# If ions in plasma are not in Maxwell-Boltzmann Distribution, what is the meaning of ion temperature?

From what I understand, if a gas has a temperature $$T$$ degrees Kelvin, it means that the probability of a gas particle to have energy $$E$$ is proportional to $$\exp(-E/kT)$$. It is the physical interpretation of temperature.

But it is only true when the gas is in Maxwell-Boltzmann Distribution. Electrons in plasma are in MB distribution. So when electron temperature is 1 eV for instance, I can know the energy distribution of electrons.

However, ions in plasma can be not in MB distribution. So, when it says ion temperature is 0.02 eV for instance, I don't know what it represents. What does ion temperature measure?

– rob
May 27, 2020 at 18:49
• @rob would you mind elaborating a bit how exactly those thermodynamic relationships should apply to a plasma? May 27, 2020 at 19:06
• @lurscher Well, I'm not totally sure that I understand this question. My understanding is that the Maxwell-Boltzman distribution (and its quantum cousins) are a consequence of a gas having a temperature, rather than a prerequisite, and that a gas with a different distribution won't have it for long. However, it is possible for a system with weakly-coupled degrees of freedom to have different temperatures in different components. I feel like electrons and ions having different temperatures in a plasma is an example, but a plasma person will be along soon.
– rob
May 27, 2020 at 22:19
• May 28, 2020 at 16:01

In a plasma, the electron temperatures and ion temperatures, when they exist, generally refer to a kinetic temperature as would be applied in a Maxwell-Boltzmann distribution for the velocities, just as the OP specifies. The temperature therefore means the same thing in both cases, although for a given temperature, the corresponding velocity distributions are different in each case because the masses of the particles may be different.

Also as the OP notes, particles in a plasma may not be thermalized, and so their velocity distributions in those cases would not be well-represented by a Maxwell-Boltzmann distribution. In principle this can be true for either the electrons, the positive ions, or both. It is also possible for the electrons and ions to be thermalized at separate temperatures.

It's also the case that the velocities of the electrons and positive ions are coupled in a variety of ways in a plasma, the simplest being two-body coulomb collisions. If the time scale for these collisions to take place is sufficiently short compared to other time scales at which the system is evolving, then the electrons and positive ions will equilibrate at a common temperature. If, however, this timescale is too long, then the two particles are not likely to equilibrate at a common temperature.

But again, the temperature is used in the same way for both collections of particles: to specify the velocity distribution when it is thermalized.

• Ahhh. Plasma physics is so simple ... May 28, 2020 at 1:30
• Sorryif it is stupid question. Plasma physics is hard. I know that temperature of a gas is related to the energy distribution of the particles, assuming the gas is in MB distribution. But when a gas is not in MB distribution, such ions in plasma, how is the temperature related to the energy distribution? If I know the temperature of a gas but the gas is not in MB distribution, then I cannot know anything about that gas from its temperature. May 28, 2020 at 16:11
• Ions in a plasma can be thermalized and follow an MB distribution. But if they're not, then strictly speaking it doesn't make sense to assign them a temperature. It would be sloppy, but I suppose one could assign a non-thermalized population of particles a temperature just as a measure of their total kinetic energy - whatever velocity distribution they happen to have, you assign them a temperature so that the total KE for that distribution is the same as would be the case for a MB distribution at that temperature. Still, I think this is a confusing decision May 28, 2020 at 16:57
• @KhantNyarPaing do you know for sure that the positive ions in the plasma you are studying are not thermalized? Is there a chance they actually are thermalized, and that is why they're being assigned a temperature? May 28, 2020 at 18:37
• Sorry for late reply. The textbook I am studying is thermal plasma. However, my question is a hypothetical. Assigning a temperature to the non MB distribution gas does not make sense for me. That is why I asked. Thanks for your answer. Jun 3, 2020 at 15:38