\parallel},\label{eq:elg2}\\
\dot{\phi} & =\frac{eB(\v R)}{mc}\equiv\omega_{c}(\v R).\label{eq:elg3}
\end{align}

\end_inset

Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:elg1"

\end_inset

 describes drifts due to 
\begin_inset Formula $\v E$
\end_inset

 and gradient and curvature of 
\begin_inset Formula $\v B$
\end_inset

.
 Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:elg2"

\end_inset

 confirms the meaning of 
\begin_inset Formula $v_{\parallel}$
\end_inset

 as the parallel velocity, and Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:elg3"

\end_inset

 of 
\begin_inset Formula $\phi$
\end_inset

 as the gyrophase with regard to the guiding-center position 
\begin_inset Formula $\v R$
\end_inset

.
 Now we would like to decouple Eqs.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:elg1"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:elg2"

\end_inset

) to get explicit expressions for 
\begin_inset Formula $\dot{\v R}$
\end_inset

 and 
\begin_inset Formula $\dot{v}_{\parallel}$
\end_inset

.
 Taking a cross product of 
\begin_inset Formula $\hat{\v b}$
\end_inset

 with Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:elg1"

\end_inset

 removes the first term and together with the BAC-CAB rule and 
\begin_inset Formula $\v b\cdot\dot{\v R}=v_{\parallel}$
\end_inset

 from Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:elg2"

\end_inset

 yields
\begin_inset Formula 
\begin{equation}
\frac{e}{c}((\hat{\v b}\cdot\v B^{\star})\dot{\v R}-v_{\parallel}\v B^{\star})=\hat{\v b}\times(\mu\nabla B-e\v E).
\end{equation}

\end_inset

Finally we obtain
\begin_inset Formula 
\begin{align}
\dot{\v R} & =\frac{v_{\parallel}\v B^{\star}+c\v E^{\star}\times\hat{\v b}}{\hat{\v b}\cdot\v B^{\star}},\label{eq:rdot}\\
\dot{v}_{\parallel} & =\frac{e}{m}\frac{\v B^{\star}\cdot\v E^{\star}}{\hat{\v b}\cdot\v B^{\star}}.
\end{align}

\end_inset

Here (convention taken from 
\begin_inset CommandInset citation
LatexCommand citep
key "lanthaler2017"
literal "false"

\end_inset

) we defined
\begin_inset Formula 
\begin{equation}
\v E^{\star}=-e^{-1}\nabla H=-\frac{\mu}{e}\nabla B-\nabla\Phi.
\end{equation}

\end_inset


\end_layout

\begin_layout Section
Plasma Models
\end_layout

\begin_layout Standard
The goal of this chapter is to show a clear path from single particle motion
 via a kinetic intermediate stage.
 Details on the kinetic part, including an alternative derivation via Liouville'
s theorem leading to a well-defined collision term are presented in the
 course 
\emph on
Plasma Kinetic Theory
\emph default
 of Winfried Kernbichler.
 More extensive descriptions of the derivations made here can be found at
 
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

http://homepages.cae.wisc.edu/~callen/book.html
\end_layout

\end_inset

 as well as 
\begin_inset Quotes eld
\end_inset

Introduction to Plasma Theory
\begin_inset Quotes erd
\end_inset

 by D.D.
 Nicholson, 
\begin_inset Quotes eld
\end_inset

Fundamentals of Plasma Physics
\begin_inset Quotes erd
\end_inset

 by P.M.
 Bellan and 
\begin_inset Quotes eld
\end_inset

Ideal MHD
\begin_inset Quotes erd
\end_inset

 by J.P.
 Freidberg.
 
\end_layout

\begin_layout Subsection
From Newton's law to the Klimontovich equation
\end_layout

\begin_layout Standard
We consider an ensemble of 
\begin_inset Formula $N$
\end_inset

 particles of this species and introduce a kinetic picture: The normalized
 number density of discrete particles at position 
\begin_inset Formula $\boldsymbol{z}_{k}(t)$
\end_inset

 in phase-space is given by placing Dirac 
\begin_inset Formula $\delta$
\end_inset

s wherever a particle lies at time 
\begin_inset Formula $t$
\end_inset

,
\begin_inset Formula 
\begin{equation}
f^{{\rm m}}(\text{\v z},t)=\frac{1}{N}\sum_{k=1}^{N}\delta(\boldsymbol{z}-\boldsymbol{z}_{k}(t)).\label{eq:fm}
\end{equation}

\end_inset

The superscript 
\begin_inset Formula ${\rm m}$
\end_inset

 stands for microscopic here as this 
\series bold
microscropic distribution function
\series default
 
\begin_inset Formula $f^{{\rm m}}$
\end_inset

 contains all details about single particle states (classically positions
 and velocities/momenta).
 Roughly speaking 
\begin_inset Formula $f^{\mathrm{m}}=0$
\end_inset

 where there is no particle and 
\begin_inset Formula $\infty$
\end_inset

 where there is a particle.
 It is important to distinguish the position in phase-space 
\begin_inset Formula $z$
\end_inset

 as an independent variable for 
\begin_inset Formula $f^{\mathrm{m}}$
\end_inset

 from the orbit curves 
\begin_inset Formula $\v z_{k}(t)$
\end_inset

 of the individual particles, that introduce the time-dependency in 
\begin_inset Formula $f^{\mathrm{m}}$
\end_inset

.
 
\end_layout

\begin_layout Standard
It is normalized, In particular the choice of the normalization by 
\begin_inset Formula $1/N$
\end_inset

 together with the properties of the Dirac 
\begin_inset Formula $\delta$
\end_inset

 leads to the 
\series bold
normalization property
\series default

\begin_inset Formula 
\begin{equation}
\int_{\Pi}\d^{6}z\,f^{{\rm m}}(\v z,t)=1.\label{eq:norm1}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Here 
\begin_inset Formula $\Pi$
\end_inset

 is the domain in phase-space where particles exist.
 If we perform an integral over a region 
\begin_inset Formula $\Pi_{k}\subset\Pi$
\end_inset

 which contains only one particle 
\begin_inset Formula $k$
\end_inset

 we obtain instead
\begin_inset Formula 
\begin{equation}
\int_{\Pi_{k}}\d^{6}z\,f^{{\rm m}}(\boldsymbol{r},\boldsymbol{v},t)=\frac{1}{N}.\label{eq:norm1-1}
\end{equation}

\end_inset

This can be interpreted as the probability to find a predefined particle
 amongst our set of 
\begin_inset Formula $N$
\end_inset

 particles near phase-space location 
\begin_inset Formula $k$
\end_inset

.
\end_layout

\begin_layout Standard
To evaluate the partial time derivative of 
\begin_inset Formula $f^{\mathrm{m}}$
\end_inset

 we use the chain rule,
\begin_inset Formula 
\begin{equation}
\frac{\partial}{\partial t}f^{{\rm m}}(\text{\v z},t)=-\frac{1}{N}\sum_{k=1}^{N}\dot{\boldsymbol{z}}_{k}(t)\cdot\nabla_{\v z}\delta(\boldsymbol{z}-\boldsymbol{z}_{k}(t)).\label{eq:fm-1}
\end{equation}

\end_inset

From the equations of motion we know relations between phase-space velocities
 
\begin_inset Formula $\dot{\v z}_{k}$
\end_inset

 and phase-space positions 
\begin_inset Formula $\v z_{k}$
\end_inset

 of the form 
\begin_inset Formula $\dot{\boldsymbol{z}}_{k}(t)=\v w(\v z_{k}(t),t)$
\end_inset

.
 In particular charged plasma particles are subject to electromagnetic forces
 excerted by electric field 
\size normal

\begin_inset Formula $\boldsymbol{E}$
\end_inset


\size default
 and magnetic field 
\begin_inset Formula $\boldsymbol{B}$
\end_inset

.
 Here we use real-space position 
\begin_inset Formula $\v r$
\end_inset

 and velocity 
\begin_inset Formula $\v v$
\end_inset

 to parameterize phase-space via 
\begin_inset Formula $\v z=(\v r,\v v)$
\end_inset

 and apply the usual Lorentz force law.
 For developing other variants like gyrokinetics one could start with the
 guiding-center Lagrangian formalism.
 
\end_layout

\begin_layout Subsubsection*
Non-interacting particles in external fields
\end_layout

\begin_layout Standard
We first start with an idealized set of 
\emph on
non-interacting
\series bold
\emph default
 
\series default
particles in 
\emph on
external
\emph default
 fields 
\begin_inset Formula $\v E$
\end_inset

 and 
\begin_inset Formula $\v B$
\end_inset

.
 The motion of a single particle 
\begin_inset Formula $k$
\end_inset

 of mass 
\begin_inset Formula $m$
\end_inset

 and charge 
\begin_inset Formula $e$
\end_inset

 using CGS units is given by 
\begin_inset Formula 
\begin{align}
\dot{\boldsymbol{r}}_{k}(t) & =\boldsymbol{v}_{k}(t),\label{eq:ma}\\
\dot{\boldsymbol{v}}_{k}(t) & =\frac{\boldsymbol{F}_{k}(\boldsymbol{r}_{k}(t),\boldsymbol{v}_{k}(t),t)}{m}=\frac{e}{m}(\boldsymbol{E}(\boldsymbol{r}_{k}(t),t)+\frac{1}{c}\boldsymbol{v}_{k}(t)\times\boldsymbol{B}(\boldsymbol{r}_{k}(t),t)).
\end{align}

\end_inset

This is of the required form 
\begin_inset Formula $\dot{\boldsymbol{z}}_{k}(t)=\v w(\v z_{k}(t),t)$
\end_inset

 with
\begin_inset Formula 
\begin{equation}
\dot{\boldsymbol{z}}_{k}(t)\equiv(\dot{\boldsymbol{r}}_{k}(t),\dot{\boldsymbol{v}}_{k}(t)),\qquad\v w(\v z_{k}(t),t)=(\boldsymbol{v}_{k}(t),\frac{\boldsymbol{F}_{k}(\boldsymbol{r}_{k}(t),t)}{m}).\label{eq:newton}
\end{equation}

\end_inset

Now we can replace 
\begin_inset Formula $\dot{\boldsymbol{z}}_{k}(t)$
\end_inset

 by 
\begin_inset Formula $\v w(\v z_{k}(t),t)$
\end_inset

 and use the rule 
\begin_inset Formula $g(\v z_{0})\delta(\v z-\v z_{0})=g(\v z)\delta(\v z-\v z_{0})$
\end_inset

 for the 
\begin_inset Formula $\delta$
\end_inset

.
 Then Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:fm-1"

\end_inset

 changes to
\begin_inset Formula 
\begin{align}
\frac{\partial}{\partial t}f^{{\rm m}}(\text{\v z},t) & =-\frac{1}{N}\sum_{k=1}^{N}\v w(\v z_{k}(t),t)\cdot\nabla_{\v z}\delta(\boldsymbol{z}-\boldsymbol{z}_{k}(t))\nonumber \\
 & =-\frac{1}{N}\sum_{k=1}^{N}\v w(\v z,t)\cdot\nabla_{\v z}\delta(\boldsymbol{z}-\boldsymbol{z}_{k}(t))\nonumber \\
 & =-\v w(\v z,t)\cdot\nabla_{\v z}f^{{\rm m}}(\text{\v z},t).\label{eq:kineq}
\end{align}

\end_inset

This is a convective derivative in phase-space where 
\begin_inset Formula $f^{\mathrm{m}}$
\end_inset

 is transported along the flow 
\begin_inset Formula $\v w$
\end_inset

 produced by the movement of particles via equations of motion evaluated
 in 
\begin_inset Formula $\v z$
\end_inset

.
 In particular for 
\begin_inset Formula $\v z=(\v r,\v v)$
\end_inset

 and Newton's equations of motion with force 
\begin_inset Formula $\v F$
\end_inset

 as in Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:newton"

\end_inset

 we obtain the 
\series bold
kinetic equation for non-interacting particles
\series default

\begin_inset Formula 
\begin{equation}
\frac{\partial}{\partial t}f^{{\rm m}}(\text{\v r},\v v,t)+\v v\cdot\nabla_{\v r}f^{{\rm m}}(\text{\v r},\v v,t)+\frac{\v F(\v r,\v v,t)}{m}\cdot\nabla_{\v v}f^{{\rm m}}(\text{\v r},\v v,t)=0.\label{eq:kin1}
\end{equation}

\end_inset

While one might say this is not a big surprise, we have achieved the complete
 elimination of individual particle orbits with dependent variables 
\begin_inset Formula $\v z_{k}(t)=(\v r_{k}(t),\v v_{k}(t))$
\end_inset

 given by 
\begin_inset Formula $2N$
\end_inset

 
\emph on
ordinary 
\emph default
differential equations of motion by a single 
\emph on
partial
\emph default
 differential equation in the microscopic distribution function 
\begin_inset Formula $f^{\mathrm{m}}$
\end_inset

 with independent variables 
\begin_inset Formula $\v z=(\v r,\v v)$
\end_inset

 and 
\begin_inset Formula $t$
\end_inset

.
 They are connected via the fact that orbits are the 
\emph on
characteristics
\emph default
 of Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:kin1"

\end_inset

 along which 
\begin_inset Formula $f^{\mathrm{m}}$
\end_inset

 propagates.
 By smoothing over a regions of phase-space with many particles we can now
 also find solutions of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:kin1"

\end_inset

 with a distribution function that is less 
\begin_inset Quotes eld
\end_inset

spiky
\begin_inset Quotes erd
\end_inset

 than the original 
\begin_inset Formula $f^{\mathrm{m}}\propto\delta$
\end_inset

.
 This will be our goal in the following sections.
 
\end_layout

\begin_layout Subsubsection*
Interacting particles
\end_layout

\begin_layout Standard
Up to now we have made our life easy by ignoring particle interactions.
 Before proceeding we have to resolve the issue that in reality the force
 
\begin_inset Formula $\v F_{k}$
\end_inset

 acting on particle 
\begin_inset Formula $k$
\end_inset

 doesn't depend on 
\begin_inset Formula $\v r_{k}$
\end_inset

 and 
\begin_inset Formula $\v v_{k}$
\end_inset

 alone, but also on other particles' positions and velocities (currents
 generating 
\begin_inset Formula $\v B$
\end_inset

 fields).
 In a plasma the motion of particle number 
\begin_inset Formula $k$
\end_inset

 is affected the microscopic fields 
\size normal

\begin_inset Formula $\boldsymbol{E}^{{\rm m}}(\boldsymbol{r}_{k},t;\{\boldsymbol{r}_{l}\},\{\boldsymbol{v}_{l}\})$
\end_inset

, 
\begin_inset Formula $\boldsymbol{B}^{m}(\boldsymbol{r}_{k},t;\{\boldsymbol{r}_{l}\},\{\boldsymbol{v}_{l}\})$
\end_inset


\size default
 that are generated by
\begin_inset Foot
status open

\begin_layout Plain Layout
or linked to, to be more precise
\end_layout

\end_inset

 charges and currents dependent on position and velocity of all the other
 particles 
\begin_inset Formula $l\neq k$
\end_inset

 (denoted by curly brackets here).
 Also (possibly explicitly time-dependent) external fields excerted on the
 plasma are included in this full description.
 We rewrite Eq.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:ma"

\end_inset

 to 
\begin_inset Formula 
\begin{align}
\dot{\boldsymbol{v}}_{k} & =\boldsymbol{F}^{{\rm m}}(\boldsymbol{r}_{k},\boldsymbol{v}_{k},t;\{\boldsymbol{r}_{l}\},\{\boldsymbol{v}_{l}\})/m\nonumber \\
 & =\frac{e}{m}\boldsymbol{E}^{{\rm m}}(\boldsymbol{r}_{k},t;\{\boldsymbol{r}_{l}\},\{\boldsymbol{v}_{l}\})+\frac{e}{mc}\boldsymbol{v}_{k}\times\boldsymbol{B}^{{\rm m}}(\boldsymbol{r}_{k},t;\{\boldsymbol{r}_{l}\},\{\boldsymbol{v}_{l}\}).
\end{align}

\end_inset

Following the same steps as in the non-interacting case we arrive at the
 
\series bold
kinetic equation for interacting particles
\series default
,
\begin_inset Formula 
\begin{equation}
\frac{\partial}{\partial t}f^{{\rm m}}(\text{\v r},\v v,t)+\v v\cdot\nabla_{\v r}f^{{\rm m}}(\text{\v r},\v v,t)+\frac{\v F^{\mathrm{m}}(\v r,\v v,t;\{\boldsymbol{r}_{l}(t)\},\{\boldsymbol{v}_{l}(t)\})}{m}\cdot\nabla_{\v v}f^{{\rm m}}(\text{\v r},\v v,t)=0.\label{eq:kin1-1}
\end{equation}

\end_inset

While formally similar to the non-interacting result 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:kin1"

\end_inset

, the interacting variant here contains all orbits 
\begin_inset Formula $\{\boldsymbol{r}_{l}(t)\},\{\boldsymbol{v}_{l}(t)\}$
\end_inset

 resulting in a complicated time-dependency of microscopic forces 
\begin_inset Formula $\v F^{\mathrm{m}}$
\end_inset

.
 This dependency cannot even be stated without knowing the solutions of
 orbits in the first place, thereby eliminating any practical value of the
 kinetic reformulation.
 For electromagnetic fields Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:kin1-1"

\end_inset

 is known as the 
\series bold
Klimontovich equation
\series default
, 
\begin_inset Formula 
\begin{align}
\frac{\partial f^{{\rm m}}}{\partial t} & +\boldsymbol{v}\cdot\nabla_{\boldsymbol{r}}f^{{\rm m}}+\frac{q}{m}(\boldsymbol{E}^{{\rm {\rm m}}}+\frac{1}{c}\boldsymbol{v}\times\boldsymbol{B}^{{\rm m}})\cdot\nabla_{\boldsymbol{v}}f^{{\rm m}}=0,\label{eq:Klimontovich}
\end{align}

\end_inset

Here 
\begin_inset Formula $\v E^{\mathrm{m}}$
\end_inset

 and 
\begin_inset Formula $\v B^{\mathrm{m}}$
\end_inset

 could in principle be found from either solving Maxwell's equations self-consis
tently together with 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Klimontovich"

\end_inset

, or finding the Liénard–Wiechert potentials for all particles (
\begin_inset Formula $N\approx10^{19}$
\end_inset

 in laboratory plasmas).
 While this is of maybe of philosophical value, we are usually not interested
 in every single particle, but rather in their average statistical behaviour.
\end_layout

\begin_layout Subsubsection*
Smoothing the distribution function
\end_layout

\begin_layout Standard
In order to make equation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Klimontovich"

\end_inset

 manageable one has to rely on some averaging procedure.
 Since we can only observe macroscopic quantities such as temperature, density
 and pressure, we average the 
\begin_inset Formula $f^{{\rm m}}$
\end_inset

 on an intermediate scale by
\begin_inset Formula 
\begin{align}
\left\langle f^{m}\right\rangle  & :=\frac{\int_{V_{x}}d^{3}x^{\prime}\int_{V_{v}}d^{3}v^{\prime}\,f^{{\rm m}}(\boldsymbol{r}^{\prime},\boldsymbol{v}^{\prime})}{\int_{V_{x}}d^{3}x^{\prime}\int_{V_{v}}d^{3}v^{\prime}}=\frac{\int_{V_{x}}d^{3}x^{\prime}\int_{V_{v}}d^{3}v^{\prime}\,f^{{\rm m}}(\boldsymbol{r}^{\prime},\boldsymbol{v}^{\prime})}{\Delta V_{\mathrm{p}}}.\label{eq:average}
\end{align}

\end_inset

The phase space volume 
\begin_inset Formula $\Delta V_{\mathrm{p}}$
\end_inset

 over which the average is taken is chosen much larger than the distance
 between individual particles and much smaller than a scale such as the
 Debye length in a plasma.
 This means that the macroscopic distribution function 
\begin_inset Formula $f=\left\langle f^{\mathrm{m}}\right\rangle $
\end_inset

 and other quantities can be treated as continuous functions with respect
 to even larger scales such as the whole plasma dimensions.
 The macroscopic distribution function retains the normalization property
 of Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:norm1"

\end_inset

 when integrating over phase-space.
\end_layout

\begin_layout Standard
We split the microscopic distribution function 
\begin_inset Formula $f^{{\rm m}}$
\end_inset

 as well as the fields into the average part and the fluctuations around
 the average,
\begin_inset Formula 
\begin{equation}
f^{{\rm m}}=f+\delta f,\,\,\,\,\,\,\boldsymbol{E}^{{\rm m}}=\boldsymbol{E}+\delta\boldsymbol{E},\,\,\,\,\,\,\boldsymbol{B}^{{\rm m}}=\boldsymbol{B}+\delta\boldsymbol{B}.
\end{equation}

\end_inset

Most importantly, 
\begin_inset Formula $\boldsymbol{E}$
\end_inset

 and 
\begin_inset Formula $\boldsymbol{B}$
\end_inset

 do not depend on the single particle positions anymore, but can still depend
 on local densities of charges 
\begin_inset Formula $\rho_{e}(\v r,t)$
\end_inset

 and currents 
\begin_inset Formula $\v j(\v r,t)$
\end_inset

 on the averaging scale.
 Applying the average 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:average"

\end_inset

 to Eq.
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Klimontovich"

\end_inset

 we obtain the 
\series bold
plasma kinetic equation
\series default

\begin_inset Formula 
\begin{align}
\frac{\partial f}{\partial t}+\boldsymbol{v}\cdot\nabla_{\boldsymbol{r}}f+\frac{e}{m}(\boldsymbol{E}+\frac{1}{c}\boldsymbol{v}\times\boldsymbol{B})\cdot\nabla_{\boldsymbol{v}}f & =-\left\langle \frac{e}{m}(\delta\boldsymbol{E}+\frac{1}{c}\boldsymbol{v}\times\delta\boldsymbol{B})\cdot\nabla_{\boldsymbol{v}}\delta f\right\rangle \equiv\left(\frac{\partial f}{\partial t}\right)_{\mathrm{c}}.\label{eq:Klimontovich-1}
\end{align}

\end_inset

Here we have used the fact that averages containing a single fluctuation
 term such as 
\begin_inset Formula $\left\langle \delta f\right\rangle $
\end_inset

 are zero.
 The remaining term containing the products of two of such terms has been
 moved to the right side.
 All the influences from microscopic fluctuations on the time evolution
 of the distribution function 
\begin_inset Formula $f$
\end_inset

 are hidden there.
 Because interactions between point-like particles apart from their collective
 behavior can be viewed as collisions it is called the collision term and
 labelled 
\begin_inset Formula $(\partial f/\partial t)_{\mathrm{c}}$
\end_inset

.
 Methods to treat this collision term by different approximations are subject
 to plasma kinetic theory.
 A consistent derivation is possible via the Lenard-Balescu equation, where
 cumulative small-angle collisions allow to write the collision term as
 a Fokker-Planck-type differential operator acting on 
\begin_inset Formula $f$
\end_inset

 (Landau collision term).
 For details, see book 
\begin_inset Quotes eld
\end_inset

Introduction to Plasma Theory
\begin_inset Quotes erd
\end_inset

 by Nicholson, or 
\begin_inset Quotes eld
\end_inset

Classical transport theory
\begin_inset Quotes erd
\end_inset

 by Balescu.
\end_layout

\begin_layout Subsection
From the plasma kinetic equation to magnetohydrodynamic (fluid) equations
\end_layout

\begin_layout Standard
With the distribution function 
\begin_inset Formula $f$
\end_inset

 defined, it is still difficult to observe it directly in experiment.
 Our goal is to get rid of velocity space dependencies in order to define
 macroscopic (fluid) observables.
 This is commonly done for three reasons.
 Firstly, we can observe distributions of quantities over position space
 
\begin_inset Formula $\v r$
\end_inset

 more easily and inuitively than over velocity space 
\begin_inset Formula $\v v$
\end_inset

.
 Secondly, in a thermodynamic equilibrium the velocity space distribution
 is fixed by a Maxwellian.
 Finally, macroscopic conservation laws in position space follow automatically
 from conservation laws in phase-space and can be of interest on their own,
 no matter what the distribution in 
\begin_inset Formula $\v v$
\end_inset

 is.
\end_layout

\begin_layout Subsubsection*
Conservation laws in phase-space
\end_layout

\begin_layout Standard
In order to find conservation laws, we first need to transform the plasma
 kinetic equation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Klimontovich-1"

\end_inset

 from its convective form into a conservative form.
 This means pulling the gradients of 
\begin_inset Formula $f$
\end_inset

 to the front so we get the divergence of a phase-flux.
 For the second term this is easy, since 
\begin_inset Formula $\v v$
\end_inset

 doesn't depend on 
\begin_inset Formula $\v r$
\end_inset

 as both are independent variables,
\begin_inset Formula 
\begin{equation}
\boldsymbol{v}\cdot\nabla_{\boldsymbol{r}}f=\nabla_{\v r}\cdot(\v vf)-(\nabla_{\v r}\cdot\v v)f=\nabla_{\v r}\cdot(\v vf).
\end{equation}

\end_inset

For the second term it works too, since 
\begin_inset Formula $\v E$
\end_inset

 doesn't depend on 
\begin_inset Formula $\v v$
\end_inset

, and
\begin_inset Formula 
\begin{align}
\nabla_{\v v}\cdot(\v v\times\v B) & =\v B\cdot(\nabla_{\v v}\times\v v)-\v v\cdot(\nabla_{\v v}\times\v B).
\end{align}

\end_inset

Taking a curl of 
\begin_inset Formula $\v v=(v_{x},v_{y},v_{z})$
\end_inset

 in velocity space yields zero, and 
\begin_inset Formula $\v B$
\end_inset

 doesn't even depend on 
\begin_inset Formula $\v v$
\end_inset

.
 Thus,
\begin_inset Formula 
\begin{equation}
\frac{e}{m}(\boldsymbol{E}+\frac{1}{c}\boldsymbol{v}\times\boldsymbol{B})\cdot\nabla_{\boldsymbol{v}}f=\nabla_{\boldsymbol{v}}\cdot\left(\frac{e}{m}(\boldsymbol{E}+\frac{1}{c}\boldsymbol{v}\times\boldsymbol{B})f\right).
\end{equation}

\end_inset

Thus we can write Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Klimontovich-1"

\end_inset

 in conservative form
\begin_inset Formula 
\begin{equation}
\frac{\partial f}{\partial t}+\nabla_{\boldsymbol{r}}\cdot(\boldsymbol{v}f)+\nabla_{\boldsymbol{v}}\cdot\left(\frac{e}{m}(\boldsymbol{E}+\frac{1}{c}\boldsymbol{v}\times\boldsymbol{B})f\right)=\left(\frac{\partial f}{\partial t}\right)_{\mathrm{c}}.\label{eq:falpha-2}
\end{equation}

\end_inset

This result is only a special case of a more general principle.
 Let's consider the advective (Lagrangian) form of any kinetic equation
 in phase space with variables 
\begin_inset Formula $\v z$
\end_inset

,
\begin_inset Formula 
\begin{equation}
\frac{Df}{Dt}=\frac{\partial f(\v z,t)}{\partial t}+\boldsymbol{w}(\v z,t)\cdot\nabla_{\boldsymbol{z}}f(\v z,t)=\left(\frac{\partial f(\v z,t)}{\partial t}\right)_{\mathrm{c}}.\label{eq:kincon}
\end{equation}

\end_inset

Again it is important to remember that 
\begin_inset Formula $\v w$
\end_inset

 and 
\begin_inset Formula $\dot{\v z}(t)$
\end_inset

 are related but different objects.
 As mentioned before, 
\begin_inset Formula $\v w(\v z,t)$
\end_inset

 are characeristics of Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:kincon"

\end_inset

, related to orbits via 
\begin_inset Formula $\dot{\v z}_{\alpha}(t)=\v w_{\alpha}(\v z_{\alpha}(t),t)$
\end_inset

.
 If 
\begin_inset Formula $\v w$
\end_inset

 follows from Hamiltonian equations of motion (including phase-space/guiding-cen
ter Lagrangian), it is called a 
\series bold
Hamiltonian vector field
\series default
 and always has the property of divergence-freeness
\begin_inset Formula 
\begin{equation}
\nabla_{\v z}\cdot\v w(\v z,t)=0,
\end{equation}

\end_inset

known as 
\series bold
Liouville's theorem
\series default
, or the conservation of phase-space volume.
 This immediately allows to write the conservative form of Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:kincon"

\end_inset

 as
\begin_inset Formula 
\begin{equation}
\frac{\partial f(\v z,t)}{\partial t}+\nabla_{\boldsymbol{z}}\cdot(\boldsymbol{w}(\v z,t)f(\v z,t))=\left(\frac{\partial f(\v z,t)}{\partial t}\right)_{\mathrm{c}}.\label{eq:kincon-1}
\end{equation}

\end_inset

Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:kincon-1"

\end_inset

 is completely analogous to the conservative Eulerian picture in fluid mechanics.
 The change of 
\begin_inset Formula $f$
\end_inset

 over time at a fixed 
\begin_inset Formula $\v z$
\end_inset

 comes from a divergence of the phase flux 
\begin_inset Formula $\Gamma_{f}=\v wf$
\end_inset

 transporting 
\begin_inset Formula $f$
\end_inset

 and a source terms on the right-hand side resulting from collisions.
\end_layout

\begin_layout Subsubsection*
Moments
\end_layout

\begin_layout Standard
In order to take influences from electrons and ions on macroscopic observables
 into account, we define an individual distribution function 
\begin_inset Formula $f_{\alpha}$
\end_inset

 for each plasma species 
\begin_inset Formula $\alpha$
\end_inset

.
 Macroscopic observables can then be calculated from the distribution functions
 
\begin_inset Formula $f_{\alpha}$
\end_inset

 of either one species 
\begin_inset Formula $\alpha$
\end_inset

 or a combination of them by removing velocity dependencies via an integral
 over the whole velocity space.
 The fluid observable 
\begin_inset Formula $\t Q(\v r,t)$
\end_inset

 is defined via its kinetic counterpart 
\begin_inset Formula $\t q(\v r,\v v,t)$
\end_inset

 by
\begin_inset Formula 
\begin{equation}
\t Q(\v r,t)\equiv\int d^{3}v\,\t q(\v r,\v v,t)f_{\alpha}(\boldsymbol{r},\boldsymbol{v},t).\label{eq:intfluid}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
where 
\begin_inset Formula $\t q$
\end_inset

 and 
\begin_inset Formula $\t Q$
\end_inset

 can be a scalar, vector, or tensor.
 The usual way to proceed is to define a number density from 
\begin_inset Formula $\t q_{0}=N_{\alpha}$
\end_inset

, momentum density 
\begin_inset Formula $\t q_{1}=m_{\alpha}N_{\alpha}\v v$
\end_inset

 and energy density 
\begin_inset Formula $\t q_{2}=m_{\alpha}N_{\alpha}v^{2}/2$
\end_inset

.
 Here 
\begin_inset Formula $N_{\alpha}$
\end_inset

 is the total number of particles of species 
\begin_inset Formula $\alpha$
\end_inset

 in the spatial domain 
\begin_inset Formula $\Omega$
\end_inset

 defined large enough that it will always contain all particles (or even
 
\begin_inset Formula $\Omega=\mathbb{R}^{3}$
\end_inset

).
 Integrals of the kind of 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:intfluid"

\end_inset

 when applied to another expression 
\begin_inset Formula $g(\v r,\v v,t)$
\end_inset

,
\begin_inset Formula 
\begin{equation}
\t M\equiv\int d^{3}v\,\t q(\v r,\v v,t)g(\v r,\v v,t)f_{\alpha}(\boldsymbol{r},\boldsymbol{v},t).\label{eq:intfluid-1}
\end{equation}

\end_inset

are called 
\begin_inset Formula $\t q$
\end_inset

-weighted 
\series bold
moments 
\series default
of 
\begin_inset Formula $g$
\end_inset

.
 We speak of a 
\begin_inset Formula $K$
\end_inset

-th moment, if 
\begin_inset Formula $\v v$
\end_inset

 appears 
\begin_inset Formula $K$
\end_inset

 times in the overall expression in the integral 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:intfluid-1"

\end_inset

.
 For our 
\begin_inset Formula $\t q_{k}$
\end_inset

 defined above, moments will be of order 
\begin_inset Formula $K=k$
\end_inset

 if no 
\begin_inset Formula $\v v$
\end_inset

 appears inside 
\begin_inset Formula $g$
\end_inset

.
\end_layout

\begin_layout Standard
In the following section we are going to take 
\begin_inset Formula $\t q_{k}$
\end_inset

-weighted moments of the plasma kinetic equation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Klimontovich-1"

\end_inset


\begin_inset Formula 
\begin{equation}
\underset{(I)}{\frac{\partial f_{\alpha}}{\partial t}}+\underset{(II)}{\nabla_{\boldsymbol{r}}\cdot(\boldsymbol{v}f_{\alpha})}+\underset{(III)}{\nabla_{\boldsymbol{v}}\cdot\left(\frac{\v F_{\alpha}}{m_{\alpha}}f_{\alpha}\right)}=\underset{(C)}{\left(\frac{\partial f_{\alpha}}{\partial t}\right)_{\mathrm{c}}}.\label{eq:falpha-2-1}
\end{equation}

\end_inset

where we have labeled single terms for later reference and denoted the Lorentz
 force as 
\begin_inset Formula $\v F_{\alpha}=\v F_{\alpha}(\v r,\v v,t)$
\end_inset

.
 Already here we see that because of 
\begin_inset Formula $\v v$
\end_inset

 appearing in 
\begin_inset Formula $(II)$
\end_inset

 the order of its moments will be of order 
\begin_inset Formula $k+1$
\end_inset

.
\end_layout

\begin_layout Subsubsection*
Continuity equation for number density
\end_layout

\begin_layout Standard
We start with the definition of the 
\series bold
number density
\series default
 
\begin_inset Formula $n_{\alpha}$
\end_inset

 that should give the average
\emph on
 local
\emph default
 particle density at 
\begin_inset Formula $\v r$
\end_inset

 at time 
\begin_inset Formula $t$
\end_inset

 via Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:intfluid"

\end_inset

 with 
\begin_inset Formula $\t q=N_{\alpha}$
\end_inset


\begin_inset Formula 
\begin{align}
n_{\alpha}(\boldsymbol{r},t) & \equiv\int d^{3}v\,N_{\alpha}f_{\alpha}(\boldsymbol{r},\boldsymbol{v},t)=N_{\alpha}\int d^{3}v\,f_{\alpha}(\boldsymbol{r},\boldsymbol{v},t).
\end{align}

\end_inset

This equation is a zeroeth moment.
 Since 
\begin_inset Formula $f_{\alpha}$
\end_inset

 is normalized with 
\begin_inset Formula 
\begin{equation}
\int_{\Omega}\d^{3}r\int\d^{3}vf_{\alpha}(\v r,\v v,t)=1,
\end{equation}

\end_inset

we can easily check that 
\begin_inset Formula 
\begin{equation}
\int_{\Omega}\d^{3}r\,n_{\alpha}(\v r,t)=N_{\alpha}
\end{equation}

\end_inset

yields the number of particles.
 Mass density 
\begin_inset Formula $\rho_{m\alpha}(\v r,t)=m_{\alpha}n_{\alpha}(\v r,t)$
\end_inset

 and charge density 
\begin_inset Formula $\rho_{e\alpha}(\v r,t)=e_{\alpha}n_{\alpha}(\v r,t)$
\end_inset

 can be computed from 
\begin_inset Formula $n_{\alpha}$
\end_inset

.
 Due to quasineutrality 
\begin_inset Formula $\sum_{\alpha}e_{\alpha}n_{\alpha}(\v r,t)=0$
\end_inset

 we will often assume 
\begin_inset Formula $n_{i}=n_{e}$
\end_inset

 in a two-species plasma of hydrogen/deuterium/tritium ions with charge
 
\begin_inset Formula $+e$
\end_inset

 and electrons with charge 
\begin_inset Formula $-e$
\end_inset

.
 Now we start taking moments 
\begin_inset Formula $N_{\alpha}\int\d^{3}v$
\end_inset

 of Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:falpha-2-1"

\end_inset

.
 Application on 
\begin_inset Formula $(I)$
\end_inset

 yields the zeroth moment
\begin_inset Formula 
\begin{equation}
N_{\alpha}\int\d^{3}v\,(I)=N_{\alpha}\int\d^{3}v\frac{\partial f_{\alpha}}{\partial t}=\frac{\partial}{\partial t}(N_{\alpha}\int\d^{3}vf_{\alpha})=\frac{\partial n_{\alpha}}{\partial t}.
\end{equation}

\end_inset

Here we could pull out the time derivative because integral boundaries are
 at 
\begin_inset Formula $\infty$
\end_inset

 and independent of time,
\begin_inset Foot
status open

\begin_layout Plain Layout
otherwise one needs the Leibniz rule
\end_layout

\end_inset

 and 
\begin_inset Formula $N_{\alpha}$
\end_inset

 is a constant, provided that the plasma is hot enought that no recombination
 occurs, and nuclear fusion is so inefficient that the relative change in
 
\begin_inset Formula $N_{\alpha}$
\end_inset

 is negligible.
\end_layout

\begin_layout Standard
Acting on 
\begin_inset Formula $(II)$
\end_inset

 yields
\begin_inset Formula 
\begin{equation}
N_{\alpha}\int\d^{3}v\,(II)=N_{\alpha}\int\d^{3}v\nabla_{\boldsymbol{r}}\cdot(\boldsymbol{v}f_{\alpha})=\nabla\cdot(N_{\alpha}\int\d^{3}v\,\boldsymbol{v}f_{\alpha})\equiv\nabla\cdot\v{\Gamma}_{n\alpha}.
\end{equation}

\end_inset

Here we have defined the 
\series bold
particle flux
\series default
 
\begin_inset Formula 
\begin{equation}
\v{\Gamma}_{n\alpha}(\v r,t)\equiv N_{\alpha}\int\d^{3}v\,\boldsymbol{v}f_{\alpha}(\v r,\v v,t)
\end{equation}

\end_inset

for species 
\begin_inset Formula $\alpha$
\end_inset

 being a 
\emph on
first
\emph default
 rather than a zeroth moment.
 Note also that from here on we denote 
\begin_inset Formula $\nabla\equiv\nabla_{\v r}$
\end_inset

 as the usual Nabla operator in position space.
\end_layout

\begin_layout Standard
Term 
\begin_inset Formula $(III)$
\end_inset

 can be transformed to a surface integral in velocity space using Gauss'
 divergence theorem,
\begin_inset Formula 
\begin{equation}
N_{\alpha}\int\d^{3}v\,(III)=N_{\alpha}\int\d^{3}v\nabla_{\boldsymbol{v}}\cdot\left(\frac{\v F_{\alpha}}{m_{\alpha}}f_{\alpha}\right)=N_{\alpha}\oint\,\d\v S_{\v v}\cdot\frac{\v F_{\alpha}}{m_{\alpha}}f_{\alpha}=0.
\end{equation}

\end_inset

Here the surface integral vanishes when taking the limit to infinite velocities,
 that are never reached by any particle.
\end_layout

\begin_layout Standard
Finally, the collisional term 
\begin_inset Formula $(C)$
\end_inset

 becomes
\begin_inset Formula 
\begin{equation}
N_{\alpha}\int\d^{3}v\,(C)=N_{\alpha}\int\d^{3}v\left(\frac{\partial f_{\alpha}}{\partial t}\right)_{\mathrm{c}}=0.\label{eq:collnum}
\end{equation}

\end_inset

This term must vanish, since collisions cannot change species, or even create
 or destroy particles, but only reshuffle them in velocity space.
 Every collision that removes a particle of species 
\begin_inset Formula $\alpha$
\end_inset

 at a certain velocity 
\begin_inset Formula $v$
\end_inset

, must add it at another velocity 
\begin_inset Formula $v$
\end_inset

.
 Since there are no further velocity-dependent weights in the integral 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:collnum"

\end_inset

, each of these contributions cancels each out identically.
 
\end_layout

\begin_layout Standard
With the above-made definitions and simplifications, we can write the conservati
on law for the particle number density 
\begin_inset Formula $n_{\alpha}$
\end_inset

 as
\begin_inset Formula 
\begin{equation}
\frac{\partial n_{\alpha}(\v r,t)}{\partial t}+\nabla\cdot\v{\Gamma}_{n\alpha}(\v r,t)=0.\label{eq:falpha-2-1-1}
\end{equation}

\end_inset

Re-writing this in the commonly used form with 
\series bold
species fluid velocity
\series default
 
\begin_inset Formula $\v V_{\alpha}$
\end_inset

 and
\begin_inset Formula 
\begin{equation}
\frac{\partial n_{\alpha}(\v r,t)}{\partial t}+\nabla\cdot(\v V_{\alpha}(\v r,t)n_{\alpha}(\v r,t))=0,\label{eq:falpha-2-1-1-1}
\end{equation}

\end_inset

yields the 
\emph on
definition
\emph default
 of 
\begin_inset Formula $\v V_{\alpha}$
\end_inset

 as
\begin_inset Formula 
\begin{equation}
\v V_{\alpha}(\v r,t)\equiv\frac{\v{\Gamma}_{n\alpha}(\v r,t)}{n_{\alpha}(\v r,t)}=\frac{\cancel{N_{\alpha}}\int\d^{3}v\,\boldsymbol{v}f_{\alpha}(\v r,\v v,t)}{\cancel{N_{\alpha}}\int d^{3}v\,f_{\alpha}(\boldsymbol{r},\boldsymbol{v},t)}.
\end{equation}

\end_inset

The fluid velocity is thus an average over the velocity distribution in
 phase-space, given by a first moment, normalized by the local particle
 density.
 As 
\begin_inset Formula $\v V_{\alpha}$
\end_inset

 is an intrinsic quantity it doesn't depend on how many particles are there
 to determine how fast they move.
 An increase in particle flux 
\begin_inset Formula $\v{\Gamma}_{n\alpha}$
\end_inset

 can be made either by increasing the local density 
\begin_inset Formula $n_{\alpha}$
\end_inset

 while keeping 
\begin_inset Formula $\v V_{\alpha}$
\end_inset

 constant, or by increasing 
\begin_inset Formula $\v V_{\alpha}$
\end_inset

 at constant 
\begin_inset Formula $n_{\alpha}$
\end_inset

.
 Thus, our family of related but different velocities has grown to 
\begin_inset Formula $\dot{\v r}_{k}(t),\v v_{k}(t),\v v,\v V_{\alpha}(\v r,t)$
\end_inset

.
 This is a good point to rest and meditate on what you have learned.
\end_layout

\begin_layout Subsubsection*
Fluid momentum law for each single species
\end_layout

\begin_layout Standard
From the continuity equation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:falpha-2-1-1"

\end_inset

 resulting from a 
\emph on
zeroth
\emph default
 moment of the plasma kinetic equation 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:kincon-1"

\end_inset

 we remember that a 
\emph on
first
\emph default
 moment popped up in form of a flux that we could translate into a fluid
 velocity 
\begin_inset Formula $\v V_{\alpha}$
\end_inset

 of species 
\begin_inset Formula $\alpha$
\end_inset

.
 Going back to a single species we now take a moment with the kinematic
 momentum 
\begin_inset Formula $m_{\alpha}N_{\alpha}\v v$
\end_inset

 as a weight, so 
\begin_inset Formula $\int\d^{3}vm_{\alpha}N_{\alpha}\v v$
\end_inset

.
 This is a 
\emph on
first
\emph default
 moment, as the velocity appears once.
 Applying it on 
\begin_inset Formula $(I)$
\end_inset

 we obtain
\begin_inset Formula 
\begin{equation}
m_{\alpha}N_{\alpha}\int\d^{3}v\v v\frac{\partial f_{\alpha}}{\partial t}=\frac{\partial}{\partial t}\left(m_{\alpha}N_{\alpha}\int\d^{3}v\v vf_{\alpha}\right)=\frac{\partial(m_{\alpha}n_{\alpha}\v V_{\alpha})}{\partial t}.
\end{equation}

\end_inset

This is the change in time of the 
\series bold
fluid
\series default
 
\series bold
momentum density
\series default
 of species 
\begin_inset Formula $\alpha$
\end_inset

,
\series bold
 
\series default

\begin_inset Formula 
\begin{equation}
\v p_{\alpha}^{\mathrm{fl}}(\v r,t)\equiv m_{\alpha}n_{\alpha}(\v r,t)\v V_{\alpha}(\v r,t).
\end{equation}

\end_inset

(remember, there was also additional electromagnetic momentum, see concepts).
 We notice that it is equal to the 
\series bold
mass flux 
\begin_inset Formula $\v{\Gamma}_{m\alpha}(\v r,t)$
\end_inset


\series default
 of species 
\begin_inset Formula $\alpha$
\end_inset

.
\end_layout

\begin_layout Standard
From 
\begin_inset Formula $(II)$
\end_inset

 we now get 
\begin_inset Formula 
\begin{equation}
m_{\alpha}N_{\alpha}\int\d^{3}v\v v(II)=m_{\alpha}N_{\alpha}\int\d^{3}v\v v\nabla_{\v r}\cdot(\v vf_{\alpha})=\nabla\cdot(m_{\alpha}N_{\alpha}\int\d^{3}v\v v\v vf_{\alpha}).\label{eq:momflux1}
\end{equation}

\end_inset

representing the spatial divergence of a 
\emph on
second 
\emph default
moment (you see this will never end if we don't stop it).
 This term contains a 
\emph on
dyad
\emph default
 
\begin_inset Formula $\v v\text{\v v}\equiv\v v\otimes\v v$
\end_inset

 with components 
\begin_inset Formula $v_{i}v_{j}$
\end_inset

 (see concepts), making it a second order tensor.
 The divergence of such a second-order tensor yields a vector field, so
 we can add this to the change of the kinematic momentum density 
\begin_inset Formula $\partial\v p_{\alpha}^{\mathrm{kin}}/\partial t$
\end_inset

 from 
\begin_inset Formula $(I)$
\end_inset

, being also a vector field.
 We leave it as it is for now.
\end_layout

\begin_layout Standard
To resolve the mass-weighted second moment of 
\begin_inset Formula $(III)$
\end_inset


\begin_inset Formula 
\begin{equation}
m_{\alpha}N_{\alpha}\int\d^{3}v\v v(III)=m_{\alpha}N_{\alpha}\int\d^{3}v\v v\left(\nabla_{\boldsymbol{v}}\cdot\frac{\v F_{\alpha}f_{\alpha}}{m_{\alpha}}\right),
\end{equation}

\end_inset

we use the identity 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:diaddiv"

\end_inset


\begin_inset Formula 
\begin{equation}
\nabla\cdot(\v a\v b)=\v a\cdot\nabla\v b+(\nabla\cdot\v a)\v b
\end{equation}

\end_inset

with 
\begin_inset Formula $\v a=\v F_{\alpha}f_{\alpha}$
\end_inset

 and 
\begin_inset Formula $\v b=\v v$
\end_inset

 , so
\begin_inset Formula 
\begin{equation}
\left(\nabla_{\boldsymbol{v}}\cdot(\v F_{\alpha}f_{\alpha})\right)\v v=\nabla_{\boldsymbol{v}}\cdot\left(f_{\alpha}\v F_{\alpha}\v v\right)-f_{\alpha}\v F_{\alpha}\cdot\underbrace{\nabla_{\v v}\v v}_{\v I}.
\end{equation}

\end_inset

Again the integral over the first term vanishes if we convert it to a boundary
 integral via the divergence theorem for tensors (one can always write it
 in components to see this, if unsure).
 The second term just yields a velocity-space average of the force,
\begin_inset Formula 
\begin{align}
m_{\alpha}N_{\alpha}\int\d^{3}v\v v(III) & =-\cancel{m_{\alpha}}N_{\alpha}\int\d^{3}v\frac{\v F_{\alpha}f_{\alpha}}{\cancel{m_{\alpha}}}\nonumber \\
 & =-N_{\alpha}e_{\alpha}\int\d^{3}v(\v E+\frac{1}{c}\v v\times\v B)f_{\alpha}\nonumber \\
 & =-e_{\alpha}(n_{\alpha}\v E+\frac{1}{c}\v{\Gamma}_{n\alpha}\times\v B)\nonumber \\
 & =-e_{\alpha}n_{\alpha}(\v E+\frac{1}{c}\v V_{\alpha}\times\v B),
\end{align}

\end_inset

This is the negative Lorentz force acting on the fluid of species 
\begin_inset Formula $\alpha$
\end_inset

 with charge density 
\begin_inset Formula $\rho_{e\alpha}=e_{\alpha}n_{\alpha}$
\end_inset

.
\end_layout

\begin_layout Standard
The collisional term 
\begin_inset Formula $(C)$
\end_inset

 yields 
\begin_inset Formula 
\begin{equation}
m_{\alpha}N_{\alpha}\int\d^{3}v\v v(C)=m_{\alpha}N_{\alpha}\int\d^{3}v\v v\left(\frac{\partial f_{\alpha}}{\partial t}\right)_{\mathrm{c}}\equiv\v R_{\alpha}.
\end{equation}

\end_inset

This contribution doesn't vanish as before, because collisions with other
 species can insert or extract momentum to the fluid of species 
\begin_inset Formula $\alpha$
\end_inset

.
 Contribution 
\begin_inset Formula $\v R_{\alpha\alpha}$
\end_inset

 from self-collisions of species 
\begin_inset Formula $\alpha$
\end_inset

 must not change its fluid momentum, so we can write
\begin_inset Formula 
\begin{equation}
\v R_{\alpha}=\sum_{\beta\neq\alpha}\v R_{\alpha\beta}.
\end{equation}

\end_inset

As global momentum conservation is still valid for elastic collisions, each
 collision can only add momentum that it removed from one species to another
 species, but not create or destroy it.
 Thus the sum of collision terms over all species must vanish,
\begin_inset Formula 
\begin{equation}
\sum_{\alpha}\v R_{\alpha}=\sum_{\alpha}\sum_{\beta\neq\alpha}\v R_{\alpha\beta}=0.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Now we can compare Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:momflux1"

\end_inset

 to the divergence of the 
\series bold
fluid momentum flux 
\series default
defined by
\begin_inset Formula 
\begin{equation}
\v{\Gamma}_{\v p\alpha}^{\mathrm{fl}}(\v r,t)\equiv\v V_{\alpha}(\v r,t)\v p_{\alpha}^{\mathrm{fl}}(\v r,t)=m_{\alpha}n_{\alpha}(\v r,t)\v V_{\alpha}(\v r,t)\v V_{\alpha}(\v r,t).
\end{equation}

\end_inset

At first sight the order of vectors in the dyadic plays a role and we have
 to write 
\begin_inset Formula $\v V_{\alpha}$
\end_inset

 that transports the vector field 
\begin_inset Formula $\v p_{\alpha}^{\mathrm{fl}}$
\end_inset

 in the front.
 In the result we see however that it doesn't matter.
\begin_inset Formula 
\begin{align}
m_{\alpha}N_{\alpha}\int\d^{3}v\v v\v vf_{\alpha} & =m_{\alpha}N_{\alpha}\int\d^{3}v(\v v-\v V_{\alpha})(\v v-\v V_{\alpha})f_{\alpha}-m_{\alpha}n_{\alpha}\v V_{\alpha}\v V_{\alpha}+\underbrace{m_{\alpha}N_{\alpha}\int\d^{3}v(\v v\v V_{\alpha}+\v V_{\alpha}\v v)f_{\alpha}}_{2m_{\alpha}n_{\alpha}\v V_{\alpha}\v V_{\alpha}}\nonumber \\
 & =m_{\alpha}N_{\alpha}\int\d^{3}v(\v v-\v V_{\alpha})(\v v-\v V_{\alpha})f_{\alpha}+\underbrace{m_{\alpha}n_{\alpha}\v V_{\alpha}\v V_{\alpha}}_{\v{\Gamma}_{\v p\alpha}^{\mathrm{fl}}}.
\end{align}

\end_inset

The remaining term apart from 
\begin_inset Formula $\v{\Gamma}_{\v p\alpha}^{\mathrm{fl}}$
\end_inset

 is called the 
\series bold
species pressure tensor 
\series default

\begin_inset Formula 
\begin{align}
\t P_{\alpha} & \equiv m_{\alpha}N_{\alpha}\int\d^{3}v(\v v-\v V_{\alpha})(\v v-\v V_{\alpha})f_{\alpha}\nonumber \\
 & =m_{\alpha}N_{\alpha}\int\d^{3}v\v v\v vf_{\alpha}-\v{\Gamma}_{\v p\alpha}^{\mathrm{fl}}.
\end{align}

\end_inset

Here we could add 
\begin_inset Quotes eld
\end_inset


\begin_inset Formula $-\v V_{\alpha}$
\end_inset


\begin_inset Quotes erd
\end_inset

 in the second term since
\begin_inset Formula 
\begin{equation}
N_{\alpha}\int\d^{3}v(\v v-\v V_{\alpha})\v V_{\alpha}f_{\alpha}=\left(N_{\alpha}\int\d^{3}v(\v v-\v V_{\alpha})f_{\alpha}\right)\v V_{\alpha}
\end{equation}

\end_inset

and
\begin_inset Formula 
\begin{equation}
N_{\alpha}\int\d^{3}v(\v v-\v V_{\alpha})f_{\alpha}=N_{\alpha}\int\d^{3}v\v vf_{\alpha}-N_{\alpha}\v V_{\alpha}\int\d^{3}vf_{\alpha}=n_{\alpha}\v V_{\alpha}-n_{\alpha}\v V_{\alpha}=0.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The pressure tensor quantifies the average deviation 
\begin_inset Formula $(\v v-\v V_{\alpha})$
\end_inset

 away from the fluid velocity.
 Finally, we write the 
\series bold
conservative momentum law for species 
\series default

\begin_inset Formula $\alpha$
\end_inset

,
\begin_inset Formula 
\begin{equation}
\frac{\partial(m_{\alpha}n_{\alpha}\v V_{\alpha})}{\partial t}+\nabla\cdot(m_{\alpha}n_{\alpha}\v V_{\alpha}\v V_{\alpha})=-\nabla\cdot\t P_{\alpha}+e_{\alpha}n_{\alpha}(\v E+\frac{1}{c}\v V_{\alpha}\times\v B)+\v R_{\alpha}.\label{eq:moma}
\end{equation}

\end_inset

Here we have moved 
\begin_inset Formula $(III)$
\end_inset

 and 
\begin_inset Formula $\nabla\cdot\t P_{\alpha}$
\end_inset

 already to the right-hand side, as it appears as a source term.
 
\end_layout

\begin_layout Subsubsection*
Single-fluid magnetohydrodynamics including multiple species
\end_layout

\begin_layout Standard
As collisions don't change masses or charges of particles either, we can
 easily extend the continuity equation of number density 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:falpha-2-1-1"

\end_inset

 to variants for charge and mass density conservation.
 For this purpose, it is more convenient to sum over species, as currents
 are a result of opposite charges moving in opposite directions, and the
 total mass density consists of all species.
 This approach is known as the 
\series bold
single-fluid
\series default
 formulation of magnetohydrodynamics, as opposed to the more complicated
 
\series bold
two-fluid
\series default
 approach (see e.g.
 book of Freidberg for more information).
\end_layout

\begin_layout Standard
Taking a sum of Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:falpha-2-1-1"

\end_inset

 over 
\begin_inset Formula $\alpha$
\end_inset

 weighted by charges 
\begin_inset Formula $e_{\alpha}$
\end_inset

, we obtain 
\series bold
charge continuity
\series default

\begin_inset Formula 
\begin{equation}
\frac{\partial\rho(\v r,t)}{\partial t}+\nabla\cdot\v J(\v r,t)=0,\label{eq:falpha-2-1-1-2}
\end{equation}

\end_inset

where the charge density in the partial time derivative 
\begin_inset Formula 
\begin{equation}
\rho(\v r,t)\equiv\sum_{\alpha}e_{\alpha}n_{\alpha}(\v r,t)
\end{equation}

\end_inset

effectively vanishes due to quasineutrality, and the 
\series bold
single-fluid current density
\series default

\begin_inset Formula 
\begin{equation}
\v J(\v r,t)\equiv\sum_{\alpha}e_{\alpha}\v{\Gamma}_{n\alpha}(\v r,t)=\sum_{\alpha}e_{\alpha}n_{\alpha}(\v r,t)\v V_{\alpha}(\v r,t)
\end{equation}

\end_inset

therefore becomes divergence-free.
 Since dynamics of electrons are usually faster than the one of ions due
 to their mass difference, 
\begin_inset Formula $\v j$
\end_inset

 will be mainly governed by electron motion 
\begin_inset Formula $\v V_{e}$
\end_inset

.
\end_layout

\begin_layout Standard
Weighting by 
\begin_inset Formula $m_{\alpha}$
\end_inset

 instead we obtain 
\series bold
single-fluid mass continuity
\series default

\begin_inset Formula 
\begin{equation}
\frac{\partial\rho_{m}(\v r,t)}{\partial t}+\nabla\cdot\v{\Gamma}_{m}(\v r,t)=0,\label{eq:falpha-2-1-1-2-1}
\end{equation}

\end_inset

where the mass density is
\begin_inset Formula 
\begin{equation}
\rho_{m}(\v r,t)\equiv\sum_{\alpha}m_{\alpha}n_{\alpha}(\v r,t).
\end{equation}

\end_inset

Since electrons are much lighter than ions, this is mainly governed by the
 ion mass, 
\begin_inset Formula $\rho_{m}\approx\rho_{mi}$
\end_inset

.
 The mass flux
\begin_inset Formula 
\begin{align}
\v{\Gamma}_{m}(\v r,t) & =\sum_{\alpha}\v{\Gamma}_{m\alpha}(\v r,t)\equiv\sum_{\alpha}m_{\alpha}\v{\Gamma}_{n\alpha}(\v r,t)\nonumber \\
 & =\sum_{\alpha}m_{\alpha}n_{\alpha}(\v r,t)\v V_{\alpha}(\v r,t)=\rho_{m}(\v r,t)\v V(\v r,t).
\end{align}

\end_inset

has the dimension of a momentum density (see concepts).
 This allows us to define the 
\begin_inset Quotes eld
\end_inset

effective
\begin_inset Quotes erd
\end_inset

 
\series bold
single-fluid velocity 
\begin_inset Formula 
\begin{equation}
\v V(\v r,t)\equiv\frac{\sum_{\alpha}m_{\alpha}n_{\alpha}(\v r,t)\v V_{\alpha}(\v r,t)}{\rho_{m}(\v r,t)}.\label{eq:vel}
\end{equation}

\end_inset


\series default
This fluid velocity is the one responsible for mass transport and is a combinati
on of ion and electron velocity due to 
\begin_inset Formula $m_{i}\gg m_{e}$
\end_inset

 but 
\begin_inset Formula $|\v V_{e}|\gg|\v V_{i}|$
\end_inset

.
 Be careful that the 
\series bold
single-fluid MHD
\series default
 
\series bold
pressure tensor
\series default
 
\begin_inset Formula $\t P$
\end_inset

 is now defined via the deviation with respect to 
\begin_inset Formula $\v V$
\end_inset

 rather than 
\begin_inset Formula $\v V_{\alpha}$
\end_inset

, so
\begin_inset Formula 
\begin{equation}
\t P\equiv\sum_{\alpha}m_{\alpha}N_{\alpha}\int\d^{3}v(\v v-\v V)(\v v-\v V)f_{\alpha}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Performing the same derivation as for 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:moma"

\end_inset

 via second moments, but now using this 
\begin_inset Formula $\t P$
\end_inset

 and taking a sum over all species 
\begin_inset Formula $\alpha$
\end_inset

 yields the 
\series bold
conservative single-fluid momentum law
\series default

\begin_inset Formula 
\begin{equation}
\frac{\partial(\rho_{m}\v V)}{\partial t}+\nabla\cdot(\rho_{m}\v V\v V)=-\nabla\cdot\t P+\rho\v E+\frac{1}{c}\v J\times\v B,\label{eq:momentum}
\end{equation}

\end_inset

where the collisional term on the right-hand side has vanished and the single-fl
uid velocity 
\begin_inset Formula $\v V$
\end_inset

 enters via its definition in Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:vel"

\end_inset

.
 Note that now the total current 
\begin_inset Formula $\v J$
\end_inset

 appears in the Lorentz force term, but it is independent from the flow
 velocity 
\begin_inset Formula $\v V$
\end_inset

, since 
\begin_inset Formula $\v J$
\end_inset

 is a difference and 
\begin_inset Formula $\v V$
\end_inset

 is a weighted sum over electron and ion velocities 
\begin_inset Formula $\v V_{\alpha}$
\end_inset

.
 As mentioned before the electrostatic force 
\begin_inset Formula $\rho_{e}\v E$
\end_inset

 is usually negligible due to quasineutrality.
\end_layout

\begin_layout Standard
Often the left-hand side of Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:momentum"

\end_inset

 is modified with help of mass continuity 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:falpha-2-1-1-2-1"

\end_inset

.
 Namely, using Leibniz' product rule and the law for the divergence of dyads
 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:diaddiv"
plural "false"
caps "false"
noprefix "false"

\end_inset

 for 
\begin_inset Formula $\nabla\cdot((\rho_{m}\v V)\v V)$
\end_inset

 we obtain
\begin_inset Formula 
\begin{align}
\frac{\partial(\rho_{m}\v V)}{\partial t}+\nabla\cdot(\rho_{m}\v V\v V) & =\frac{\partial\rho_{m}}{\partial t}\v V+\rho_{m}\frac{\partial\v V}{\partial t}+(\rho_{m}\v V\cdot\nabla)\v V+(\nabla\cdot(\rho_{m}\v V))\v V\nonumber \\
 & =\rho_{m}\frac{\partial\v V}{\partial t}+\rho_{m}\v V\cdot\nabla\v V+\left(\frac{\partial\rho_{m}}{\partial t}+\nabla\cdot(\rho_{m}\v V)\right)\v V.
\end{align}

\end_inset

The term in brackets vanishes due to Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:falpha-2-1-1-2-1"

\end_inset

 and we are left with the 
\series bold
convective single-fluid momentum law
\series default

\begin_inset Formula 
\begin{equation}
\rho_{m}\frac{\partial\v V}{\partial t}+\rho_{m}\v V\cdot\nabla\v V=-\nabla\cdot\t P+\rho_{e}\v E+\frac{1}{c}\v J\times\v B.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
\paragraph_spacing other 0.9
\noindent
This corresponds to 
\series bold
Euler's equation
\series default
 of inviscious fluid mechanics with a scalar perssure if 
\begin_inset Formula $\t P$
\end_inset

 if it is a diagonal tensor field,
\begin_inset Formula 
\begin{equation}
\t P=\left(\begin{array}{ccc}
p & 0 & 0\\
0 & p & 0\\
0 & 0 & p
\end{array}\right)=p\t I
\end{equation}

\end_inset

then 
\begin_inset Formula $\nabla\cdot\t P=\nabla p$
\end_inset

.
 More generally we define the 
\series bold
scalar pressure
\series default
 
\begin_inset Formula $p$
\end_inset

 as the isotropic part of 
\begin_inset Formula $\t P$
\end_inset

 over the trace
\begin_inset Formula 
\begin{equation}
p\equiv\frac{1}{3}\mathrm{Tr}\,\t P.
\end{equation}

\end_inset

The rest we call the 
\series bold
anistropic part of the pressure tensor
\series default

\begin_inset Formula 
\begin{equation}
\v{\Pi}\equiv\t P-p\t I,
\end{equation}

\end_inset

which is responsible for viscosity.
 It can e.g.
 arise from gyration (
\series bold
gyroviscosity
\series default
), but for the remaining lecture we set 
\begin_inset Formula $\v{\Pi}=0$
\end_inset

, as it is often done in practice.
 Instead of going on to equations for higher moments above the second moment
 
\begin_inset Formula $p$
\end_inset

, we make the ad-hoc assumption that
\begin_inset Formula 
\begin{align}
\frac{\d}{\d t}\left(\frac{p}{\rho_{m}^{\,\gamma}}\right) & =0,
\end{align}

\end_inset

being the 
\series bold
adiabatic equation
\series default
.

\series bold
 
\series default
Take 
\begin_inset Formula $\gamma=5/3$
\end_inset

 for a monoatomic gas.
 
\end_layout

\begin_layout Standard
This is justified if the process of interest happens much faster than heat
 can flow in the system which is usually the case for waves and instabilities.
 One can see this approximation as anaalogue to the starting point to derive
 acoustic waves, but with more wave types, including electromagnetic interaction.
 A more complete derivation is done via the energy flux law and its limiting
 cases (see e.g.
 the book 
\begin_inset Quotes eld
\end_inset

Fundamentals of Plasma Physics
\begin_inset Quotes erd
\end_inset

 by Paul Bellan for a good explanation).
 The opposite to the adiabatic law is the isothermal condition that can
 be used for processes much slower than the time-scale of heat condution
 across the system.
\end_layout

\begin_layout Section
Plasma Electrodynamics
\end_layout

\begin_layout Subsection*
6.1 The Vlasov-Maxwell-system
\end_layout

\begin_layout Standard
Before we continue entirely in the magnetohydrodynamic/fluid picture we
 look at how one would generally derive a 
\series bold
plasma response
\series default
 to an externally imposed electromagnetic field including multi-species
 collisionless kinetic effects.
 This is known as the 
\series bold
Vlasov-Maxwell system
\series default
.
 It is given by the following equations,
\end_layout

\begin_layout Standard
\paragraph_spacing other 0.3
\noindent
\begin_inset Formula 
\begin{align}
\text{Vlasov:}\qquad\frac{\partial f_{\alpha}}{\partial t}+\boldsymbol{v}\cdot\nabla_{\boldsymbol{r}}f_{\alpha}+\frac{e_{\alpha}}{m_{\alpha}}(\v E+\frac{1}{c}\v v\times\v B)\cdot\nabla_{\boldsymbol{v}}f_{\alpha} & =0.\label{eq:falpha-2-1-2-1-1-1}
\end{align}

\end_inset


\begin_inset Formula 
\begin{align}
\text{Induced charge density:}\qquad\rho^{\mathrm{ind}} & =\sum_{\alpha}e_{\alpha}n_{\alpha},\\
\text{Induced current density:}\qquad\boldsymbol{J}^{\mathrm{ind}} & =\sum_{\alpha}e_{\alpha}n_{\alpha}\v V_{\alpha}.
\end{align}

\end_inset


\begin_inset Formula 
\begin{align}
\text{Gauss \textbf{E}:}\qquad\nabla\cdot\v E & =4\pi(\rho^{\mathrm{ind}}+\rho^{\mathrm{ext}}),\\
\text{Ampère/Maxwell:}\qquad\nabla\times\v B & =\frac{4\pi}{c}(\boldsymbol{J}^{\mathrm{ind}}+\v J^{\mathrm{ext}})+\frac{1}{c}\frac{\partial\v E}{\partial t},\\
\text{Faraday:}\qquad\nabla\times\v E & =-\frac{1}{c}\frac{\partial\v B}{\partial t},\\
\text{Gauss \textbf{B}:}\qquad\nabla\cdot\v B & =0.
\end{align}

\end_inset


\end_layout

\begin_layout Standard
\noindent
An interesting detail is the question of the continuity equation.
 From taking moments of the Vlasov equation we know that continuity holds
 for induced charges and currents with.
\begin_inset Formula 
\begin{equation}
\frac{\partial\rho^{\mathrm{ind}}}{\partial t}+\nabla\cdot\boldsymbol{J}^{\mathrm{ind}}=0.
\end{equation}

\end_inset

Continuity of total 
\begin_inset Formula $\rho=\rho^{\mathrm{ind}}+\rho^{\mathrm{ext}}$
\end_inset

 and 
\begin_inset Formula $\v J=\boldsymbol{J}^{\mathrm{ind}}+\v J^{\mathrm{ext}}$
\end_inset

 is 
\emph on
required
\emph default
 in order to be compatible with Maxwell (combine Gauss E with 
\begin_inset Formula $\nabla\cdot$
\end_inset

Amp/Max), so
\begin_inset Formula 
\begin{equation}
\frac{\partial\rho}{\partial t}+\nabla\cdot\boldsymbol{J}=0.
\end{equation}

\end_inset

As a consequence, there 
\emph on
must
\emph default
 be continuity of external charges and currents,
\begin_inset Formula 
\begin{equation}
\frac{\partial\rho^{\mathrm{ext}}}{\partial t}+\nabla\cdot\boldsymbol{J}^{\mathrm{ext}}=0,
\end{equation}

\end_inset

if we want to call them as such.
 This is clearly fulfilled if also 
\begin_inset Formula $\rho^{\mathrm{ext}}$
\end_inset

 and 
\begin_inset Formula $\boldsymbol{J}^{\mathrm{ext}}$
\end_inset

 ultimately come from particles carrying charges and currents.
 The deeper conceptual fact is though that Maxwell's equations couldn't
 be fulfilled by a more 
\begin_inset Quotes eld
\end_inset

unnatural
\begin_inset Quotes erd
\end_inset

 type of source fields.
 One could say that the continuity equation in 
\begin_inset Formula $\rho$
\end_inset

 and 
\begin_inset Formula $\v J$
\end_inset

 is a compatibility condition for Maxwell's equations and can (must) always
 be assumed together with them.
\end_layout

\begin_layout Standard
As a side note, if one takes the 
\series bold
stationary
\series default
 limit, Maxwell's equations reduce to
\begin_inset Formula 
\begin{align}
\text{Gauss \textbf{E}:}\qquad\nabla\cdot\v E & =4\pi\rho,\\
\text{Ampère/Maxwell:}\qquad\nabla\times\v B & =\frac{4\pi}{c}\boldsymbol{J},\\
\text{Faraday:}\qquad\nabla\times\v E & =0,\\
\text{Gauss \textbf{B}:}\qquad\nabla\cdot\v B & =0.
\end{align}

\end_inset

In that case, the continuity condition is reduced to divergence-freeness
 of currents,
\begin_inset Formula 
\begin{equation}
\nabla\cdot\boldsymbol{J}=0.
\end{equation}

\end_inset

For numerical approximations one must choose an adequate discrete basis
 (
\begin_inset Formula $H^{\mathrm{div}}$
\end_inset

 or Raviart-Thomas Elements) to correctly represent 
\begin_inset Formula $\v J$
\end_inset

 for this purpose – it is not enough to just discretize each vector component
 individually.
\end_layout

\begin_layout Subsection*
6.2 Linear plasma response
\end_layout

\begin_layout Standard
The 
\series bold
goal
\series default
 of this section is to derive the 
\series bold
response fields 
\series default

\begin_inset Formula $\v E$
\end_inset

 and 
\begin_inset Formula $\v B$
\end_inset

 inside a plasma subject to externally imposed 
\series bold
source fields 
\series default

\begin_inset Formula $\rho^{\mathrm{ext}},\v J^{\mathrm{ext}}$
\end_inset

, taking also the reaction of induced charges and currents 
\begin_inset Formula $\rho^{\mathrm{ind}},\boldsymbol{J}^{\mathrm{ind}}$
\end_inset

 into account.
\end_layout

\begin_layout Standard
Our 
\series bold
strategy
\series default
 will be to relate 
\begin_inset Formula $\rho^{\mathrm{ind}},\boldsymbol{J}^{\mathrm{ind}}$
\end_inset

 to 
\begin_inset Formula $\v E$
\end_inset

 and 
\begin_inset Formula $\v B$
\end_inset

 and move the resulting expressions to the left-hand side of Maxwell's equations
, leaving only 
\begin_inset Formula $\rho^{\mathrm{ext}},\v J^{\mathrm{ext}}$
\end_inset

 as source terms.
 This is similar to splitting induced and free charges in the classical
 derivation of Maxwell's equation in a material with permittivity 
\begin_inset Formula $\varepsilon$
\end_inset

 and permeability 
\begin_inset Formula $\mu$
\end_inset

.
 To relate 
\begin_inset Formula $\boldsymbol{J}^{\mathrm{ind}}$
\end_inset

 to 
\begin_inset Formula $\rho^{\mathrm{ind}}$
\end_inset

 we already know that we can use 
\series bold
charge continuity
\series default
.
 A relation between 
\begin_inset Formula $\boldsymbol{J}^{\mathrm{ind}}$
\end_inset

 and 
\begin_inset Formula $\v E$
\end_inset

 can be made without specific assumptions via a 
\series bold
generalized Ohm's law
\series default
.
\end_layout

\begin_layout Standard
The main 
\series bold
assumption 
\series default
we make is that 
\begin_inset Formula $\v E$
\end_inset

 and 
\begin_inset Formula $\v B$
\end_inset

 of the plasma are linearly related to 
\begin_inset Formula $\rho^{\mathrm{ind}},\boldsymbol{J}^{\mathrm{ind}}$
\end_inset

 such that we can assume a generalized Ohm's law at the first place.
 This would have to be justified from analyzing the solution of the plasma
 kinetic (Vlasov) equation in the given circumstances, which is not a trivial
 task.
 As a consequence, since Maxwell's equations are linear, we also obtain
 a linear response in 
\begin_inset Formula $\rho^{\mathrm{ext}},\v J^{\mathrm{ext}}$
\end_inset

.
 To obtain an explicit solution, we have to make the assumption
\series bold
\emph on
 
\series default
\emph default
that our plasma can be treated as an infinite and homogeneous system.
 In that case, we can work in the frequency/wavenumber space where differential
 operators and convolution integrals turn into simple products.
\end_layout

\begin_layout Subsubsection*
Generalized Ohm's law and plasma response
\end_layout

\begin_layout Standard
We would like to generalize Ohm's law with conductivity 
\begin_inset Formula $\sigma$
\end_inset

,
\begin_inset Formula 
\begin{equation}
\boldsymbol{J}^{\mathrm{ind}}=\sigma\v E.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The response of a medium (here: plasma) is said to be 
\series bold
linear
\series default
 if it follows 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\boldsymbol{J}^{\mathrm{ind}}(\v r,t)=\int_{t_{0}}^{t}\d t^{\prime}\int_{\Omega}\d^{3}r^{\prime}\,\v{\sigma}(\v r,t,\v r^{\prime},t^{\prime})\cdot\v E(\v r^{\prime},t^{\prime})\equiv\hat{\v{\sigma}}\v E.\label{eq:Jind}
\end{equation}

\end_inset

with a rank-2 
\series bold
conductivity tensor
\series default
 
\begin_inset Formula $\v{\sigma}(t,\tau,\v r,\v{\xi})$
\end_inset

 that may introduce 
\series bold
anisotropy
\series default
.
 In the short-hand notation the law is a 
\series bold
linear
\series default
 integral operator 
\begin_inset Formula 
\begin{equation}
\hat{\v{\sigma}}\equiv\int_{t_{0}}^{t}\d t^{\prime}\int_{\Omega}\d^{3}r^{\prime}\,\v{\sigma}(\v r,t,\v r^{\prime},t^{\prime})\cdot
\end{equation}

\end_inset

acting on 
\begin_inset Formula $\v E$
\end_inset

.
 Linearity means that
\begin_inset Formula 
\begin{equation}
\hat{\v{\sigma}}(\alpha\v E_{1}+\beta\v E_{2})=\alpha\hat{\v{\sigma}}\v E_{1}+\beta\hat{\v{\sigma}}\v E_{2},
\end{equation}

\end_inset

where 
\begin_inset Formula $\alpha$
\end_inset

 and 
\begin_inset Formula $\beta$
\end_inset

 are constants.
 The response is 
\series bold
non-local 
\series default
in space and time since it relates 
\series bold
response
\series default
 current 
\begin_inset Formula $\boldsymbol{J}^{\mathrm{ind}}(\v r,t)$
\end_inset

 to 
\series bold
excitation 
\series default
field 
\begin_inset Formula $\v E(\v r^{\prime},t^{\prime})$
\end_inset

 at any other point 
\begin_inset Formula $(\v r^{\prime},t^{\prime})$
\end_inset

 in space/time.
 It is 
\series bold
causal
\series default
 since integration over 
\begin_inset Formula $t^{\prime}$
\end_inset

 goes until 
\begin_inset Formula $t$
\end_inset

, but not in the future.
 Formally we can relate 
\begin_inset Formula $\boldsymbol{J}^{\mathrm{ind}}$
\end_inset

 and 
\begin_inset Formula $\rho^{\mathrm{ind}}$
\end_inset

 via continuity
\begin_inset Formula 
\begin{align}
\rho^{\mathrm{ind}}(\v r,t) & =\rho^{\mathrm{ind}}(\v r,t_{0})-\int_{t_{0}}^{t}\nabla\cdot\boldsymbol{J}^{\mathrm{ind}}(\v r,t)\,\d t^{\prime}\nonumber \\
 & =\rho^{\mathrm{ind}}(\v r,t_{0})-\nabla\cdot\int_{t_{0}}^{t}\boldsymbol{J}^{\mathrm{ind}}(\v r,t)\,\d t^{\prime}\nonumber \\
 & \equiv\rho^{\mathrm{ind}}(\v r,t_{0})+\nabla\cdot(\hat{\rho}\boldsymbol{J}^{\mathrm{ind}})=\rho^{\mathrm{ind}}(\v r,t_{0})+\nabla\cdot(\hat{\rho}\hat{\v{\sigma}}\v E).
\end{align}

\end_inset

We can assume 
\begin_inset Formula $\rho^{\mathrm{ind}}(\v r,t_{0})=0$
\end_inset

 because perturbation is switched off before a certain time 
\begin_inset Formula $t_{0}$
\end_inset

.
 Then 
\begin_inset Formula $\rho^{\mathrm{ind}}$
\end_inset

 is again given by a linear operator
\begin_inset Formula 
\begin{equation}
\rho^{\mathrm{ind}}(\v r,t)=\nabla\cdot\hat{\rho}\hat{\v{\sigma}}\v E
\end{equation}

\end_inset

acting on 
\begin_inset Formula $\v E$
\end_inset

 by combining divergence 
\begin_inset Formula $\nabla\cdot$
\end_inset

, charge accumulation 
\begin_inset Formula $\hat{\rho}$
\end_inset

 and conductivity 
\begin_inset Formula $\hat{\v{\sigma}}$
\end_inset

.
 Inserting these relations into the first two Maxwell equations yields
\begin_inset Formula 
\begin{align}
\text{Gauss \textbf{E}:}\qquad\nabla\cdot(\t I-4\pi\hat{\rho}\hat{\v{\sigma}})\v E & =4\pi\rho^{\mathrm{ext}},\\
\text{Ampère/Maxwell:}\qquad\nabla\times\v B-\frac{1}{c}\left(\frac{\partial}{\partial t}+4\pi\hat{\v{\sigma}}\right)\v E & =\frac{4\pi}{c}\v J^{\mathrm{ext}}.
\end{align}

\end_inset

To obtain a generalized Gauss 
\series bold
E
\series default
 law 
\begin_inset Formula $\nabla\cdot(\hat{\v{\varepsilon}}\v E)=4\pi\rho^{\mathrm{ext}}$
\end_inset

 for plasma seen as a dielectric materials we can define a tensorial permittivit
y operator
\begin_inset Formula 
\begin{equation}
\hat{\v{\varepsilon}}\equiv\t I-4\pi\hat{\rho}\hat{\v{\sigma}}.\label{eq:permi}
\end{equation}

\end_inset

Taking a time derivative of 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
Ampère/Maxwell and inserting Faraday for 
\begin_inset Formula $\frac{\partial\v B}{\partial t}$
\end_inset

 yields
\begin_inset Formula 
\begin{equation}
-c^{2}\nabla\times(\nabla\times\v E)-\left(\frac{\partial^{2}}{\partial t^{2}}+4\pi\frac{\partial}{\partial t}\hat{\v{\sigma}}\right)\v E=4\pi\frac{\partial}{\partial t}\v J^{\mathrm{ext}}.
\end{equation}

\end_inset

Combining everything on the left-hand side to a single operator yields our
 final system
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\xout default
\uuline default
\uwave default
\noun default
\color inherit

\begin_inset Formula 
\begin{align}
\nabla\cdot(\t I-4\pi\hat{\rho}\hat{\v{\sigma}})\v E & =4\pi\rho^{\mathrm{ext}},\label{eq:gauss}\\
\left(\frac{\partial^{2}}{\partial t^{2}}+c^{2}\nabla\times\nabla\times+4\pi\frac{\partial}{\partial t}\hat{\v{\sigma}}\right)\v E & =-4\pi\frac{\partial}{\partial t}\v J^{\mathrm{ext}}.\label{eq:wave}
\end{align}

\end_inset

Compare this result to the derivation of electromagnetic waves in a homogeneous,
 charge- and current-free vacuum where 
\begin_inset Formula $\rho^{\mathrm{ext}}=0$
\end_inset

, 
\begin_inset Formula $\v J^{\mathrm{ext}}=0$
\end_inset

,
\begin_inset Formula $\hat{\v{\sigma}}=0$
\end_inset

.
 There we obtain divergence-free 
\begin_inset Formula $\v E$
\end_inset

 and the electromagnetic wave equation,
\begin_inset Formula 
\begin{align}
\nabla\cdot\v E & =0,\\
\left(\frac{\partial^{2}}{\partial t^{2}}+c^{2}\nabla\times\nabla\times\right)\v E & =0.
\end{align}

\end_inset

There the curl-curl operator is usually replaced by a vector Laplacian defined
 as 
\begin_inset Formula $\Delta\v E\equiv\nabla(\nabla\cdot\v E)-\nabla\times(\nabla\times\v E)$
\end_inset

, since the first term vanishes in vacuum due to charge-freeness in Gauss
 
\series bold
E
\series default
.
 In the plasma, even for vanishing excitations 
\begin_inset Formula $\rho^{\mathrm{ext}}$
\end_inset

 and 
\begin_inset Formula $\v J^{\mathrm{ext}}$
\end_inset

 this is not necessarily the case due to the additional plasma response
 terms
\begin_inset Formula 
\begin{equation}
-4\pi\hat{\rho}\hat{\v{\sigma}},\quad\text{and}\quad4\pi\frac{\partial}{\partial t}\hat{\v{\sigma}}.
\end{equation}

\end_inset

Those will modify the usual electromagnetic waves depending on the plasma
 conductivity encoded inside 
\begin_inset Formula $\hat{\v{\sigma}}$
\end_inset

.
 Such linear response laws with modified electromagnetic wave equations
 are also applicable to other materials besides plasma.
\end_layout

\begin_layout Subsubsection*
Solution in Fourier space for a homogeneous plasma
\end_layout

\begin_layout Standard
If our plasma is infinite and homogeneous, there can be no material parameters
 with dependencies on 
\emph on
absolute
\emph default
 
\begin_inset Formula $\v r,t$
\end_inset

 but only 
\emph on
relative
\emph default
 distances 
\begin_inset Formula $(\v r-\v r^{\prime},t-t^{\prime})$
\end_inset

 we can write Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Jind"

\end_inset

 as
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\boldsymbol{J}^{\mathrm{ind}}(\v r,t)=\int_{t_{0}}^{t}\d t^{\prime}\int_{\mathbb{R}^{3}}\d^{3}r^{\prime}\,\v{\sigma}(\v r-\v r^{\prime},t-t^{\prime})\cdot\v E(\v r^{\prime},t^{\prime}).\label{eq:Jconv}
\end{equation}

\end_inset

This is a 
\series bold
convolution
\series default
 integral.
 In case we have an infinite homogeneous space we can 
\series bold
Fourier transform 
\series default
where quantities are set up as 
\begin_inset Formula $f(\v r,t)\propto f_{\omega\v k}e^{i(\v k\cdot\v r-\omega t)}$
\end_inset

 (watch the minus, if something is finite and periodic, use a Fourier series
 instead).
 This makes life easier with products instead of differential operators,
 
\begin_inset Formula 
\begin{align}
\frac{\partial}{\partial t}\, & \rightarrow\,-i\omega,\qquad\nabla\,\rightarrow\,i\v k,\\
\nabla\cdot\, & \rightarrow\,i\v k\cdot,\qquad\nabla\times\,\rightarrow\,i\v k\times.
\end{align}

\end_inset

Convolution integrals transform like
\begin_inset Formula 
\begin{equation}
\int_{-\infty}^{\infty}\d t^{\prime}\int_{\mathbb{R}^{3}}f(\v r-\v r^{\prime},t-t^{\prime})g(\v r^{\prime},t^{\prime})\quad\rightarrow\quad f_{\omega\v k}g_{\omega\v k}.
\end{equation}

\end_inset

Looking more carefully at Ohm's law 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Jconv"

\end_inset

, we should treat causality correctly in time, which can start at 
\begin_inset Formula $t_{0}=-\infty$
\end_inset

 but should end at 
\begin_inset Formula $t$
\end_inset

 and not 
\begin_inset Formula $+\infty$
\end_inset

.
 Actually we need a 
\emph on
half-sided
\emph default
 Fourier transform or 
\emph on
complex Laplace transform
\emph default
.
 This concerns mainly the back-transformation but all other rules for Fourier
 transforms that we use remain the same.
 This is why we ignore this peculiarity in the frequency domain for now,
 but should keep it in mind for later (
\begin_inset Formula $\rightarrow$
\end_inset

 Landau damping).
 The generalized non-local 
\series bold
Ohm's law
\series default
 for a homogeneous (not necessarily isotropic) medium in 
\begin_inset Formula $\omega\v k$
\end_inset

 space thus becomes
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\boldsymbol{J}_{\omega\v k}^{\mathrm{ind}}=\v{\sigma}_{\omega\v k}\cdot\v E_{\omega\v k}.\label{eq:Jconv-1}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard

\series bold
Continuity
\series default
 becomes
\end_layout

\begin_layout Standard
\noindent
\begin_inset Formula 
\begin{equation}
i\omega\rho_{\omega\v k}^{\mathrm{ind}}=i\v k\cdot\boldsymbol{J}_{\omega\v k}^{\mathrm{ind}}.\label{eq:conti-2}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard

\series bold
Maxwell's equations 
\series default
become (leaving out 
\begin_inset Formula $\omega\v k$
\end_inset

 subscripts for readiliby)
\begin_inset Formula 
\begin{align}
\text{Gauss \textbf{E}:}\qquad i\v k\cdot\v E & -4\pi\rho^{\mathrm{ind}}=4\pi\rho^{\mathrm{ext}},\\
\text{Ampère/Maxwell:}\qquad i\v k\times\v B & -\frac{4\pi}{c}\boldsymbol{J}^{\mathrm{ind}}+\frac{i\omega}{c}\v E=\frac{4\pi}{c}\v J^{\mathrm{ext}},\\
\text{Faraday:}\qquad i\v k\times\v E & =\frac{i\omega}{c}\v B,\\
\text{Gauss \textbf{B}:}\qquad i\v k\cdot\v B & =0.
\end{align}

\end_inset

In 
\begin_inset Formula $\omega\v k$
\end_inset

 space Eqs.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:gauss"

\end_inset

-
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:wave"

\end_inset

) can either be derived from those results or directly transformed to become
 the 
\series bold
linear plasma response in frequency space
\series default

\begin_inset Formula 
\begin{align}
i\v k\cdot(\t I-\frac{4\pi}{i\omega}\v{\sigma}\cdot)\v E & =4\pi\rho^{\mathrm{ext}},\\
i\omega\left(1+\frac{c^{2}}{\omega^{2}}\v k\times\v k\times-\frac{4\pi}{i\omega}\v{\sigma}\cdot\right)\v E & =4\pi\v J^{\mathrm{ext}}.
\end{align}

\end_inset

The second equation being a wave equation in 
\begin_inset Formula $\omega\v k$
\end_inset

 space is often written in a different form where the double cross-product
 is replaced by
\begin_inset Formula 
\begin{equation}
\v k\times\v k\times\v E=\v{kk}\cdot\v E-k^{2}\v E.
\end{equation}

\end_inset

Defining further the 
\series bold
wave vector
\series default

\begin_inset Formula 
\begin{equation}
\v N\equiv\frac{c\v k}{\omega}
\end{equation}

\end_inset

we write the 
\series bold
modified EM wave equation
\series default

\begin_inset Formula 
\begin{equation}
i\omega\left(\v{NN}-N^{2}\t I+\t I-\frac{4\pi}{i\omega}\v{\sigma}\right)\cdot\v E=4\pi\v J^{\mathrm{ext}}.\label{eq:wave-1}
\end{equation}

\end_inset

Transforming the permittivity operator
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:permi"

\end_inset

 into 
\begin_inset Formula $\omega\v k$
\end_inset

 space we obtain the 
\series bold
dielectric tensor
\series default

\begin_inset Formula 
\begin{equation}
\v{\varepsilon}\equiv\t I-\frac{4\pi}{i\omega}\v{\sigma}.
\end{equation}

\end_inset

The total expression in front of 
\begin_inset Formula $\v E$
\end_inset

 in the wave equation called the 
\series bold
Maxwell tensor
\series default

\begin_inset Formula 
\begin{equation}
\v{\Lambda}\equiv\v{NN}-N^{2}\t I+\v{\varepsilon}
\end{equation}

\end_inset

We call modes that follow purely from Gauss 
\series bold
E 
\series default
in the plasma the 
\series bold
longitudinal modes 
\series default

\begin_inset Formula 
\begin{equation}
i\v k\cdot\v{\varepsilon}\cdot\v E=4\pi\rho^{\mathrm{ext}}.
\end{equation}

\end_inset

Those from the modified EM wave equation are the 
\series bold
transverse modes
\series default

\begin_inset Formula 
\begin{equation}
i\omega\v{\Lambda}\cdot\v E=4\pi\v J^{\mathrm{ext}}.
\end{equation}

\end_inset

In a region without sources the problem reduces to an eigenvalue problem
 of the 
\series bold
dispersion equation
\series default
 
\begin_inset Formula 
\begin{equation}
\mathrm{det}\v{\Lambda}=0.
\end{equation}

\end_inset

Since 
\begin_inset Formula $\v{\Lambda}$
\end_inset

 depends on both, 
\begin_inset Formula $\omega$
\end_inset

 and 
\begin_inset Formula $\v k$
\end_inset

, this gives a dispersion relation
\begin_inset Formula 
\begin{equation}
\omega=\omega^{\sigma}(\v k)
\end{equation}

\end_inset

between the two for each mode 
\begin_inset Formula $\sigma$
\end_inset

, determining the 
\series bold
polarization
\series default
 of the mode.
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "writeups"
options "plainnat"

\end_inset


\end_layout

\end_body
\end_document