ELI5: Imagine you have a group of very tiny particles, like the ones you can't even see with your eyes. These particles can have positive or negative charges, like little electric magnets. When you have a lot of these particles together, they form something called a plasma.

Now, the special thing about a plasma is that the particles can move around freely. It's like they can dance and wiggle without bumping into each other too much. This makes the plasma behave differently from other things we see every day, like liquids or solids.

But sometimes, when scientists want to understand how plasmas work, they need to simplify things. They want to make it easier to study and calculate. So they use something called the plasma approximation.

The plasma approximation is like pretending that all the particles in the plasma are not individual particles anymore. Instead, they act together as a big group, almost like a fluid. It's like thinking of the plasma as a whole, like one big thing, instead of lots of tiny particles.

This approximation helps scientists understand and predict how the plasma behaves in simpler ways. They can use equations and models that work well for fluids, like liquids, to describe how the plasma moves and changes.

So, the plasma approximation is a way of thinking about a plasma as a single group or fluid, even though it's actually made up of many tiny particles with charges. This helps scientists study and understand plasmas in a simpler and easier way.

However, the plasma approximation may become less valid under certain conditions. For example:

Low-density plasmas: In plasmas with very low densities or when the particles are widely spaced, individual particle interactions become more significant, and the collective behavior of the plasma may be less dominant.
High-temperature plasmas: At extremely high temperatures, the thermal motion of individual particles becomes more pronounced, and their individual interactions can have a greater impact on the plasma behaviour.
Strongly interacting or non-equilibrium plasmas: In situations where the plasma is strongly affected by phenomena such as particle collisions, plasma waves, or high-energy particle beams, the plasma approximation may not accurately capture the detailed behaviour of the system.

In deriving the velocity of ion waves, we assume neutrality condition, ni =ne while allowing E to be finite. This is plasma Approximation. To find the error due to such Approximation. Let ni ≠ ne.

The linearized poisson equation is

The electron density is given by linearized Boltzmann relation as

Using equation 2 in equation 1, we get

The ion density is given by linearized ion continuity equation as

The linearized equation of motion for ion is

This is same as we obtained in equation of motion (i.e ion waves) except for the factor (1+ k²𝜆²D) . Our assumption ni=ne has given rise to an error of order.

Since 𝜆D is very small in most experiment the plasma approximation is valid for all except shortest wavelength waves. When k²𝜆²D is small, it suggests that the plasma is in a quasi-neutral state, with the positive and negative charges being relatively balanced. In this case, the collective behavior of the plasma dominates, and the plasma can be treated as a fluid-like medium.

However, when k²𝜆²D is not small, it indicates that the spatial variations in the plasma properties or the screening effects are significant on the scale of the system. In such situations, the plasma approximation may not be valid, and more detailed models, such as kinetic or particle-in-cell (PIC) simulations, may be required to accurately describe the plasma behavior.

It's important to note that the validity of the plasma approximation should be assessed case by case, considering the specific parameters and conditions of the plasma system under investigation. The plasma approximation provides a useful starting point for understanding plasmas, but its limitations should be considered, especially when dealing with strongly non-neutral or highly localized plasma systems.

This note is a part of the Physics Repository.

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