Expand knowledge base

6 files changed, 639 insertions, 5 deletions

diff --git a/content/know/concept/boltzmann-relation/index.pdc b/content/know/concept/boltzmann-relation/index.pdc
index ddaa22f..b892745 100644
--- a/content/know/concept/boltzmann-relation/index.pdc
+++ b/content/know/concept/boltzmann-relation/index.pdc
@@ -39,16 +39,16 @@ $$\begin{aligned}
     = - k_B T_e \nabla n_e
 \end{aligned}$$
 
-At equilibrium, we demand that $\vb{f}_e = \vb{f}_p$,
+At equilibrium, we demand that $\vb{f}_e = - \vb{f}_p$,
 and isolate this equation for $\nabla n_e$, yielding:
 
 $$\begin{aligned}
     k_B T_e \nabla n_e
-    = q_e n_e \nabla \phi
+    = - q_e n_e \nabla \phi
     \quad \implies \quad
     \nabla n_e
-    = \frac{q_e \nabla \phi}{k_B T_e} n_e
-    = \nabla \bigg( \frac{q_e \phi}{k_B T_e} \bigg) n_e
+    = - \frac{q_e \nabla \phi}{k_B T_e} n_e
+    = - \nabla \bigg( \frac{q_e \phi}{k_B T_e} \bigg) n_e
 \end{aligned}$$
 
 This equation is straightforward to integrate,
diff --git a/content/know/concept/ion-sound-waves/index.pdc b/content/know/concept/ion-sound-waves/index.pdc
new file mode 100644
index 0000000..c56188b
--- /dev/null
+++ b/content/know/concept/ion-sound-waves/index.pdc
@@ -0,0 +1,265 @@
+---
+title: "Ion sound waves"
+firstLetter: "I"
+publishDate: 2021-10-31
+categories:
+- Physics
+- Plasma physics
+
+date: 2021-10-31T09:38:14+01:00
+draft: false
+markup: pandoc
+---
+
+# Ion sound waves
+
+In a plasma, electromagnetic interactions allow
+compressional longitudinal waves to propagate
+at lower temperatures and pressures
+than would be possible in a neutral gas.
+
+We start from the [two-fluid model's](/know/concept/two-fluid-equations/) momentum equations,
+rewriting the [electric field](/know/concept/electric-field/) $\vb{E} = - \nabla \phi$
+and the pressure gradient $\nabla p = \gamma k_B T \nabla n$,
+and arguing that $m_e \approx 0$ because $m_e \ll m_i$:
+
+$$\begin{aligned}
+    m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
+    &= - q_i n_i \nabla \phi - \gamma_i k_B T_i \nabla n_i
+    \\
+    0
+    &= - q_e n_e \nabla \phi - \gamma_e k_B T_e \nabla n_e
+\end{aligned}$$
+
+Note that we neglect ion-electron collisions,
+and allow for separate values of $\gamma$.
+We split $n_i$, $n_e$, $\vb{u}_i$ and $\phi$ into an equilibrium
+(subscript $0$) and a perturbation (subscript $1$):
+
+$$\begin{aligned}
+    n_i
+    = n_{i0} + n_{i1}
+    \qquad
+    n_e
+    = n_{e0} + n_{e1}
+    \qquad
+    \vb{u}_i
+    = \vb{u}_{i0} + \vb{u}_{i1}
+    \qquad
+    \phi
+    = \phi_0 + \phi_1
+\end{aligned}$$
+
+Where the perturbations $n_{i1}$, $n_{e1}$, $\vb{u}_{i1}$ and $\phi_1$ are tiny,
+and the equilibrium components $n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$
+by definition satisfy:
+
+$$\begin{aligned}
+    \pdv{n_{i0}}{t} = 0
+    \qquad
+    \frac{\mathrm{D} \vb{u}_{i0}}{\mathrm{D} t} = 0
+    \qquad
+    \nabla n_{i0} = \nabla n_{e0} = 0
+    \qquad
+    \vb{u}_{i0} = 0
+    \qquad
+    \phi_0 = 0
+\end{aligned}$$
+
+Inserting this decomposition into the momentum equations
+yields new equations.
+Note that we will implicitly use $\vb{u}_{i0} = 0$
+to pretend that the [material derivative](/know/concept/material-derivative/)
+$\mathrm{D}/\mathrm{D} t$ is linear:
+
+$$\begin{aligned}
+    m_i (n_{i0} \!+\! n_{i1}) \frac{\mathrm{D} (\vb{u}_{i0} \!+\! \vb{u}_{i1})}{\mathrm{D} t}
+    &= - q_i (n_{i0} \!+\! n_{i1}) \nabla (\phi_0 \!+\! \phi_1) - \gamma_i k_B T_i \nabla (n_{i0} \!+\! n_{i1})
+    \\
+    0
+    &= - q_e (n_{e0} \!+\! n_{e1}) \nabla (\phi_0 \!+\! \phi_1) - \gamma_e k_B T_e \nabla (n_{e0} \!+\! n_{e1})
+\end{aligned}$$
+
+Using the defined properties of the equilibrium components
+$n_{i0}$, $n_{e0}$, $\vb{u}_{i0}$ and $\phi_0$,
+and neglecting all products of perturbations for being small,
+this reduces to:
+
+$$\begin{aligned}
+    m_i n_{i0} \pdv{\vb{u}_{i1}}{t}
+    &= - q_i n_{i0} \nabla \phi_1 - \gamma_i k_B T_i \nabla n_{i1}
+    \\
+    0
+    &= - q_e n_{e0} \nabla \phi_1 - \gamma_e k_B T_e \nabla n_{e1}
+\end{aligned}$$
+
+Because we are interested in linear waves,
+we make the following plane-wave ansatz:
+
+$$\begin{aligned}
+    n_{i1}(\vb{r}, t)
+    &= n_{i1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
+    \\
+    n_{e1}(\vb{r}, t)
+    &= n_{e1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
+    \\
+    \vb{u}_{i1}(\vb{r}, t)
+    &= \vb{u}_{i1} \exp\!(i \vb{k} \cdot \vb{r}  - i \omega t)
+    \\
+    \phi_1(\vb{r}, t)
+    &= \phi_1 \,\,\exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
+\end{aligned}$$
+
+Which we then insert into the momentum equations for the ions and electrons:
+
+$$\begin{aligned}
+    - i \omega m_i n_{i0} \vb{u}_{i1}
+    &= - i \vb{k} q_i n_{i0} \phi_1 - i \vb{k} \gamma_i k_B T_i n_{i1}
+    \\
+    0
+    &= - i \vb{k} q_e n_{e0} \phi_1 - i \vb{k} \gamma_e k_B T_e n_{e1}
+\end{aligned}$$
+
+The electron equation can easily be rearranged
+to get a relation between $n_{e1}$ and $n_{e0}$:
+
+$$\begin{aligned}
+     i \vb{k} \gamma_e k_B T_e n_{e1}
+     = - i \vb{k} q_e n_{e0} \phi_1
+     \quad \implies \quad
+     n_{e1}
+     = - \frac{q_e \phi_1}{\gamma_e k_B T_e} n_{e0}
+\end{aligned}$$
+
+Due to their low mass, the electrons' heat conductivity
+can be regarded as infinite compared to the ions'.
+In that case, all electron gas compression is isothermal,
+meaning it obeys the ideal gas law $p_e = n_e k_B T_e$, so that $\gamma_e = 1$.
+Note that this yields the first-order term of a Taylor expansion
+of the [Boltzmann relation](/know/concept/boltzmann-relation/).
+
+At equilibrium, quasi-neutrality demands that $n_{i0} = n_{e0} = n_0$,
+so we can rearrange the above relation to $n_0 = - k_B T_e n_{e1} / (q_e \phi_1)$,
+which we insert into the ion equation to get:
+
+$$\begin{gathered}
+    i \omega m_i \frac{k_B T_e n_{e1}}{q_e \phi_1} \vb{u}_{i1}
+    = - i q_i \frac{k_B T_e n_{e1}}{q_e \phi_1} \phi_1 \vb{k} - i \gamma_i k_B T_i n_{i1} \vb{k}
+    \\
+    \implies \qquad
+    \omega m_i \frac{T_e n_{e1}}{q_e \phi_1} \vb{k} \cdot \vb{u}_{i1}
+    = T_e n_{e1} |\vb{k}|^2 - \gamma_i T_i n_{i1} |\vb{k}|^2
+\end{gathered}$$
+
+Where we have taken the dot product with $\vb{k}$,
+and used that $q_i / q_e = -1$.
+In order to simplify this equation,
+we turn to the two-fluid ion continuity relation:
+
+$$\begin{aligned}
+    0
+    &= \pdv{(n_{i0} \!+\! n_{i1})}{t} + \nabla \cdot \Big( (n_{i0} \!+\! n_{i1}) (\vb{u}_{i0} \!+\! \vb{u}_{i1}) \Big)
+    \approx \pdv{n_{i1}}{t} + n_{i0} \nabla \cdot \vb{u}_{i1}
+\end{aligned}$$
+
+Then we insert our plane-wave ansatz,
+and substitute $n_{i0} = n_0$ as before, yielding:
+
+$$\begin{aligned}
+    0
+    = - i \omega n_{i1} + i n_{i0} \vb{k} \cdot \vb{u}_{i1}
+    \quad \implies \quad
+    \vb{k} \cdot \vb{u}_{i1}
+    = \omega \frac{n_{i1}}{n_{i0}}
+    = \omega \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}}
+\end{aligned}$$
+
+Substituting this in the ion momentum equation
+leads us to a dispersion relation $\omega(\vb{k})$:
+
+$$\begin{gathered}
+    \omega^2 m_i \frac{T_e n_{e1}}{q_e \phi_1} \frac{q_e n_{i1} \phi_1}{k_B T_e n_{e1}}
+    = \omega^2 m_i \frac{n_{i1}}{k_B}
+    = |\vb{k}|^2 \big( T_e n_{e1} - \gamma_i T_i n_{i1} \big)
+    \\
+    \implies \qquad
+    \omega^2
+    = \frac{|\vb{k}|^2}{m_i} \Big( k_B T_e \frac{n_{e1}}{n_{i1}} - \gamma_i k_B T_i \Big)
+\end{gathered}$$
+
+Finally, we would like to find an expression for $n_{e1} / n_{i1}$.
+It cannot be $1$, because then $\phi_1$ could not be nonzero,
+according to [Gauss' law](/know/concept/maxwells-equations/).
+Nevertheless, authors often ignore this fact,
+thereby making the so-called **plasma approximation**.
+We will not, and therefore turn to Gauss' law:
+
+$$\begin{aligned}
+    \varepsilon_0 \nabla \cdot \vb{E}
+    = - \varepsilon_0 \nabla^2 \phi_1
+    = q_i n_i - q_e n_e
+    = - q_e (n_{i1} - n_{e1})
+\end{aligned}$$
+
+One final time, we insert our plane-wave ansatz,
+and use our Boltzmann-like relation between $n_{e1}$ and $n_{e0}$
+to substitute $\phi_1 = - k_B T_e n_{e1} / (q_e n_{e0})$:
+
+$$\begin{gathered}
+    q_e (n_{e1} - n_{i1})
+    = |\vb{k}|^2 \varepsilon_0 \phi_1
+    = - |\vb{k}|^2 \varepsilon_0 \frac{k_B T_e n_{e1}}{q_e n_{e0}}
+    \\
+    \implies \qquad
+    n_{i1}
+    = n_{e1} + |\vb{k}|^2 \varepsilon_0 \frac{k_B T_e n_{e1}}{q_e^2 n_{e0}}
+    = n_{e1} \big( 1 + |\vb{k}|^2 \lambda_{De}^2 \big)
+\end{gathered}$$
+
+Where $\lambda_{De}$ is the electron [Debye length](/know/concept/debye-length/).
+We thus reach the following dispersion relation,
+which governs **ion sound waves** or **ion acoustic waves**:
+
+$$\begin{aligned}
+    \boxed{
+        \omega^2
+        = \frac{|\vb{k}|^2}{m_i} \bigg( \frac{k_B T_e}{1 + |\vb{k}|^2 \lambda_{De}^2} + \gamma_i k_B T_i \bigg)
+    }
+\end{aligned}$$
+
+The aforementioned plasma approximation is valid if $|\vb{k}| \lambda_{De} \ll 1$,
+which is often reasonable,
+in which case this dispersion relation reduces to:
+
+$$\begin{aligned}
+    \omega^2
+    = \frac{|\vb{k}|^2}{m_i} \bigg( k_B T_e + \gamma_i k_B T_i \bigg)
+\end{aligned}$$
+
+The phase velocity $v_s$ of these waves,
+i.e. the speed of sound, is then given by:
+
+$$\begin{aligned}
+    \boxed{
+        v_s
+        = \frac{\omega}{k}
+        = \sqrt{\frac{k_B T_e}{m_i} + \frac{\gamma_i k_B T_i}{m_i}}
+    }
+\end{aligned}$$
+
+Curiously, unlike a neutral gas,
+this velocity is nonzero even if $T_i = 0$,
+meaning that the waves still exist then.
+In fact, usually the electron temperature $T_e$ dominates $T_e \gg T_i$,
+even though the main feature of these waves
+is that they involve ion density fluctuations $n_{i1}$.
+
+
+
+## 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.
diff --git a/content/know/concept/langmuir-waves/index.pdc b/content/know/concept/langmuir-waves/index.pdc
new file mode 100644
index 0000000..cd9b449
--- /dev/null
+++ b/content/know/concept/langmuir-waves/index.pdc
@@ -0,0 +1,260 @@
+---
+title: "Langmuir waves"
+firstLetter: "L"
+publishDate: 2021-10-30
+categories:
+- Physics
+- Plasma physics
+
+date: 2021-10-15T20:31:46+02:00
+draft: false
+markup: pandoc
+---
+
+# Langmuir waves
+
+In plasma physics, **Langmuir waves** are oscillations in the electron density,
+which may or may not propagate, depending on the temperature.
+
+Assuming no [magnetic field](/know/concept/magnetic-field/) $\vb{B} = 0$,
+no ion motion $\vb{u}_i = 0$ (since $m_i \gg m_e$),
+and therefore no ion-electron momentum transfer,
+the [two-fluid equations](/know/concept/two-fluid-equations/)
+tell us that:
+
+$$\begin{aligned}
+    m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
+    = q_e n_e \vb{E} - \nabla p_e
+    \qquad \quad
+    \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) = 0
+\end{aligned}$$
+
+These are the electron momentum and continuity equations.
+We also need [Gauss' law](/know/concept/maxwells-equations/):
+
+$$\begin{aligned}
+    \varepsilon_0 \nabla \cdot \vb{E}
+    = q_e (n_i - n_e)
+\end{aligned}$$
+
+We split $n_e$, $\vb{u}_e$ and $\vb{E}$ into a base component
+(subscript $0$) and a perturbation (subscript $1$):
+
+$$\begin{aligned}
+    n_e
+    = n_{e0} + n_{e1}
+    \qquad \quad
+    \vb{u}_e
+    = \vb{u}_{e0} + \vb{u}_{e1}
+    \qquad \quad
+    \vb{E}
+    = \vb{E}_0 + \vb{E}_1
+\end{aligned}$$
+
+Where the perturbations $n_{e1}$, $\vb{u}_{e1}$ and $\vb{E}_1$ are very small,
+and the equilibrium components $n_{e0}$, $\vb{u}_{e0}$ and $\vb{E}_0$
+by definition satisfy:
+
+$$\begin{aligned}
+    \pdv{n_{e0}}{t} = 0
+    \qquad
+    \pdv{\vb{u}_{e0}}{t} = 0
+    \qquad
+    \nabla n_{e0} = 0
+    \qquad
+    \vb{u}_{e0} = 0
+    \qquad
+    \vb{E}_0 = 0
+\end{aligned}$$
+
+We insert this decomposistion into the electron continuity equation,
+arguing that $n_{e1} \vb{u}_{e1}$ is small enough to neglect, leading to:
+
+$$\begin{aligned}
+    0
+    &= \pdv{(n_{e0} \!+\! n_{e1})}{t} + \nabla \cdot \Big( (n_{e0} \!+\! n_{e1}) \: (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \Big)
+    \\
+    &= \pdv{n_{e1}}{t} + \nabla \cdot \Big( n_{e0} \vb{u}_{e1} + n_{e1} \vb{u}_{e1} \Big)
+    \\
+    &\approx \pdv{n_{e1}}{t} + \nabla \cdot (n_{e0} \vb{u}_{e1})
+    = \pdv{n_{e1}}{t} + n_{e0} \nabla \cdot \vb{u}_{e1}
+\end{aligned}$$
+
+Likewise, we insert it into Gauss' law,
+and use the plasma's quasi-neutrality $n_i = n_{e0}$ to get:
+
+$$\begin{aligned}
+    \varepsilon_0 \nabla \cdot \big( \vb{E}_0 \!+\! \vb{E}_1 \big)
+    = q_e (n_i - n_{e0} \!-\! n_{e1} )
+    \quad \implies \quad
+    \varepsilon_0 \nabla \cdot \vb{E}_1
+    = - q_e n_{e1}
+\end{aligned}$$
+
+Since we are looking for linear waves,
+we make the following ansatz for the perturbations:
+
+$$\begin{aligned}
+    n_{e1}(\vb{r}, t)
+    &= n_{e1} \exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
+    \\
+    \vb{u}_{e1}(\vb{r}, t)
+    &= \vb{u}_{e1} \exp\!(i \vb{k} \cdot \vb{r}  - i \omega t)
+    \\
+    \vb{E}_1(\vb{r}, t)
+    &= \vb{E}_1 \:\exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
+\end{aligned}$$
+
+Inserting this into the continuity equation and Gauss' law yields, respectively:
+
+$$\begin{aligned}
+    - i \omega n_{e1} = - i n_{e0} \vb{k} \cdot \vb{u}_{e1}
+    \qquad \quad
+    i \varepsilon_0 \vb{k} \cdot \vb{E}_1 = q_e n_{e1}
+\end{aligned}$$
+
+However, there are three unknowns $n_{e1}$, $\vb{u}_{e1}$ and $\vb{E}_1$,
+so one more equation is needed.
+
+
+## Cold Langmuir waves
+
+We therefore turn to the electron momentum equation.
+For now, let us assume that the electrons have no thermal motion,
+i.e. the electron temperature $T_e = 0$, so that $p_e = 0$, leaving:
+
+$$\begin{aligned}
+    m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
+    = q_e n_e \vb{E}
+\end{aligned}$$
+
+Inserting the decomposition then gives the following,
+where we neglect $(\vb{u}_{e1} \cdot \nabla) \vb{u}_{e1}$
+because $\vb{u}_{e1}$ is so small by assumption:
+
+$$\begin{gathered}
+    m_e (n_{e0} \!+\! n_{e1}) \Big( \pdv{(\vb{u}_{e0} \!+\! \vb{u}_{e1})}{t}
+    + \big( (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \cdot \nabla \big) (\vb{u}_{e0} \!+\! \vb{u}_{e1}) \Big)
+    = q_e \big( n_{e0} \!+\! n_{e1} \big) \big( \vb{E}_0 \!+\! \vb{E}_1 \big)
+    \\
+    \implies \qquad
+    q_e \vb{E}_1
+    = m_e \Big( \pdv{\vb{u}_{e1}}{t} + \big(\vb{u}_{e1} \cdot \nabla \big) \vb{u}_{e1} \Big)
+    \approx m_e \pdv{\vb{u}_{e1}}{t}
+\end{gathered}$$
+
+And then inserting our plane-wave ansatz yields
+the third equation we were looking for:
+
+$$\begin{aligned}
+    -i \omega m_e \vb{u}_{e1} = q_e \vb{E}_1
+\end{aligned}$$
+
+Solving this system of three equations for $\omega^2$
+gives the following dispersion relation:
+
+$$\begin{aligned}
+    \omega^2
+    = \frac{\omega n_{e0}}{n_{e1}} \vb{k} \cdot \vb{u}_{e1}
+    = \frac{i \omega n_{e0} q_e}{\omega m_e n_{e1}} \vb{k} \cdot \vb{E}_1
+    = \frac{i n_{e0} n_{e1} q_e^2}{i \varepsilon_0 m_e n_{e1}}
+    = \frac{n_{e0} q_e^2}{\varepsilon_0 m_e}
+\end{aligned}$$
+
+This result is known as the **plasma frequency** $\omega_p$,
+and describes the frequency of **cold Langmuir waves**,
+otherwise known as **plasma oscillations**:
+
+$$\begin{aligned}
+    \boxed{
+        \omega_p
+        = \sqrt{\frac{n_{0e} q_e^2}{\varepsilon_0 m_e}}
+    }
+\end{aligned}$$
+
+Note that this is a dispersion relation $\omega(k) = \omega_p$,
+but that $\omega_p$ does not contain $k$.
+This means that cold Langmuir waves do not propagate:
+the oscillation is "stationary".
+
+
+## Warm Langmuir waves
+
+Next, we generalize this result to nonzero $T_e$,
+in which case the pressure $p_e$ is involved:
+
+$$\begin{aligned}
+    m_e n_{e0} \pdv{\vb{u}_{e1}}{t}
+    = q_e n_{e0} \vb{E}_1 - \nabla p_e
+\end{aligned}$$
+
+From the two-fluid thermodynamic equation of state,
+we know that $\nabla p_e$ can be written as:
+
+$$\begin{aligned}
+    \nabla p_e
+    = \gamma k_B T_e \nabla n_e
+    = \gamma k_B T_e \nabla (n_{e0} + n_{e1})
+    = \gamma k_B T_e \nabla n_{e1}
+\end{aligned}$$
+
+With this, insertion of our plane-wave ansatz
+into the electron equation results in:
+
+$$\begin{aligned}
+    -i \omega m_e n_{e0} \vb{u}_{e1} = q_e n_{e0} \vb{E}_1 - i \gamma k_B T_e n_{e1} \vb{k}
+\end{aligned}$$
+
+Which once again closes the system of three equations.
+Solving for $\omega^2$ then gives:
+
+$$\begin{aligned}
+    \omega^2
+    = \frac{\omega n_{e0}}{n_{e1}} \vb{k} \cdot \vb{u}_{e1}
+    &= \frac{i \omega n_{e0}}{\omega n_{e0} m_e n_{e1}} \vb{k} \cdot \Big( q_e n_{e0} \vb{E}_1 - i \gamma k_B T_e n_{e1} \vb{k} \Big)
+    \\
+    &= \frac{n_{e0} q_e^2}{\varepsilon_0 m_e} - \frac{i \omega}{\omega m_e n_{e1}} i \gamma k_B T_e n_{e1} \big(\vb{k} \cdot \vb{k}\big)
+\end{aligned}$$
+
+Recognizing the first term as the plasma frequency $\omega_p^2$,
+we therefore arrive at the **Bohm-Gross dispersion relation** $\omega(\vb{k})$
+for **warm Langmuir waves**:
+
+$$\begin{aligned}
+    \boxed{
+        \omega^2
+        = \omega_p^2 + \frac{\gamma k_B T_e}{m_e} |\vb{k}|^2
+    }
+\end{aligned}$$
+
+This expression is typically quoted for 1D oscillations,
+in which case $\gamma = 3$ and $k = |\vb{k}|$:
+
+$$\begin{aligned}
+    \omega^2
+    = \omega_p^2 + \frac{3 k_B T_e}{m_e} k^2
+\end{aligned}$$
+
+Unlike for $T_e = 0$, these "warm" waves do propagate,
+carrying information at group velocity $v_g$,
+which, in the limit of large $k$, is given by:
+
+$$\begin{aligned}
+    v_g
+    = \pdv{\omega}{k}
+    \to \sqrt{\frac{3 k_B T_e}{m_e}}
+\end{aligned}$$
+
+This is the root-mean-square velocity of the
+[Maxwell-Boltzmann speed distribution](/know/concept/maxwell-boltzmann-distribution/),
+meaning that information travels at the thermal velocity for large $k$.
+
+
+
+## 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.
diff --git a/content/know/concept/sigma-algebra/index.pdc b/content/know/concept/sigma-algebra/index.pdc
index 6e90fcb..690c4cc 100644
--- a/content/know/concept/sigma-algebra/index.pdc
+++ b/content/know/concept/sigma-algebra/index.pdc
@@ -42,6 +42,9 @@ Likewise, a **sub-$\sigma$-algebra**
 is a sub-family of a certain $\mathcal{F}$,
 which is a valid $\sigma$-algebra in its own right.
 
+
+## Notable examples
+
 A notable $\sigma$-algebra is the **Borel algebra** $\mathcal{B}(\Omega)$,
 which is defined when $\Omega$ is a metric space,
 such as the real numbers $\mathbb{R}$.
diff --git a/content/know/concept/two-fluid-equations/index.pdc b/content/know/concept/two-fluid-equations/index.pdc
index df45e73..9ae9dbf 100644
--- a/content/know/concept/two-fluid-equations/index.pdc
+++ b/content/know/concept/two-fluid-equations/index.pdc
@@ -129,7 +129,7 @@ $$\begin{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),
+of a gas with constant heat capacity (i.e. a *calorically perfect* gas),
 it turns out that:
 
 $$\begin{aligned}
@@ -168,6 +168,22 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
+Note that from the relation $p = C n^\gamma$,
+we can calculate the $\nabla p$ term in the momentum equation,
+using simple differentiation and the ideal gas law:
+
+$$\begin{aligned}
+    p = C n^\gamma
+    \quad \implies \quad
+    \nabla p
+    = \gamma \frac{C n^{\gamma}}{n} \nabla n
+    = \gamma p \frac{\nabla n}{n}
+    = \gamma k_B T \nabla n
+\end{aligned}$$
+
+Note that the ideal gas law was not used immediately,
+to allow for $\gamma \neq 1$.
+
 
 ## Fluid drifts
 
diff --git a/content/know/concept/wiener-process/index.pdc b/content/know/concept/wiener-process/index.pdc
new file mode 100644
index 0000000..49aebfb
--- /dev/null
+++ b/content/know/concept/wiener-process/index.pdc
@@ -0,0 +1,90 @@
+---
+title: "Wiener process"
+firstLetter: "W"
+publishDate: 2021-10-29
+categories:
+- Physics
+- Mathematics
+
+date: 2021-10-21T19:40:02+02:00
+draft: false
+markup: pandoc
+---
+
+# Wiener process
+
+The **Wiener process** is a stochastic process that provides
+a pure mathematical definition of the physical phenomenon of **Brownian motion**,
+and hence is also called *Brownian motion*.
+
+A Wiener process $B_t$ is defined as any
+time-indexed [random variable](/know/concept/random-variable/)
+$\{B_t: t \ge 0\}$ (i.e. stochastic process)
+that has the following properties:
+
+1.  Initial condition $B_0 = 0$.
+2.  Each **increment** of $B_t$ is independent of the past:
+    given $0 \le s < t \le u < v$, then
+    $B_t \!-\! B_s$ and $B_v \!-\! B_u$ are independent random variables.
+3.  The increments of $B_t$ are Gaussian with mean $0$
+    and variance $h$, where $h$ is the time step,
+    such that $B_{t+h} \!-\! B_t \sim \mathcal{N}(0, h)$.
+4.  $B_t$ is a continuous function of $t$.
+
+There exist stochastic processes that satisfy these requirements,
+infinitely many in fact.
+In other words, Brownian motion exists,
+and can be constructed in various ways.
+
+Since the variance of an increment is expressed in units of time $t$,
+the physical unit of the Wiener process is the square root of time $\sqrt{t}$.
+
+Brownian motion is **self-similar**:
+if we define a rescaled $W_t = \sqrt{\alpha} B_{t/\alpha}$ for some $\alpha$,
+then $W_t$ is also a valid Wiener process,
+meaning that there are no fundemental scales.
+A consequence of this is that:
+$\mathbf{E}|B_t|^p = \mathbf{E}|\sqrt{t} B_1|^p = t^{p/2} \mathbf{E}|B_1|^p$.
+Another consequence is invariance under "time inversion",
+by defining $\sqrt{\alpha} = t$, such that $W_t = t B_{1/t}$.
+
+Despite being continuous by definition,
+the **total variation** $V(B)$ of $B_t$ is infinite
+(informally, the curve is infinitely long).
+For $t_i \in [0, 1]$ in $n$ steps of maximum size $\Delta t$:
+
+$$\begin{aligned}
+    V_t
+    = \lim_{\Delta t \to 0} \sup \sum_{i = 1}^n \big|B_{t_i} - B_{t_{i-1}}\big|
+    = \infty
+\end{aligned}$$
+
+However, curiously, the **quadratic variation**, written as $[B]_t$,
+turns out to be deterministically finite and equal to $t$,
+while a differentiable function $f$ would have $[f]_t = 0$:
+
+$$\begin{aligned}
+    \:[B]_t
+    = \lim_{\Delta t \to 0} \sum_{i = 1}^n \big|B_{t_i} - B_{t_{i - 1}}\big|^2
+    = t
+\end{aligned}$$
+
+Therefore, despite being continuous by definition,
+the Wiener process is not differentiable,
+not even in the mean square, because:
+
+$$\begin{aligned}
+    \frac{B_{t+h} - B_t}{h}
+    \sim \frac{1}{h} \mathcal{N}(0, h)
+    \sim \mathcal{N}\Big(0, \frac{1}{h}\Big)
+    \qquad \quad
+    \lim_{h \to 0} \mathbf{E} \bigg|\mathcal{N}\Big(0, \frac{1}{h}\Big) \bigg|^2
+    = \infty
+\end{aligned}$$
+
+
+
+## References
+1.  U.F. Thygesen,
+    *Lecture notes on diffusions and stochastic differential equations*,
+    2021, Polyteknisk Kompendie.