From 3170fc4b5c915669cf209a521e551115a9bd0809 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 20 Oct 2021 11:50:20 +0200
Subject: Expand knowledge base

---
 .../know/concept/archimedes-principle/index.pdc    |   6 +-
 content/know/concept/boltzmann-relation/index.pdc  |  96 +++++++++++
 content/know/concept/coulomb-logarithm/index.pdc   |   9 +-
 content/know/concept/debye-length/index.pdc        | 156 ++++++++++++++++++
 content/know/concept/euler-equations/index.pdc     |   6 +-
 .../know/concept/hydrostatic-pressure/index.pdc    |   2 +-
 content/know/concept/lindhard-function/index.pdc   |   7 +-
 .../know/concept/maxwell-bloch-equations/index.pdc |  15 +-
 content/know/concept/maxwells-equations/index.pdc  |  54 +++++++
 .../know/concept/rutherford-scattering/index.pdc   |   3 +-
 content/know/concept/two-fluid-equations/index.pdc | 178 +++++++++++++++++++++
 11 files changed, 517 insertions(+), 15 deletions(-)
 create mode 100644 content/know/concept/boltzmann-relation/index.pdc
 create mode 100644 content/know/concept/debye-length/index.pdc
 create mode 100644 content/know/concept/two-fluid-equations/index.pdc

(limited to 'content/know/concept')

diff --git a/content/know/concept/archimedes-principle/index.pdc b/content/know/concept/archimedes-principle/index.pdc
index 0837cc9..3a063ec 100644
--- a/content/know/concept/archimedes-principle/index.pdc
+++ b/content/know/concept/archimedes-principle/index.pdc
@@ -47,9 +47,9 @@ $$\begin{aligned}
     = - \int_V \nabla p \dd{V}
 \end{aligned}$$
 
-The last step follows from Gauss' theorem.
-We replace $\nabla p$ by assuming
-[hydrostatic equilibrium](/know/concept/hydrostatic-pressure/),
+Where we have used the divergence theorem.
+Assuming [hydrostatic equilibrium](/know/concept/hydrostatic-pressure/),
+we replace $\nabla p$,
 leading to the definition of the **buoyant force**:
 
 $$\begin{aligned}
diff --git a/content/know/concept/boltzmann-relation/index.pdc b/content/know/concept/boltzmann-relation/index.pdc
new file mode 100644
index 0000000..ddaa22f
--- /dev/null
+++ b/content/know/concept/boltzmann-relation/index.pdc
@@ -0,0 +1,96 @@
+---
+title: "Boltzmann relation"
+firstLetter: "B"
+publishDate: 2021-10-18
+categories:
+- Physics
+- Plasma physics
+
+date: 2021-10-18T15:25:39+02:00
+draft: false
+markup: pandoc
+---
+
+# Boltzmann relation
+
+In a plasma where the ions and electrons are both in thermal equilibrium,
+and in the absence of short-lived induced electromagnetic fields,
+their densities $n_i$ and $n_e$ can be predicted.
+
+By definition, a particle in an [electric field](/know/concept/electric-field/) $\vb{E}$
+experiences a [Lorentz force](/know/concept/lorentz-force/) $\vb{F}_e$.
+This corresponds to a force density $\vb{f}_e$,
+such that $\vb{F}_e = \vb{f}_e \dd{V}$.
+For the electrons, we thus have:
+
+$$\begin{aligned}
+    \vb{f}_e
+    = q_e n_e \vb{E}
+    = - q_e n_e \nabla \phi
+\end{aligned}$$
+
+Meanwhile, if we treat the electrons as a gas
+obeying the ideal gas law $p_e = k_B T_e n_e$,
+then the pressure $p_e$ leads to another force density $\vb{f}_p$:
+
+$$\begin{aligned}
+    \vb{f}_p
+    = - \nabla p_e
+    = - k_B T_e \nabla n_e
+\end{aligned}$$
+
+At equilibrium, we demand that $\vb{f}_e = \vb{f}_p$,
+and isolate this equation for $\nabla n_e$, yielding:
+
+$$\begin{aligned}
+    k_B T_e \nabla n_e
+    = q_e n_e \nabla \phi
+    \quad \implies \quad
+    \nabla n_e
+    = \frac{q_e \nabla \phi}{k_B T_e} n_e
+    = \nabla \bigg( \frac{q_e \phi}{k_B T_e} \bigg) n_e
+\end{aligned}$$
+
+This equation is straightforward to integrate,
+leading to the following expression for $n_e$,
+known as the **Boltzmann relation**,
+due to its resemblance to the statistical Boltzmann distribution
+(see [canonical ensemble](/know/concept/canonical-ensemble/)):
+
+$$\begin{aligned}
+    \boxed{
+        n_e(\vb{r})
+        = n_{e0} \exp\!\bigg( \!-\! \frac{q_e \phi(\vb{r})}{k_B T_e} \bigg)
+    }
+\end{aligned}$$
+
+Where the linearity factor $n_{e0}$ represents
+the electron density for $\phi = 0$.
+We can do the same for ions instead of electrons,
+leading to the following ion density $n_i$:
+
+$$\begin{aligned}
+    \boxed{
+        n_i(\vb{r})
+        = n_{i0} \exp\!\bigg( \!-\! \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg)
+    }
+\end{aligned}$$
+
+However, due to their larger mass,
+ions are much slower to respond to fluctuations in the above equilibrium.
+Consequently, after a perturbation,
+the ions spend much more time in a transient non-equilibrium state
+than the electrons, so this formula for $n_i$ is only valid
+if the perturbation is sufficiently slow,
+allowing the ions to keep up.
+Usually, electrons do not suffer the same issue,
+thanks to their small mass and fast response.
+
+
+## References
+1.  P.M. Bellan,
+    *Fundamentals of plasma physics*,
+    1st edition, Cambridge.
+2.  M. Salewski, A.H. Nielsen,
+    *Plasma physics: lecture notes*,
+    2021, unpublished.
diff --git a/content/know/concept/coulomb-logarithm/index.pdc b/content/know/concept/coulomb-logarithm/index.pdc
index 649806b..71b13a8 100644
--- a/content/know/concept/coulomb-logarithm/index.pdc
+++ b/content/know/concept/coulomb-logarithm/index.pdc
@@ -143,8 +143,8 @@ We know that the deflection grows for smaller $b$,
 so it would be reasonable to choose $b_\mathrm{large}$ as the lower limit.
 For very large $b$, the plasma shields the particles from each other,
 thereby nullifying the deflection,
-so as upper limit
-we choose the Debye length $\lambda_D$,
+so as upper limit we choose
+the [Debye length](/know/concept/debye-length/) $\lambda_D$,
 i.e. the plasma's self-shielding length.
 We thus find:
 
@@ -157,12 +157,15 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Here, $\ln\!(\Lambda)$ is known as the **Coulomb logarithm**,
-with $\Lambda$ defined as follows:
+with the **plasma parameter** $\Lambda$ defined below,
+equal to $9/2$ times the number of particles
+in a sphere with radius $\lambda_D$:
 
 $$\begin{aligned}
     \boxed{
         \Lambda
         \equiv \frac{\lambda_D}{b_\mathrm{large}}
+        = 6 \pi n \lambda_D^3
     }
 \end{aligned}$$
 
diff --git a/content/know/concept/debye-length/index.pdc b/content/know/concept/debye-length/index.pdc
new file mode 100644
index 0000000..19d7188
--- /dev/null
+++ b/content/know/concept/debye-length/index.pdc
@@ -0,0 +1,156 @@
+---
+title: "Debye length"
+firstLetter: "D"
+publishDate: 2021-10-18
+categories:
+- Physics
+- Plasma physics
+
+date: 2021-10-15T20:28:31+02:00
+draft: false
+markup: pandoc
+---
+
+# Debye length
+
+If a charged object is put in a plasma,
+it repels like charges and attracts opposite charges,
+leading to a **Debye sheath** around the object's surface
+with a net opposite charge.
+This has the effect of **shielding** the object's presence
+from the rest of the plasma.
+
+We start from [Gauss' law](/know/concept/maxwells-equations/)
+for the [electric field](/know/concept/electric-field/) $\vb{E}$,
+expressing $\vb{E}$ as the gradient of a potential $\phi$,
+i.e. $\vb{E} = -\nabla \phi$,
+and splitting the charge density into ions $n_i$ and electrons $n_e$:
+
+$$\begin{aligned}
+    \nabla^2 \phi(\vb{r})
+    = - \frac{1}{\varepsilon_0} \Big( q_i n_i(\vb{r}) + q_e n_e(\vb{r}) + q_t \delta(\vb{r}) \Big)
+\end{aligned}$$
+
+The last term represents a *test particle*,
+which will be shielded.
+This particle is a point charge $q_t$,
+whose density is simply a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta(\vb{r})$,
+and is not included in $n_i$ or $n_e$.
+
+For a plasma in thermal equilibrium,
+we have the [Boltzmann relations](/know/concept/boltzmann-relation/)
+for the densities:
+
+$$\begin{aligned}
+    n_i(\vb{r})
+    = n_{i0} \exp\!\bigg( \!-\! \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg)
+    \qquad \quad
+    n_e(\vb{r})
+    = n_{e0} \exp\!\bigg( \!-\! \frac{q_e \phi(\vb{r})}{k_B T_e} \bigg)
+\end{aligned}$$
+
+We assume that electrical interactions are weak compared to thermal effects,
+i.e. $k_B T \gg q \phi$ in both cases.
+Then we Taylor-expand the Boltzmann relations to first order:
+
+$$\begin{aligned}
+    n_i(\vb{r})
+    \approx n_{i0} \bigg( 1 - \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg)
+    \qquad \quad
+    n_e(\vb{r})
+    \approx n_{e0} \bigg( 1 - \frac{q_e \phi(\vb{r})}{k_B T_e} \bigg)
+\end{aligned}$$
+
+Inserting this back into Gauss' law,
+we arrive at the following equation for $\phi(\vb{r})$,
+where we have assumed quasi-neutrality such that $q_i n_{i0} = q_e n_{e0}$:
+
+$$\begin{aligned}
+    \nabla^2 \phi
+    &= - \frac{1}{\varepsilon_0}
+    \bigg( q_i n_{i0} - n_{i0} \frac{q_i^2 \phi}{k_B T_i} + q_e n_{e0} - n_{e0} \frac{q_e^2 \phi}{k_B T_e} + q_t \delta(\vb{r}) \bigg)
+    \\
+    &= \bigg( \frac{n_{i0} q_i^2}{\varepsilon_0 k_B T_i} + \frac{n_{e0} q_e^2}{\varepsilon_0 k_B T_e} \bigg) \phi
+    - \frac{q_t}{\varepsilon_0} \delta(\vb{r})
+\end{aligned}$$
+
+We now define the **ion** and **electron Debye lengths**
+$\lambda_{Di}$ and $\lambda_{De}$ as follows:
+
+$$\begin{aligned}
+    \boxed{
+        \frac{1}{\lambda_{Di}^2}
+        \equiv \frac{n_{i0} q_i^2}{\varepsilon_0 k_B T_i}
+    }
+    \qquad \quad
+    \boxed{
+        \frac{1}{\lambda_{De}^2}
+        \equiv \frac{n_{e0} q_e^2}{\varepsilon_0 k_B T_e}
+    }
+\end{aligned}$$
+
+And then the **total Debye length** $\lambda_D$ is defined as the sum of their inverses,
+and gives the rough thickness of the Debye sheath:
+
+$$\begin{aligned}
+    \boxed{
+        \frac{1}{\lambda_D^2}
+        \equiv \frac{1}{\lambda_{Di}^2} + \frac{1}{\lambda_{De}^2}
+        = \frac{n_{i0} q_i^2 T_e + n_{e0} q_e^2 T_i}{\varepsilon_0 k_B T_i T_e}
+    }
+\end{aligned}$$
+
+With this, the equation can be put in the form below,
+suggesting exponential decay:
+
+$$\begin{aligned}
+    \nabla^2 \phi(\vb{r})
+    &= \frac{1}{\lambda_D^2} \phi(\vb{r})
+    - \frac{q_t}{\varepsilon_0} \delta(\vb{r})
+\end{aligned}$$
+
+This has the following solution,
+known as the **Yukawa potential**,
+which decays exponentially,
+representing the plasma's **self-shielding**
+over a characteristic distance $\lambda_D$:
+
+$$\begin{aligned}
+    \boxed{
+        \phi(r)
+        = \frac{q_t}{4 \pi \varepsilon_0 r} \exp\!\Big( \!-\!\frac{r}{\lambda_D} \Big)
+    }
+\end{aligned}$$
+
+Note that $r$ is a scalar,
+i.e. the potential depends only on the radial distance to $q_t$.
+This treatment only makes sense
+if the plasma is sufficiently dense,
+such that there is a large number of particles
+in a sphere with radius $\lambda_D$.
+This corresponds to a large [Coulomb logarithm](/know/concept/coulomb-logarithm/) $\ln\!(\Lambda)$:
+
+$$\begin{aligned}
+    1 \ll \frac{4 \pi}{3} n_0 \lambda_D^3 = \frac{2}{9} \Lambda
+\end{aligned}$$
+
+The name *Yukawa potential* originates from particle physics,
+but can in general be used to refer to any potential (electric or energetic)
+of the following form:
+
+$$\begin{aligned}
+    V(r)
+    = \frac{A}{r} \exp\!(-B r)
+\end{aligned}$$
+
+Where $A$ and $B$ are scaling constants that depend on the problem at hand.
+
+
+
+## References
+1.  P.M. Bellan,
+    *Fundamentals of plasma physics*,
+    1st edition, Cambridge.
+2.  M. Salewski, A.H. Nielsen,
+    *Plasma physics: lecture notes*,
+    2021, unpublished.
diff --git a/content/know/concept/euler-equations/index.pdc b/content/know/concept/euler-equations/index.pdc
index 0088d4f..b531260 100644
--- a/content/know/concept/euler-equations/index.pdc
+++ b/content/know/concept/euler-equations/index.pdc
@@ -57,7 +57,7 @@ Next, we want to find another expression for $\va{f^*}$.
 We know that the overall force $\va{F}$ on an arbitrary volume $V$ of the fluid
 is the sum of the gravity body force $\va{F}_g$,
 and the pressure contact force $\va{F}_p$ on the enclosing surface $S$.
-Using Gauss' theorem, we then find:
+Using the divergence theorem, we then find:
 
 $$\begin{aligned}
     \va{F}
@@ -91,7 +91,7 @@ $$\begin{aligned}
 The last ingredient is **incompressibility**:
 the same volume must simultaneously
 be flowing in and out of an arbitrary enclosure $S$.
-Then, by Gauss' theorem:
+Then, by the divergence theorem:
 
 $$\begin{aligned}
     0
@@ -131,7 +131,7 @@ but the size of their lumps does not change (incompressibility).
 To update the equations, we demand conservation of mass:
 the mass evolution of a volume $V$
 is equal to the mass flow through its boundary $S$.
-Applying Gauss' theorem again:
+Applying the divergence theorem again:
 
 $$\begin{aligned}
     0
diff --git a/content/know/concept/hydrostatic-pressure/index.pdc b/content/know/concept/hydrostatic-pressure/index.pdc
index 001a198..d47d77f 100644
--- a/content/know/concept/hydrostatic-pressure/index.pdc
+++ b/content/know/concept/hydrostatic-pressure/index.pdc
@@ -54,7 +54,7 @@ $$\begin{aligned}
 
 If we now consider a *closed* surface,
 which encloses a "blob" of the fluid,
-then we can use Gauss' theorem to get a volume integral:
+then we can use the divergence theorem to get a volume integral:
 
 $$\begin{aligned}
     \va{F}
diff --git a/content/know/concept/lindhard-function/index.pdc b/content/know/concept/lindhard-function/index.pdc
index d38dc2e..96244c9 100644
--- a/content/know/concept/lindhard-function/index.pdc
+++ b/content/know/concept/lindhard-function/index.pdc
@@ -13,6 +13,11 @@ markup: pandoc
 
 # Lindhard function
 
+The **Lindhard function** describes the response of an electron gas
+to an external perturbation,
+and can be regarded as a quantum-mechanical
+alternative to the [Drude model](/know/concept/drude-model/).
+
 We start from the [Kubo formula](/know/concept/kubo-formula/)
 for the electron density operator $\hat{n}$,
 which describes the change in $\expval{\hat{n}}$
@@ -396,7 +401,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Therefore, by inserting all the above expressions,
-we arrive at the following dielectric function $\varepsilon_r$
+we arrive at the Lindhard dielectric function $\varepsilon_r$
 for a non-interacting electron gas in a uniform potential:
 
 $$\begin{aligned}
diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc
index ae7d119..020a120 100644
--- a/content/know/concept/maxwell-bloch-equations/index.pdc
+++ b/content/know/concept/maxwell-bloch-equations/index.pdc
@@ -393,9 +393,18 @@ It is trivial to show that $\vb{E}$ and $\vb{P}$
 can be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$.
 It is also simple to convert
 the dipole $\vb{p}^{+}$ and inversion $d$
-into their macroscopic versions $\vb{P}^{+}$ and $D$,
-simply by averaging over the atoms per unit of volume.
-We thus arrive at the **Maxwell-Bloch equations**:
+into their macroscopic versions $\vb{P}^{+}$ and $D$:
+
+$$\begin{aligned}
+    \vb{P}^{+}(\vb{r}, t)
+    = \sum_{n} \vb{p}^{+}_n \: \delta(\vb{r} \!-\! \vb{r}_n)
+    \qquad \quad
+    D(\vb{r}, t)
+    = \sum_{n} d_n \: \delta(\vb{r} \!-\! \vb{r}_n)
+\end{aligned}$$
+
+We thus arrive at the **Maxwell-Bloch equations**,
+which are relevant for laser theory:
 
 $$\begin{aligned}
     \boxed{
diff --git a/content/know/concept/maxwells-equations/index.pdc b/content/know/concept/maxwells-equations/index.pdc
index 1551311..967372d 100644
--- a/content/know/concept/maxwells-equations/index.pdc
+++ b/content/know/concept/maxwells-equations/index.pdc
@@ -209,3 +209,57 @@ $$\begin{aligned}
     = \vb{J}_M + \vb{J}_P
     = \nabla \cross \vb{M} + \pdv{\vb{P}}{t}
 \end{aligned}$$
+
+
+## Redundancy of Gauss' laws
+
+In fact, both of Gauss' laws are redundant,
+because they are already implied by Faraday's and Ampère's laws.
+Suppose we take the divergence of Faraday's law:
+
+$$\begin{aligned}
+    0
+    = \nabla \cdot \nabla \cross \vb{E}
+    = - \nabla \cdot \pdv{\vb{B}}{t}
+    = - \pdv{t} (\nabla \cdot \vb{B})
+\end{aligned}$$
+
+Since the divergence of a curl is always zero,
+the right-hand side must vanish.
+We know that $\vb{B}$ can vary in time,
+so our only option to satisfy this is to demand that $\nabla \cdot \vb{B} = 0$.
+We thus arrive arrive at Gauss' law for magnetism from Faraday's law.
+
+The same technique works for Ampère's law.
+Taking its divergence gives us:
+
+$$\begin{aligned}
+    0
+    = \frac{1}{\mu_0} \nabla \cdot \nabla \cross \vb{B}
+    = \nabla \cdot \vb{J} + \varepsilon_0 \pdv{t} (\nabla \cdot \vb{E})
+\end{aligned}$$
+
+We integrate this over an arbitrary volume $V$,
+and apply the divergence theorem:
+
+$$\begin{aligned}
+    0
+    &= \int_V \nabla \cdot \vb{J} \dd{V} + \pdv{t} \int_V \varepsilon_0 \nabla \cdot \vb{E} \dd{V}
+    \\
+    &= \oint_S \vb{J} \cdot \dd{S} + \pdv{t} \int_V \varepsilon_0 \nabla \cdot \vb{E} \dd{V}
+\end{aligned}$$
+
+The first integral represents the current (charge flux)
+through the surface of $V$.
+Electric charge is not created or destroyed,
+so the second integral *must* be the total charge in $V$:
+
+$$\begin{aligned}
+    Q
+    = \int_V \varepsilon_0 \nabla \cdot \vb{E} \dd{V}
+    \quad \implies \quad
+    \nabla \cdot \vb{E}
+    = \frac{\rho}{\varepsilon_0}
+\end{aligned}$$
+
+And we thus arrive at Gauss' law from Ampère's law and charge conservation.
diff --git a/content/know/concept/rutherford-scattering/index.pdc b/content/know/concept/rutherford-scattering/index.pdc
index 81bb133..c89b477 100644
--- a/content/know/concept/rutherford-scattering/index.pdc
+++ b/content/know/concept/rutherford-scattering/index.pdc
@@ -14,7 +14,8 @@ markup: pandoc
 # Rutherford scattering
 
 **Rutherford scattering** or **Coulomb scattering**
-is an elastic pseudo-collision of two electrically charged particles.
+is an [elastic pseudo-collision](/know/concept/elastic-collision/)
+of two electrically charged particles.
 It is not a true collision, and is caused by Coulomb repulsion.
 
 The general idea is illustrated below.
diff --git a/content/know/concept/two-fluid-equations/index.pdc b/content/know/concept/two-fluid-equations/index.pdc
new file mode 100644
index 0000000..cd77f5e
--- /dev/null
+++ b/content/know/concept/two-fluid-equations/index.pdc
@@ -0,0 +1,178 @@
+---
+title: "Two-fluid equations"
+firstLetter: "T"
+publishDate: 2021-10-19
+categories:
+- Physics
+- Plasma physics
+
+date: 2021-10-18T10:12:20+02:00
+draft: false
+markup: pandoc
+---
+
+# Two-fluid equations
+
+The **two-fluid model** describes a plasma as two separate but overlapping fluids,
+one for ions and one for electrons.
+Instead of tracking individual particles,
+it gives the dynamics of fluid elements $\dd{V}$ (i.e. small "blobs").
+These blobs are assumed to be much larger than
+the [Debye length](/know/concept/debye-length/),
+such that electromagnetic interactions between nearby blobs can be ignored.
+
+From Newton's second law, we know that the velocity $\vb{v}$
+of a particle with mass $m$ and charge $q$ is as follows,
+when subjected only to the [Lorentz force](/know/concept/lorentz-force/):
+
+$$\begin{aligned}
+    m \dv{\vb{v}}{t}
+    = q (\vb{E} + \vb{v} \cross \vb{B})
+\end{aligned}$$
+
+From here, the derivation is similar to that of the
+[Navier-Stokes equations](/know/concept/navier-stokes-equations/).
+We replace the time derivative with a
+[material derivative](/know/concept/material-derivative/) $\mathrm{D}/\mathrm{D}t$,
+and define a blob's velocity $\vb{u}$
+as the average velocity of the particles inside it, leading to:
+
+$$\begin{aligned}
+    m n \frac{\mathrm{D} \vb{u}}{\mathrm{D} t}
+    = q n (\vb{E} + \vb{u} \cross \vb{B})
+\end{aligned}$$
+
+Where we have multiplied by the number density $n$ of the particles.
+Due to particle collisions in the fluid,
+stresses become important. Therefore, we include
+the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $\hat{P}$,
+leading to the following two equations:
+
+$$\begin{aligned}
+        m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
+        &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) + \nabla \cdot \hat{P}_i{}^\top
+        \\
+        m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
+        &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top
+\end{aligned}$$
+
+Where the subscripts $i$ and $e$ refer to ions and electrons, respectively.
+Finally, we also account for momentum transfer between ions and electrons
+due to [Rutherford scattering](/know/concept/rutherford-scattering/),
+leading to these **two-fluid momentum equations**:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
+            &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) + \nabla \cdot \hat{P}_i{}^\top - f_{ie} m_i n_i (\vb{u}_i - \vb{u}_e)
+            \\
+            m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
+            &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top - f_{ei} m_e n_e (\vb{u}_e - \vb{u}_i)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+Where $f_{ie}$ is the mean frequency at which an ion collides with electrons,
+and vice versa for $f_{ei}$.
+For simplicity, we assume that the plasma is isotropic
+and that shear stresses are negligible,
+in which case the stress term can be replaced
+by the gradient $- \nabla p$ of a scalar pressure $p$:
+
+$$\begin{aligned}
+    m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
+    &= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) - \nabla p_i - f_{ie} m_i n_i (\vb{u}_i - \vb{u}_e)
+    \\
+    m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
+    &= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) - \nabla p_e - f_{ei} m_e n_e (\vb{u}_e - \vb{u}_i)
+\end{aligned}$$
+
+Next, we demand that matter is conserved.
+In other words, the rate at which particles enter/leave a volume $V$
+must be equal to the flux through the enclosing surface $S$:
+
+$$\begin{aligned}
+    0
+    &= \pdv{t} \int_V n \dd{V} + \oint_S n \vb{u} \cdot \dd{\vb{S}}
+    = \int_V \Big( \pdv{n}{t} + \nabla \cdot (n \vb{u}) \Big) \dd{V}
+\end{aligned}$$
+
+Where we have used the divergence theorem.
+Since $V$ is arbitrary, we can remove the integrals,
+leading to the following **continuity equations**:
+
+$$\begin{aligned}
+    \boxed{
+        \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i)
+        = 0
+        \qquad \quad
+        \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e)
+        = 0
+    }
+\end{aligned}$$
+
+Currently, we have 8 equations (2 scalar continuity, 2 vector momentum),
+but 16 unknowns $\vb{u}_i$, $\vb{u}_e$, $\vb{E}$, $\vb{B}$, $n_i$, $n_e$, $p_i$ and $p_e$.
+We would like to close this system, so we need 8 more.
+An obvious choice is [Maxwell's equations](/know/concept/maxwells-equations/),
+in particular Faraday's and Ampère's law
+(since Gauss' laws are in fact redundant; see the article on Maxwell's equations):
+
+$$\begin{aligned}
+    \boxed{
+        \nabla \cross \vb{E} = - \pdv{\vb{B}}{t}
+        \qquad \quad
+        \nabla \cross \vb{B} = \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big)
+    }
+\end{aligned}$$
+
+Now we have 14 equations, so we need 2 more, for the pressures $p_i$ and $p_e$.
+This turns out to be the thermodynamic **equation of state**:
+for quasistatic, reversible, adiabatic compression
+of a gas with constant heat capacities (i.e. a *calorically perfect* gas),
+it turns out that:
+
+$$\begin{aligned}
+    \dv{t} \big( p V^\gamma \big) = 0
+    \qquad \quad
+    \gamma
+    \equiv \frac{C_P}{C_V}
+    = \frac{N + 2}{N}
+\end{aligned}$$
+
+Where $\gamma$ is the *heat capacity ratio*,
+and can be calculated from the number of degrees of freedom $N$
+of each particle in the gas.
+In a fully ionized plasma, $N = 3$.
+
+The density $n \propto 1/V$,
+so since $p V^\gamma$ is constant in time,
+for some (known) constant $C$:
+
+$$\begin{aligned}
+    \dv{t} \Big( \frac{p}{n^\gamma} \Big) = 0
+    \quad \implies \quad
+    p = C n^\gamma
+\end{aligned}$$
+
+In the two-fluid model, we thus have the following two equations of state,
+giving us a set of 16 equations for 16 unknowns:
+
+$$\begin{aligned}
+    \boxed{
+        p_i = C_i n_i^\gamma
+        \qquad \quad
+        p_e = C_e n_e^\gamma
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  F.F. Chen,
+    *Introduction to plasma physics and controlled fusion*,
+    3rd edition, Springer.
+2.  M. Salewski, A.H. Nielsen,
+    *Plasma physics: lecture notes*,
+    2021, unpublished.
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