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1 Kinetic Equations: From Newton to Boltzmann
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space mass density f of a single particle rarified gas with identical, perfectly
spherical, elastically colliding molecules reads:
∂ F
f (x, v, t)+ v .grad f (x, v, t)+ .grad f (x, v, t)
∂t x m v

= B(v, w, n) f (x, v , t) f (x, w , t)− f (x, v, t) f (x, w, t) dndw := Q ( f , f ).

∗
∗
3 S 2
3
2
Here S is the unit sphere in R and B = B(v, w, n)standsfor thecollision kernel
representing the microscopic properties of the molecular collisions. Typically B
only depends on the relative post-collisional velocity V = w−v and on the angle
between the vectors n and V.
As mentioned above, the integral operator Q ( f , f )onthe righthandsideis
usually referred to as collision integral. It represents the statistics of all possible
collision events leading to the post-collisional velocity v (first integrand term,
called gain term after the integrations) and of all possible outgoing collisions
(second integrand term, called loss term after the integrations). Note that the
bilinear nature of the collision integral is due to the fact that only binary molec-
ular collisions are taken into account by the Boltzmann equation. Collisions of
three and more molecules are neglected, which implies that the considered gas
has to be sufficiently rarified.
External interactions of the gas, e.g. with the hull of the re-entrant space
shuttle mentioned above, have to be modeled by appropriate boundary condi-
tions.
For a wealth of mathematical detail, including a derivation from multi-
particle physics and important mathematical properties of the Boltzmann equa-
tion we refer to the books [4] and [16].
A fascinating reading about the life of Ludwig Boltzmann, to whom the
equation is attributed, at least in the case of hard sphere molecules with
B(v, w, n) = const. |V.n|, after his celebrated paper from 1872 [3], the role
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of Maxwell in the derivation of the Boltzmann equation and about the cultural-
scientific background of their times, is provided by [5].
Formathematicalpurposesitisconvenienttorewrite theBoltzmann equa-
tion in terms of dimensionless quantites. Then a parameter ε,the so called
Knudsen number (= normalized particle mean free path), appears:

F
ε f t + v .grad f + .grad f = Q ( f , f ).
x v
m
(for the sake of simplicity we use the same notations for the dimensionless
quantities). Note that the case ‘ε large’ corresponds to ballistic transport, ‘ε of
order 1’ represents a dynamic balance between free transport and collisional
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http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Maxwell.html