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Diffstat (limited to 'source/know/concept/boltzmann-relation')
| -rw-r--r-- | source/know/concept/boltzmann-relation/index.md | 28 |
1 files changed, 14 insertions, 14 deletions
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, |