Differences between "cold" and "collisionless" plasma

Side Notes and/or Definitions:

• Definition of cold plasma: https://physics.stackexchange.com/a/219081/59023
• Index of refraction dependence on temperature discussion: https://physics.stackexchange.com/a/265731/59023
• Different ion and electron temperatures: https://physics.stackexchange.com/a/268594/59023
• Speed of sound in space: https://physics.stackexchange.com/a/179057/59023

I just seem to not have clear the difference between the definitions of "cold plasma approximation" and "collisionless plasma" for what regards the two-fluid plasma modeling.

The term cold should be taken literally in many cases. That is, you do the normal derivations but in the limit as the temperatures go to zero. The point is to make approximations easy but it is applicable to situations where thermal effects are not critical or other factors dominate.

The term collisionless should be taken relatively, as there can be no such thing as truly collisionless. The particles within such systems suffer insufficient collisions for that mechanisms to be effective at regulating any dynamics. For instance, the solar wind is considered weakly collisional because collisions do occur but they are very infrequent (e.g., ~one per day near Earth's orbit about sun). Shock waves that occur in similar regions are considered collisionless because the crossing time of a particle is orders of magnitude smaller than the collisional time, thus the name collisionless shocks.

Could someone help me with a clarification on what exactly makes a plasma fall in the "cold approximation", and what's the difference between the fluid equations of a cold plasma and a collisionless plasma?

Sure, a collisionless plasma can have any temperature including zero. It's just another type of approximation where you ignore Coulomb collisions, i.e., you integrate the Vlasov equation to get the fluid equations of motion.

The cold plasma approximation, therefore, can be just a subset of the collisionless plasma system. It is possible to think of a cold plasma system and kludge-in some collisional drag terms in the fluid dynamics, but this is, as I say, a kludge, since Coulomb collisions are energy-dependent and cold here implies monoenergetic beams/populations.

My reference is J.A. Bittencourt's book Fundamentals of Plasma Physics.

Cold plasma

One understands under a cold plasma a particular choice of closure for the infinite series of moments of the Boltzmann equation.

In the momentum equation appears the pressure dyad (energy density). In a cold plasma this term is taken to be zero (no limit implied) and hence the series is truncated. As a result such a plasma fluid model only consists of mass and momentum equations.

This means however, that there is no thermal motion of particles (e.g. particle flux due to temperature gradient) or forces due to the divergence of the kinetic pressure tensor. But there are still collisions for momentum transfer. In fact it follows that the plasma temperature is zero and the velocity distribution function is a Dirac $delta$-function centered at the macroscopic flow velocity $u(r,t)$ $$f_alpha(r,v,t)=delta(v-u(r,t))$$

Warm plasma

As a side note the next easier closure is the heat flux vector in the energy density equation to be taken zero. This is also called the adiabatic approximation.

Collisionless plasma

Now, the term collisionless plasma is studied in relation to propagation of electromagnetic waves in the plasma. In this context the collision frequency $nu$ appears in the dielectric constant of the medium. Here, collisionless usually means that the collision frequency is much less than the wave frequency $omega$. $$nu llomega$$. If this holds, the resulting dispersion relations can be significantly simplified.

However, to properly study the propagation of electromagnetic waves in the plasma, one has to have a model for the medium first, meaning one has to pick a closure relation for the momemt equations.

E.g., as another side note, the theory for wave propagation in a cold plasma (as defined above) is known as the magnetoionic theory.

As a result one has dispersion relations for a cold plasma with and without collisions as well as for a warm plasma and so on.