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1 Kinetic Equations: From Newton to Boltzmann
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where T t denotes the Newtonian flow map, which maps a point in phase space
into the state of the Newtonian trajectory at time t. Here we assumed that the
Newtonian map is defined globally for t> 0.
Anothercomplicationarisesfromthefactthatinthemostimportantphysical
cases there is not only one isolated particle to observe but instead a swarm
consisting of a large number of particles, which interact with each other. Two
types of interactions are distinguished, namely long range and short range
interactions. Typical long range interactions are either given by the repulsive
Coulomb force of electrodynamics, occurring in charged particle transport,
or the attractive gravitational force, e.g. occurring in the modeling of galaxy
motion. Short range interactions can be classified as particle collisions, they will
be discussed in detail later.
In the small coupling thermodynamical limit (i.e. small force, number of par-
ticles tends to infinity) long range forces typically lead to nonlocal nonlinearities
in the effective Liouville equation, when the total chaos assumption (Hartree
ansatz) is made (see [11]). In the Coulomb/gravitational case we obtain (after
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appropriate scaling) the so called Vlasov–Poisson system (see [7]):
f t + v .grad f −grad V .grad f = 0 (1.5)
x v
± ΔV = n (1.6)

n = fdv . (1.7)
d
Here f is the effective particle number density on 2d-dimensional phase
space, dependent on time t of course, V is the mean field Coulomb/gravitational
potential (the + sign in front of the Laplacian in the Poisson equation (1.6)
corresponds to the gravitational case and the − sign to the Coulomb case), and
n denotes the position space number density.
The existence and uniqueness of smooth solutions of the Vlasov–Poisson in
the 6-dimensional phase space case was a longstanding open problem, finally
answered positively in [14] and shortly afterwards in [10].
Short range interactions (so called particle collisions) typically lead to ad-
ditional terms in Liouville-type kinetic equations, which are nonlocal in the
velocity variable. In the absence of an external potential and neglecting long
range interactions, the number (or mass) phase space density of a particle
swarm undergoing collisional events, satisfies a kinetic equation posed in the
2d dimensional phase space (after the Boltzmann–Grad limit):

f t + v .grad f = Q ( f , f ) , (1.8)
x
where Q ( f , f ) is the nonlocal collision operator (typically an integral operator).
Theequation(1.8)modelsadynamicbalancebetweenthefreestreamingparticle
motion, represented by the left hand side of the equation, and the collisions.
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http://relativity.livingreviews.org/open?pubNo=lrr-2002-7&page=articlesu2.html