Categories: Physics, Plasma physics.

# Boltzmann relation

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 $$\vb{E}$$ 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 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):

\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 M. Salewski, A.H. Nielsen, Plasma physics: lecture notes, 2021, unpublished.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.