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1 Kinetic Equations: From Newton to Boltzmann
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To understand the modeling hierarchy consider a space shuttle orbiting the
earth outside its atmospheric layer, i.e. in a vacuum. Obviously, the shuttle
moves there in ballistic motion free of interactions. Then, when the shuttle
starts its re-entry phase, interactions of the shuttle hull with the molecules
of the upper atmosphere will start to take place. Since the upper atmosphere
is highly rarified, only few interactions will occur within a given time unit.
After a short time, when the shuttle enters more deeply into the earth’s at-
mospheric layer, the effect of these gas molecule-shuttle hull collisions will
dynamically balance the free-streaming shuttle motion. Deeper down in the
atmosphere, close to the surface of the earth, the air becomes even less rar-
ified such that there will be many collisions of air particles with the shuttle
hull (and many air molecule-air molecule collisions) within a given unit of
time such that they start to dominate the shuttle motion. This re-entry pro-
cess shows a typical transition from ballistic motion (infinite mean free path
between consecutive collisions) to ballistic-collision equilibrated motion (or-
der 1 mean free path) to collision dominated motion (small mean free path).
Theformer regimehas averysimplemathematicaldescription (namely free
streaming phase space flow, i.e. the Liouville equation without external force),
the latter is precisely the fluid regime covered by macroscopic flow equations and
the middle regime requires a microscopic molecular-based model, as presented
below.
Let f = f (x, v, t) be the expected mass density in (position x,velocity v)
phase space of the gas at time t, i.e. the expected mass per unit volume, at time t,
in the six-dimensional phase space. Assume that the gas consists of perfectly
spherical identical molecules of diameter D. Now consider two gas molecules,
immediately after a collision event between themselves, with states (x, v)and,
resp., (x − Dn, w), where n is the unit vector along the directions connecting
the centers of the spheres. Then, immediately prior to this collision, the phase
space states of these two molecules were (x, v )and,resp.,(x − Dn, w ), where
∗
∗
the pre-collisional velocities v and w satisfy momentum conservation
∗
∗
m(v + w) = m(v + w )
∗
∗
and energy conservation
∗ 2
∗ 2
2
2

m(v + w ) = m (v ) +(w )
in the collision process. Here m denotes the mass of the gas molecules. The
pre-collisional velocities read:
v = v + n .(w − v)n , w = w + n .(v − w)n .
∗
∗
Then the celebrated Boltzmann equation, named after the Viennese Physicist
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Ludwig Boltzmann (1844–1906) describing the temporal evolution of the phase
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http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Boltzmann.html