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.

Taking into account the conservation of the particles, the phase space density is conserved as well- also known as the Liouville Theorem:

ddtF(x,v,t)=0=(∂∂t+v→⋅∇x+dv→dt⋅∇v)F(x,v,t)

Where dv→dt is given by the equation of motion for a particle under the action of electromagnetic fields

ddtv→i(t)=qm[E→m(x→i(t),t)+v→i(t)×B→m(x→i(t),t)]

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

E→m(x→,v→,t)=E→(x→,v→,t)+δE→(x→,v→,t)B→m(x→,v→,t)=B→(x→,v→,t)+δB→(x→,v→,t)

By substitution we get

∂f∂t+v→⋅∇x→f+qm(E→+v→×B→)⋅∇v→f=−qm⟨(δE→+v→×δB→)⋅∇v→δF⟩

This can be written in the generalized form of the Boltzman equation:

∂f∂t+v⋅∇f+Fm⋅∂f∂v=(∂f∂t)c

where F is the force acting on the particles and ∂f/∂tc is the time rate of change of f due to collisions @chenIntroductionPlasmaPhysics2016

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

∂f∂t+v→⋅∇x→f+qm(E→+v→×B→)⋅∇v→f=0∇×B→m(x→,t)=μ0j→m(x→,t)+ϵ0μ0∂∂tE→m(x→,t)∇×E→m(x→,t)=−∂∂tB→m(x→,t)∇⋅E→m(x→,t)=1ϵ0ρm(x→,t)∇⋅B→m(x→,t)=0