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1 Kinetic Equations: From Newton to Boltzmann
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effects and ‘ε small’ is the case of collision-dominated transport (cf. the space
shuttle example cited above!). The operator Qhas 5 so-called collisioninvariants,
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i.e. when it is multiplied by the 5-vector (1, v, |v| ) and integrated over velocity
space, then (at least formally) zero is obtained. This corresponds to the classical
physical requirements of mass, momentum und energy conservation in gas
flows, which are thus verified on a formal level. Also the Boltzmann equation
preserves positivity, i.e. solutions f with nonnegative initial data f (t = 0) will
remain nonnegative during the time evolution (as long as they exist), as required
for probability densities.
It is a trivial exercise to show that the post-collisional velocities become the
pre-collisional ones when another collision process is applied to them, i.e. the
map from (v, w)to(v , w ) is an involution. This implies that each individual col-
∗
∗
lision process is reversible, i.e. when the post-collisional velocities are reversed
and put through a collision then the negative original pre-collisional velocities
are obtained. Boltzmann realized that this micro-reversibility does not imply
reversibility of the gas flow. On the contrary the Boltzmann equation is non-
reversible (although based on reversible binary micro-collisions!). In particular,
the Boltzmann equation dissipates the convex functional H given by

H ( f ):= f log fdvdx .

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This means that along a sufficiently smooth nonnegative solution f of the
Boltzmann equation the inequality

d
H f (t) =: S ( f (t)) ≤ 0
dt
holds, with equality iff f is a local Maxwellian function:

ρ(x, t) |u(x, t)− v| 2
exp − ,
f (x, v, t) =
3/2
2πT(x, t) 2T(x, t)
where ρ, u, T are the position density, mean velocity and, resp., temperature
associated to f .
The quantity H(f ) is the (negative) physical entropy of the phase space
density f and S(f ) is its dissipation generated by the time evolution of the
Boltzmann equation. This explains also – again on a formal level – the tendency
of solutions of the Boltzmann equation to converge to (global) Maxwellians in
the large time limit and the connection to the macroscopic Euler and Navier–
Stokes equations obtained (formally!) by the limit procedure ‘ε → 0’ and by
assuming that f is a local Maxwellian.
The Boltzmann equation has been a great challenge for mathematicians
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(a long list starts with David Hilbert ) but some important analytical results are
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http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Hilbert.html