Kinetic approach of plasma

The behavior of plasma can be described through several frameworks, depending on the assumptions and its characteristics. A well known approach is the magnetohydrodynamic one, which treats the plasma as a conducting fluid, with macroscopic quanities such as density, velocity and pressure. This approach is valid when the plasma is collisional, meaning that the mean free path of the particles is much smaller than the characteristic length scale of the system.

However, in the magnetosphere, the plasma is often collisionless, while there are also regions that exhibit strong wave-particle interactions or non-Maxwellian distributions. In these cases, the kinetic approach is more appropriate, where the distribution function of the particles is considered.

When referring to the distribution function here, we are considering the probability to find a particle in each point of phase space, rather than the exact position and velocity of each particle. This is imposed by the complexity of solving for the exact phase space number densities coupled with the microscopic Maxwell equations, which in turn rely on the exact positions and velocities of the particles. To simplify this problem, one can define the ensemble averaged phase space density as $⟨F(x,v,t)⟩=f(x,v,t)$. The exact phase space density would then be

where $⟨\delta F⟩=0$, as an ensemble average over a fluctuation.

Taking into account the conservation of the particles, the phase space density is conserved as well:

Where $\frac{d\overrightarrow{v}}{dt}$ is given by the equation of motion for a particle under the action of electromagnetic fields

where the fields E and B are given by the microscopic Maxwell equations. We can again express the fields as sums of averages and fluctuations

By substitution we get

Since we are interested in collissionless plasma, the right hand term can be neglected, leading to the Vlasov equation: