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Question: why ions and electrons are at different temperatures in plasma?
Answer: Consider a plasma that just's been formed and then left alone. Being far from equilibrium, the plasma will evolve towards an equilibrium state. At this stage, it's not very useful to characterize the plasma with a temperature because the velocity distribution would bear little resemblance to a Boltzmann distribution, or really any kind of distribution function with meaningful moments. 

Let's assume that the plasma is being driven towards equilibrium by collision events. The particles will lose energy during each inelastic collision event, with some being more inelastic than others. The degree of inelasticity will depend significantly on the mass ratio: ion-ion collisions and electron-electron collisions will be inelastic compared to ion-electron collisions.

Since the $i-i$ and $e-e$ collisions are relatively inelastic, it often occurs that the ions and electrons quickly develop their own "temperatures". The ion velocity distribution rapidly approaches a Boltzmann distribution with temperature, and the electrons also rapidly approach a Boltzmann distribution with some different temperature. If you were to superpose the two, you'd get a funny-looking distribution with a tall top and wide tails, i.e. not global equilibrium. 

The "centeredness" or "tailedness" (basically kurtosis) of the total distribution tells you how far from equilibrium the plasma is. The further the system is from equilibrium, the faster it will try to equilibrate via collisions. So if the temperatures are very different, the $i-e$ collision frequency will be high. If the temperatures are not too different, the $i-e$ will be lower, but still higher than the $e-e$ and $i-i$ systems that have already approximately equilibrated with themselves.

Note: In laboratory experiments, it's common to produce plasmas whose ions and electrons do not settle to a Boltzmann distribution. In that case, the kurtosis argument flies out the window. Another major complication arises if the electrons and ions have different flow speeds, where so-called "two-stream" or "bump-on-tail" instabilities become important in the equilibration physics.