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1 Kinetic Equations: From Newton to Boltzmann
6

Fig. 1.1. Airplane departing from Rio de Janeiro’s City Airport

is theprobability of finding theparticleattime t = 0inthe set A. Then, instead of
calculating the evolution of the phase space trajectories we can try to compute
the location probability density f = f (x, v, t), evolving out of f 0 . For this we
impose the condition that f remains constant along the Newtonian trajectories:

d
f x(t; x 0 , v 0 ), v(t; x 0 , v 0 ), t = 0.
dt
Carrying out the differentiation with respect to time, taking into account the
Newtonian equations (1.1, 1.2) and renaming coordinates gives the so called
Liouville equation:

1
d
d
f t + v .grad f + F .grad f = 0, x ∈ R , v ∈ R ;, t> 0 , (1.3)
x v
m
subject to the initial condition
f (t = 0) = f 0 . (1.4)
Note that the Liouville equation is a linear hyperbolic PDE, whose charcter-
istics are precisely the Newtonian trajectories. Thus, its solution can be written
as

f (x, v, t) = f 0 T −t (x, v) ,