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+---
+title: "Boltzmann relation"
+date: 2021-10-18
+categories:
+- Physics
+- Plasma physics
+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.
+
+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.
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