---
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.