10.1.1. Liouville’s Theorem
The statistical description of a system in thermal equilibrium begins with the
assumption that the measured values of its macroscopic physical properties are long
time averages of the corresponding instantaneous quantities. This provides the link
between the thermodynamic and the atomic properties of the system. However, time
averages are difficult to calculate. The next crucial step in the theory is to replace
long time averages with the so-called ensemble averages.
In classical mechanics, the instantaneous state of a system of N particles in 3-D space
can be represented as a point in a 6N-D manifold called the phase space  (see §3.7).
It is convenient to denote the 3N generalized coordinates Q   q1,
generalized momenta P   p1,
X  Q, P    q1,
, q3N  and 3N
, p3N  as a 6N-D coordinate
, q3N , p1,
, p3N 
An ensemble average is a weighted average of a quantity f(X) at time t given by
f
t
  d 6N X   X , t  f  X 
(10.1)

where   X , t  is a probability density for finding at time t the system near state X
(in volume d 6 N X centered about X ). Physically,   X , t  denotes the fraction of
systems expected to be found near state X at time t in a Gibbs ensemble. By “Gibbs
ensemble” we mean a collection of all possible distinct systems that can be
constructed, out of a given set of constituents, to obey a given set of macroscopic
boundary conditions. In other words, every instantaneous state consistent with the
macroscopic boundary conditions is represented by exactly one member in the Gibbs
ensemble.
The evolution of   X , t  can be treated as the flow of an incompressible
‘probability fluid’ in phase space.
The current density of this fluid is simply
ji  X i   X , t 
Since there is no sink or source, the equation of continuity gives
 6 N ji  6 N 



 Xi   0
t i 1 X i t i 1 X i
(10.2)
From Hamilton’s equations, we have
X i 3 N  qi pi  3 N   H
 H 
 



  
0
pi  i 1  qi pi pi qi 
i 1 X i
i 1  qi
6N
(10.3)
so that (10.2) can be written as
 6 N
  3N  
 
  Xi

   qi
 pi

t i 1 X i t i 1  qi
pi 


i
 3N  H  H  
 

0
t i 1  pi qi qi pi 

 H
t
(10.3a)
(10.4)
where the Liouville operator is defined by
H  H 
iH 

(3.19)
pi qi qi pi
Eq(10.4) is called the Liouville equation. [ A proof of this for a special form of 
was already given in §3.4 ]. Now, to an observer that moves with the probability
fluid flow ( or follows the evolution of a given member system ), the time rate of
change of   X , t  is given by
d   6 N


  Xi
0
dt
t i 1 X i
(10.5)
where (10.3a) was used. Hence, the probability of finding a particular system in the
ensemble is the same for all time, which is why the probability fluid is called
incompressible. This is known as Liouville’s theorem.
For a system in equilibrium, all averages
f
t
should be independent of time.
According to (10.1), we have
d f
dt
t

  X , t 
d
f X   0
d 6N X   X , t  f  X    d 6N X

t
dt 

for all measurable f. This can be true only if
H   i H ,  P  0

 0 , which, by (10.4), requires
t
In other words,  can depend only on conserved quantities of the system. Obviously,
the quantities that are conserved are selected by the boundary conditions that specify
how the system interacts with its environment. As will be shown shortly, once the
boundary conditions are specified, the construction of the appropriate  is quite
straightforward.