wind, the Van Allen radiations belts, and galactic nebulae.

From a mathematical perspective, the fundamental Lorentz-invariant equations which describe the time evolution of a collisionless plasma are given by the relativistic Vlasov-Maxwell system: $$\label{RVM} \tag{RVM} \left \{ \begin{gathered} \partial_t f + \hat{v} \cdot \nabla_x f + \left (E + \hat{v} \times
B \right ) \cdot \nabla_v f = 0 \\  \rho(t,x) = \int f(t,x,v) \ dv, \quad j(t,x)= \int \hat{v}  f(t,x,v) \ dv \\  \partial_t E = \nabla \times B - j, \qquad \nabla \cdot E = \rho \\  \partial_t B = - \nabla \times E,
\qquad \nabla \cdot B = 0. \\ \end{gathered} \right.$$ Here, $f$ represents the distribution of (positively-charged) ions in the plasma, while $\rho$ and $j$ are the charge and current density, and $E$ and $B$ represent electric and magnetic fields generated by the charge and current. The independent variables, $t \geq 0$ and $x,v \in \bfR^3$ represent time, position, and momentum, respectively, and physical constants, such as the charge and mass of particles, as well as, the speed of light, have been normalized to one. The structure of the velocity terms $\hat{v}$ in (\[RVM\]) arise due to relativistic corrections, and this quantity is defined by 