# Electron Gas (Plasma) to Muon

• I
When do we use the Boltzmann equation for density in a Fermi plasma?
n in [cm-3]
and when do we use the ρ=m/V, ρ in [Kg/m3 ]
(this is not an example, I just added the equations to make my question more understandable)

Is the ideal gas only when we have electron and ions? Is the Boltzmann equation applicable when we change the electrons to other particles like ions, protons or muons?

## Answers and Replies

mfb
Mentor
The particle density n and the mass density ρ always exist, that is independent of the particle type you consider.
In general laws of physics are independent of the type of particle in your system.

jasonRF
Science Advisor
Gold Member
I find your questions to be a little confusing... but here we go.
When do we use the Boltzmann equation for density in a Fermi plasma?
Is the Boltzmann equation applicable when we change the electrons to other particles like ions, protons or muons?

The Boltzmann equation describes the evolution of the particle distribution function ##f_\alpha(\mathbf{r},\mathbf{p},t)## of a single species (denoted ##\alpha##) in the classical limit (see https://en.wikipedia.org/wiki/Boltzmann_equation). Is this what you are talking about? if so, it is derived using classical mechanics, so is valid whenever you can ignore quantum effects. In the classical limit we use it for all kinds of particles. The title of your post mentions plasmas - we certainly use the Boltzman equation to describe the dynamics of each species in garden variety plasmas such as planetary magnetospheres (electrons, protons, Helium ions, etc.). I personally do no know anything about muons.

However, you may be asking about the Boltzman distribution (https://en.wikipedia.org/wiki/Boltzmann_distribution). If so, then it is valid whenever the energies of the fermions are >> kT. Again, I know nothing about muons.

and when do we use the ρ=m/V, ρ in [Kg/m3 ]
What do you mean by your symbols? If ##\rho## is average density, then it is always true that the average density of "stuff" in a volume V is given by ##\rho = m/V##, where ##m## is the total mass of stuff in that volume. This is true no matter what you are talking about.

However, if you are thinking about a plasma and are interested in ##\rho(\mathbf{r},t)## as a function of space and time, then it is given in terms of the distribution functions as ##\rho(\mathbf{r},t) = \sum_\alpha m_\alpha \int d^3\mathbf{p} \, f_\alpha(\mathbf{r},\mathbf{p},t)##, where ##m_\alpha## is the mass of each particle of species ##\alpha##.

jason