Categories: Physics, Plasma physics.

# Boltzmann relation

In a plasma where the ions and electrons are in thermal equilibrium, in the absence of short-lived induced electromagnetic fields, the densities $n_i$ and $n_e$ can be predicted.

By definition, a charged particle in an electric field $\vb{E} = - \nabla \phi$ experiences a 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 electrons:

$\begin{aligned} \vb{f}_e = q_e n_e \vb{E} = - q_e n_e \nabla \phi \end{aligned}$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):

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 $n_i$:

$\begin{aligned} \boxed{ n_i(\vb{r}) = n_{i0} \exp\!\bigg( \!-\! \frac{q_i \phi(\vb{r})}{k_B T_i} \bigg) } \end{aligned}$But due to their large mass, ions respond much slower to fluctuations in the above equilibrium. Consequently, after a perturbation, the ions spend more time in a non-equilibrium state than the electrons, so this formula for $n_i$ is only valid if the perturbation is sufficiently slow, such that the ions can keep up. Usually, electrons do not suffer the same issue, thanks to their small mass and hence fast response.

## References

- P.M. Bellan,
*Fundamentals of plasma physics*, 1st edition, Cambridge. - M. Salewski, A.H. Nielsen,
*Plasma physics: lecture notes*, 2021, unpublished.