The purpose of kinetic theory is to model the time evolution
of a collection of particles. The particles may be entirely
different objects depending on the physical situation. For
instance, the particles are atoms and molecules in a neutral gas
or electrons and ions in a plasma. In stellar dynamics the
particles are stars and in a cosmological case they are galaxies
or even clusters of galaxies. Mathematical models of particle
systems are most frequently described by kinetic or fluid
equations. A characteristic feature of kinetic theory is that its
models are statistical and the particle systems are described by
distribution functions
*f*
=
*f*
(*t*,
*x*,
*p*), which represent the density of particles with given spacetime
position
and momentum
. A distribution function contains a wealth of information, and
macroscopical quantities are easily calculated from this
function. In a fluid model the quantities that describe the
system do not depend on the momentum
*p*
but only on the spacetime point (*t*,
*x*). A choice of model is usually made with regard to the physical
properties of interest for the system or with regard to numerical
considerations. It should be mentioned that a fluid model that is
too naive may give rise to shell-crossing singularities, which
are unphysical. In a kinetic description such phenomena are ruled
out.

The time evolution of the system is determined by the interactions between the particles which depend on the physical situation. For instance, the driving mechanism for the time evolution of a neutral gas is the collision between particles (the relativistic Boltzmann equation). For a plasma the interaction is through the electric charges (the Vlasov-Maxwell system), and in the stellar and cosmological cases the interaction is gravitational (the Einstein-Vlasov system). Of course, combinations of interaction processes are also considered but in many situations one of them is strongly dominating and the weaker processes are neglected.

- 1.1 The relativistic Boltzmann equation
- 1.2 The Vlasov-Maxwell and Vlasov-Poisson systems
- 1.3 The Einstein-Vlasov system

The Einstein-Vlasov System/Kinetic Theory
Håkan Andréasson
http://www.livingreviews.org/lrr-2002-7
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