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It is well known that for a collisionless plasma, we have: $$ \Lambda = n_e \lambda_{De}^3 \gg 1 $$

which means that the Debye sphere is densely populated. Whereas it is the opposite for a collisional plasma, where the Debye sphere is sparsely populated. Isn't this quite counter - intuitive? Since we expect collisions to play a bigger role for a more densely populated region?

Also, why would collective behavior be more dominant in the case of a densely populated Debye sphere?

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  • $\begingroup$ It seems that the definition of coupling parameter $\Gamma$ in the mentioned Wiki-article gives the answer for your question $\endgroup$ Dec 15, 2021 at 10:26
  • $\begingroup$ It is not counter-intuitive. Indeed, from the definition of $\Gamma$ it is clear that in case of $\Lambda\gg 1$ we have wekly-coupled plasma: interaction between particles is weak, therefore it can be negligible if one consider reponse on external perturbation. In this sense $\Lambda\gg 1$ corresponds to dominance of collective behavior $\endgroup$ Dec 15, 2021 at 10:30

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What is the Debye length from physical point of view? It modifies the Coulomb interaction in plasma, producing Yukawa-like potential, $$V(r)\sim \frac{e^{-r/\lambda_D}}{r}.$$ The case of large $\lambda_D$ corresponds to weak long-range interaction of electrons in plasma, whereas in case of small Debye length electrons are interacting on long-range scale. Then, consider the plasma parameter, $$\Lambda = n\lambda_D^3.$$ With fixed $n$ and large $\lambda_D$ we deal with wekly-interacting electrons. It means that it is possible to use mean-field description: you can approximate electron-electron interactions as interaction of electron with external field. Roughly speaking, you simply can write $$\frac{e^2}{r}\sim eE_{\text{mean-field}}.$$ In such set-up, we deal with collective behavior: a response on external field is the simply sum of response of individual electron. In addition, the hydrodynamic description is valid.

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  • $\begingroup$ Does that mean $\lambda_D$ has nothing to do with the "compactness" of electrons within a region? I was thinking along more classical binary collisions where a larger number of electrons in a region (in this case a Debye sphere), would result in a larger number of collisions as the chances of them colliding is larger since there are more electrons in that region. $\endgroup$
    – D. Soul
    Dec 15, 2021 at 14:18
  • $\begingroup$ The case of large λD corresponds to weak long-range interaction of electrons in plasma, whereas in case of small Debye length electrons are interacting on long-range scale. If we fix certain r and see what potential (as given by your Eq. 1) it is for, say, λD=1000 and 100, then it turns out to be 0.000367 and 4.54*10^-8. So, for large λD we have large value of V compared with what we have for small value of λD for constant r. Which in turn means that range of potential large for large λD where range is small 4 small value of λD. Doesn’t it contradict the statement that you have made? Thanks $\endgroup$
    – sreeraj t
    Mar 7, 2023 at 1:55

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