Phase space evolution in linear instabilities

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Abstract
A simple and powerful way to investigate the linear evolution of particle distribution functions in kinetic instabilities in a homogeneous collisionless plasma is presented. The method can be applied to any kind of instability, provided the characteristics (growth rate, frequency, wave vector, and polarization) of the mode are known and can also be used to estimate the amplitude of the waves at the end of the linear phase of growth. Two didactic examples are used to illustrate the versatility of the technique: the Alfvén Ion Cyclotron (AIC) instability, which is electromagnetic, and the Electron Ion Cyclotron (EIC) instability, which is electrostatic.