1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
45
O s,t (() =
t
s
O(S
u
)du, O s,t =
1
t − s
t
s
O(S
u
)du
(1.173)
the solution of Eq. (1.169) can be formally written as:
f
(t)
(() = exp
− −t,0 (()
f
(0)
(S
−t
)
= exp
f
(0)
−t,0 (()
f
(0)
(()
(1.174)
Then, a density f
(0) evolves, unless
f
(0) identically vanishes. Thanks to these expressions, one may compute the time dependent phase space averages of phase functions,
obtaining:
O =
M
O((){
()
(()d =
M
O((){
()
(S
−
)J
−
(()d
(1.175)
where J
−t
(() = exp
− −t,0 (()
. Then, introducing = S
t Y we get:
M
O(S
t Y ) f 0 (Y )J
−t
(S
t Y )J
t
(Y )dY
(1.176)
where J
t
(Y ) is the variation of phase space volumes along a trajectory starting at Y
and proceeding for a time t, and J
−t
(S
t Y ) is the variation of the backward evolution
coming back from S
t Y , which can be easily shown to obey J
−t
(S
t Y ) = 1/J
t
(Y ).
15
One then gets:
O =
M
O((){
()
(()d =
M
O(S
){
()
(()d = =O ◦ S
(1.177)
proving that evolution of observables can equivalently be obtained evolving either the
microscopic quantity itself or the probability density. This is the classical analog of
the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics.
1.6.1 Boltzmann Equation and H-Theorem
Let us specify the above general treatment of probabilities in phase space, to the case
of N identical spherical hard particles of mass 1, that are subjected to no external
forces, and that only interact elastically when they collide. We may think of their
15 Note, S −t traces backward the phase space trajectory; it is not the time reversed trajectory, cf.
(1.195) below.