# Plasma physics when plasma conditions don't apply

I've started to look into the field of plasma physics and I saw that not any gas of charges can be considered a plasma. There are 3 main conditions for something to be considered a plasma if I understood correctly:

1. The Debye length needs to be much smaller than the relevant dimensions of the system $$\lambda_D \ll L$$

2. The number of particles in the Debye sphere needs to be much greater than 1, $$\frac{4\pi}{3} n \lambda_D^3 \gg 1$$.

3. A third relation about time scales which isn't really relevant to my question (and frankly, I don't really understand yet).

My question is: say you take air and you magically heat it up so quickly that the density of particles doesn't change but the Debye length becomes around a nano-meter. In that case, condition 2 doesn't apply, but I do have a soup of ions and electrons, so how does one treat these kind of situations? What is the relevant area in physics that knows how to deal with such cases?

Follow up question: the same thing but with condition 1, what do I need to do when I'm trying to study something so hot that it has many ions but on the scale of shorter than a Debye length?

If I had understood the third condition, I would have asked a the same question about it as well.

So to sum up: how does one deal with macroscopic quantities of charged particles when the conditions for plasma don't apply?

Thank you very much in advance!

• Note that the Debye theory is in most practical cases not adequate as it assumes thermodynamic equilibrium i.e. collisions dominating all other physical processes. This means the collision frequency must for instance be higher than the plasma frequency, which would only hold for extremely high plasma densities like in fluids, solids or the interior of the sun. In all other cases the charge distribution will be determined by (collisionless) collective plasma effects rather than kinetic collisions (as the Debye theory assumes) Feb 23, 2021 at 18:58
• @Thomas - I think you are confusing something else with the idea of Debye shielding. Debye shielding works perfectly fine in a collisionless plasma. In fact, most textbooks start with a simple model of electrons and ions having some Gaussian profile and derive the Debye length from a simple PDE. Unless you are pointing out the fact that plasmas rarely have Maxwellian velocity distributions, in which case, I agree but that doesn't significantly change the Debye length. Mar 4, 2021 at 22:06
• @honeste_vivere The derivation of the Debye length assumes a Maxwell-Boltzmann distribution for the particles. And a Maxwell-Boltzmann distribution requires (elastic) collisions. In this case, the elastic collisions provide the pressure force that cancels the force due the global field of the perturbing charge. But if there are no collisions, that's not going to happen. There won't be any Maxwell-Boltzmann distribution and there won't be any pressure force. The global field of the excess charge will in this case be neutralized by collective plasma displacements (plasma polarization fields) Mar 5, 2021 at 21:52
• @Thomas - Are you referring to something like: doi.org/10.1063/1.5091949 Mar 5, 2021 at 21:55
• @honeste_vivere It is intuitively obvious: assume a quasi neutral plasma sphere with all charges at rest (the ions assumed infinitely heavy). If you now put a positive excess charge at the center, all electrons will start accelerating towards it until their mutual repulsion becomes strong enough so that they bounce back, and so on. So there will be a radial plasma oscillation which, on average, results in a zero electric field throughout the sphere (the net positive charge is now at its surface). And for most plasmas this occurs much quicker than any collision/pressure effect could occur. Mar 6, 2021 at 14:44

The exact definition of a plasma is really pretty arbitrary-- the reason behind the definition you wrote down (which I believe is from Chen if I'm not mistaken) is pretty much just that matter in that regime is easier to analyze so that's what the plasma physicists of old concentrated on.

More specifically, the reason behind conditions 1 & 2 is pretty much that it enables us to neglect the effects of strong particle correlations. When coming up with classical descriptions of plasmas, the starting point is usually some sort of distribution function that tells you how many particles you would expect to have in a certain place with a certain velocity (and interestingly enough, by playing around with some weird mathematical tools you can even do a similar thing for quantum systems). You then have to come up with some sort of equation (called the Boltzmann equation) that describes how this distribution function of particles evolves, and if certain ordering conditions are met you can take averages of quantities over velocity space to recover the fluid like equations most people think of when they think of plasma physics.

But the usual treatment of this distribution function in classical plasma physics assumes certain things about how these particles are correlated before and after collisions-- namely, the assumption of Debye screening lets us only consider binary collisions in the Boltzmann equation because charge screening essentially places a limit on the range that a single particle can affect a different particle which makes everything a lot more tractable. If your system is as big as the Debye length, or if condition 2 fails and Debye screening doesn't happen at all though, the long range nature of the coulomb force means that a bunch of infinities start popping up and you can't take the same 2 body, perturbative approach to the Boltzmann equation.

When you start getting into this regime, you have to be lot more careful about how you deal with particle distribution functions and you have to take into account a more collective picture that's very reminiscent of the field theories used by condensed matter physicists. This document here lays out some foundations for how you might deal with "plasmas" where conditions 1 & 2 aren't met.

My question is: say you take air and you magically heat it up so quickly that the density of particles doesn't change but the Debye length becomes around a nano-meter. In that case, condition 2 doesn't apply, but I do have a soup of ions and electrons, so how does one treat these kind of situations? What is the relevant area in physics that knows how to deal with such cases?

This is just considered a weakly ionized plasma. This is not really surprising or special, it's just a specific type of plasma-like phenomena. In these scenarios, collisions with neutral particles must be taken into account and can be a dominant term in the equations of motion, depending on the scenario. This is an issue routinely encountered in some lab plasmas and something folks care greatly about in the ionosphere.

Follow up question: the same thing but with condition 1, what do I need to do when I'm trying to study something so hot that it has many ions but on the scale of shorter than a Debye length?

I am not sure I follow. If you have "many ions," what do you mean? If the number density of ions and electrons is large, then you can still have a plasma, in principle. The constraint about the system size relative to the Debye length is defined because at scales below the Debye length, there can be electric fields and the system is not quasi-neutral, i.e., $$n_{e} \neq n_{i}$$. Generally to be a plasma, the first condition is necessary so the plasma can exhibit collective behavior and maintain quasi-neutrality. The collective behavior bit is related to how plasmas behave like a fluid but are controlled by long-range forces (i.e., Coulomb fields) rather than binary particle collisions.

So to sum up: how does one deal with macroscopic quantities of charged particles when the conditions for plasma don't apply?

If condition 1 is not satisfied, the system will indeed behave in a chaotic manner and not be well described by the usual models and approximations. This is because there will be time-varying electric fields actively doing work to get rid of themselves. I am not sure if anyone has ever tried to tango with such a scenario as there are very difficult to ascertain assumptions necessary.

Debye length increases with temperature. If the electron and ion temperatures are equal, $$T_e = T_i = T$$, then

$$\lambda_D = \sqrt{\frac{\epsilon_0 k_B T /e^2}{n_e + z_i^2n_i}}$$

Increasing the temperature will drive the plasma parameter/Debye number up, in line with physical intuition: At high temperature, particles will be more able to thermally fluctuate into ionic repulsion with one another. The system remains a plasma, albeit a more strongly coupled one.