7.5 The Liouville and Boltzmann Equations
385
= −
sN
j =1
f
(N )
∂ ˙
q j
∂q j
+
∂ ˙
p j
∂p j
+
∂f (N )
∂q j
˙
q j +
∂f (N )
∂p j
˙
p j
δqδp .
We shall now utilize Hamilton’s equations (7.5.1) to help simplify this result: the
first term on the right-hand side of our net result vanishes, and we can similarly
simplify the remaining two terms to give
d(δN )
dt
= −
sN
j =1
∂f (N )
∂q j
∂H
∂p j
−
∂f (N )
∂p j
∂H
∂q j
(7.5.5)
for the final result.
3. We are now ready to take the final step in this procedure: the change in the
number of phase points passing through the phase volume element δqδp divided
by the volume element must represent the change in the phase-point density f (N )
with time for that phase volume, i.e.,
∂f (N )
∂t
= −
sN
j =1
∂f (N )
∂q j
∂H
∂p j
−
∂f (N )
∂p j
∂H
∂q j
.
It is conventional to rearrange this final result slightly and to write it as
∂f (N )
∂t
+
sN
j =1
∂f (N )
∂q j
∂H
∂p j
−
∂f (N )
∂p j
∂H
∂q j
= 0 .
(7.5.6)
Equation (7.5.6), referred to as the Liouville equation, is one of the most fundamental equations of classical statistical mechanics because it provides the starting point
for most theories of nonequilibrium statistical mechanics.
In terms of a Cartesian representation of the positions and linear momenta of the
molecules in the gas, we may write q = r ≡ (r 1 , · · · , r N ) for the positions, and
p ≡ (p 1 , · · · , p N ) = m(˙ r 1 , · · · , ˙
r N ) for the linear momenta of the N molecules.
Moreover, as the Hamiltonian for an atomic system is
H =
N
i=1
p 2
i
2m i
+ V (r 1 , · · · , r N ) ,
(7.5.7)
Hamilton’s equations of motion reduce to
∂H
∂p i
=
1
m i
p i , and
∂H
∂r i
=
∂V
∂r i
≡ − F i .
(7.5.8)