From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/boltzmann-relation/index.md | 28 ++++++++++++-------------
 1 file changed, 14 insertions(+), 14 deletions(-)

(limited to 'source/know/concept/boltzmann-relation')

diff --git a/source/know/concept/boltzmann-relation/index.md b/source/know/concept/boltzmann-relation/index.md
index b7f82b7..d5409d2 100644
--- a/source/know/concept/boltzmann-relation/index.md
+++ b/source/know/concept/boltzmann-relation/index.md
@@ -10,12 +10,12 @@ layout: "concept"
 
 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.
+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 $\vb{F}_e = \vb{f}_e \dd{V}$.
+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:
 
 $$\begin{aligned}
@@ -25,8 +25,8 @@ $$\begin{aligned}
 \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 $\vb{f}_p$:
+obeying the ideal gas law $$p_e = k_B T_e n_e$$,
+then the pressure $$p_e$$ leads to another force density $$\vb{f}_p$$:
 
 $$\begin{aligned}
     \vb{f}_p
@@ -34,8 +34,8 @@ $$\begin{aligned}
     = - 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:
+At equilibrium, we demand that $$\vb{f}_e = - \vb{f}_p$$,
+and isolate this equation for $$\nabla n_e$$, yielding:
 
 $$\begin{aligned}
     k_B T_e \nabla n_e
@@ -47,7 +47,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This equation is straightforward to integrate,
-leading to the following expression for $n_e$,
+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/)):
@@ -59,10 +59,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where the linearity factor $n_{e0}$ represents
-the electron density for $\phi = 0$.
+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 $n_i$:
+leading to the following ion density $$n_i$$:
 
 $$\begin{aligned}
     \boxed{
@@ -75,7 +75,7 @@ However, due to their larger mass,
 ions are much slower to respond to fluctuations in the above equilibrium.
 Consequently, after a perturbation,
 the ions spend much more time in a transient non-equilibrium state
-than the electrons, so this formula for $n_i$ is only valid
+than the electrons, so this formula for $$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,
-- 
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