### 3.6 Basics of Vlasov–Boltzmann theory

The coronal expansion and solar wind acceleration are a complex processes, which require a kinetic
description if the detailed particle velocity distributions are to be evaluated. The coronal magnetic field
guides the outflow of plasma out to the Alfvénic critical surface, where the ram pressure of the wind starts
exceeding the magnetic pressure of the coronal field, and where thus the solar wind is ultimately released
from the Sun. The subsequent almost spherical expansion and the large-scale inhomogeneity continuously
compel the solar wind plasma to attain a variable state of dynamic statistical equilibrium between the
particles and electromagnetic field fluctuations. In principle, all these kinetic processes are fully
described by the Boltzmann–Vlasov equation for the phase-space distribution for each species,
, which is a measure of the number of particles at time in a volume surrounding
position and with velocities in a certain range around . The kinetic equation reads:
with the interplanetary magnetic field, , electric field, , and the Sun’s gravitational
acceleration, . Coulomb collisions or wave-particle interactions are also included and described by the
Fokker–Planck collision integral or the quasilinear diffusion operator on the right hand side of Equation (9).
Here the particle charge is , its mass , speed , and space coordinate . The speed of light
in vacuo is . In addition to Equation (9), Maxwell’s equations have to be solved with the
self-consistent current density and charge density of the multi-component plasma of the solar
corona and wind, which are calculated from the velocity moments of Equation (9). This complex
problem has not been solved for the corona, but solutions of simplified versions of this problem,
restricted to a single particle species and special geometries, do exist, and will be discussed in
Section 7.