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5
1. Kinetic Equations:
From Newton to Boltzmann

Consider a mass particle, which moves under the action of a force. Let the
positive constant m be theparticlemassand F = F(x, t) the force field. Here x
d
in R is the position variable (d = 3 in physical space but there is at this point
no mathematical reason why m cannot be an arbitrary positive integer) and
d
t> 0 the time. The force field is a d-dimensional vector field on R ,possibly
d
time dependent. By v ∈ R we denote the velocity variable. Then the motion
d
d
of the particle is characterized by the Newtonian phase space (i.e. R × R )
v
x
trajectories, which satisfy the system of ordinary differential equations (ODEs):
˙ x = v (1.1)
1
˙ v = F(x, t) . (1.2)
m
Note that the first equation simply states that the particle’s velocity is the time
1
derivative of its position and the second equation is just Newton’s celebrated
second law, stating

force = mass · acceleration .

If the field F is sufficiently smooth, then by standard ODE theory we conclude
that, given an initial state

2d
x(t = 0), v(t = 0) = (x 0 , v 0 ) ∈ R
there exists a locally defined, unique and smooth trajectory (x(t; x 0 , v 0 ),
v(t; x 0 , v 0 )). Thus, giventhe force and the initial positionand velocity, the motion
of the mass particle is – in the framework of classical Newtonian mechanics –
completely determined. However, in many applications, there are additional
complications …
Assume at first that the intial state (x 0 , v 0 )isnot knownapriorily, instead
let f 0 = f 0 (x, v) be a given probability distribution of the initial state, i.e. f 0 is
non-negative, its integral over the whole phase space is 1 and, for any measurable
subset A of the phase space,

f 0 (x, v)dxdv = : P 0 (A)
A
1
We refer to the webpage http://scienceworld.wolfram.com/biography/Newton.html for a biogra-
phy of Isaac Newton (1642–1727).