# Cold plasma approximation leads to infinite coupling

In plasma physics, there is an approximation called cold plasma approximation according to which, the temperature of a plasma species in an electron-ion plasma, say ions ( $$T_i$$ ) is assumed to be zero. As far as I have understood this approximation is valid when the average temperature of electrons ($$T_e$$) is much larger than that of ions ( $$T_e>>T_i$$).

For any plasma species, one can define Coulomb coupling parameter($$\Gamma$$), $$\Gamma=\frac{q^2}{4\pi\epsilon_0ak_BT_i},$$ which is basically the ratio of average interparticle potential energy to average thermal energy, where $$q,\;a\;\text{and} \;T_i$$ denotes charge, average interparticle distance and average temperature of ions respectively.

So, if one assumes ions to be cold, he/she gets infinite coupling for ions ( as $$T_i\to0$$, $$\Gamma\to\infty).$$

I don't understand the physical meaning of infinte coupling. Is this result ( i.e, $$\Gamma\to\infty$$ as $$T_i\to 0)$$ correct physically ?

• I would imagine that the coupling in a $T=0$ plasma would require certain adjustments in its definition (i.e., using a $\Gamma$ derived under one set of assumptions may not be valid for another set). For instance, rather than using $T_i$ it might be required to use $T_e$ as any propagation of signals is through the electrons. Commented Sep 7, 2023 at 15:48
• But $\Gamma$ can be separately defined for ions and electrons. That means $\Gamma$ for ions cannot be replaced by $\Gamma$ for electrons. Of course for some phenomena one may need $\Gamma$ for electrons and for others one will need that of ions. Commented Sep 7, 2023 at 17:11
• Those ratios usually just indicate the relative strength of certain terms in a partial differential equation. If you parameter goes to zero or infinity, it just means that one term loses against the other in determining the time evolution of the system. Commented Sep 8, 2023 at 22:13

As far as I have understood this approximation is valid when the average temperature of electrons ($$T_{e}$$) is much larger than that of ions ($$T_{e} \gg T_{i}$$).

No, this is not really necessary. The cold plasma approximation is more of a statement that the finite temperature effects are not important for the phenomena in question. Sometimes this is thought of in terms of the plasma beta, defined as: $$\beta_{s} = \frac{ 2 \ \mu_{o} \ n_{s} \ k_{B} \ T_{s} }{ B_{o}^{2} } \tag{0}$$ where $$\mu_{o}$$ is the permeability of free space [$$T \ m \ A^{-1}$$], $$n_{s}$$ is the number density [$$\# \ m^{-3}$$] of species $$s$$, $$k_{B}$$ is the Boltzmann constant [$$J \ K^{-1}$$], $$T_{s}$$ is the average temperature [$$K$$] of species $$s$$, and $$B_{o}$$ is the quasi-static magnetic field magnitude [$$T$$].

In a plasma where $$\beta_{s} \ll 1$$, one can usually assume that thermal contributions are not as important as magnetic ones (e.g., this works well for whistler waves).

I don't understand the physical meaning of infinte coupling. Is this result (i.e., $$\Gamma \rightarrow \infty$$ as $$T_{i} \rightarrow 0$$) correct physically?

Physically correct? No, not really. Though when $$\Gamma \ll 1$$ (e.g., most space plasmas), the approximation of an ideal gas is much more accurate for a plasma than when $$\Gamma \gg 1$$ (e.g., many types of high energy lab plasmas like inertial fusion experiments). The reason being is that the weakly coupled plasma cares more about the electrostatic and electromagnetic interactions than strict binary particle-particle collisions. Or another way to think about it is that when $$\Gamma \gg 1$$, you are packing the particle constituents so tightly together that it disrupts the usual collective plasma behavior because the Coulomb fields are so strong, they dominate the particle dynamics.

• As you have said cold plasma approximation does not necessarily mean $T_i << T_e.$ But in a One Component Plasma ( for example, a system of Coulomb interacting ions embedded in a uniform neutralizing background) does cold plasma approximation mean $T_i\to 0$ ? Commented Sep 8, 2023 at 19:58
• If there is only one ionized charge specie, then it's not a plasma... Commented Sep 8, 2023 at 20:06
• Please have a look at Setsuo Ichimaru's monograph. Here plasma is defined as : " A plasma may be defined as any statistical system containing mobile charged particles. Vague as it may sound the forgoing statement is sufficient to define what is known as plasma in science and engineering. " Also, plasma description by taking a single charged particle species by assuming the other species to be Boltzmann distributed is used to describe many comlex plasma phenomena. For example see, journals.aps.org/prl/pdf/10.1103/PhysRevLett.93.155002 Commented Sep 8, 2023 at 20:32