7.5 The Liouville and Boltzmann Equations
383
determined by the location of the phase point at some initial time t = t 0 . Of course,
this cannot actually be accomplished in practice.
We shall now introduce a classical microcanonical phase space ensemble to aid
us in developing the distribution function concept. Consider a large number N of
isolated systems, each having the same values of the macroscopic variables N , V ,
and E ≡ U . As each N-particle system in this ensemble will have a representative
phase point in the same phase space, the entire ensemble will make up a cloud of
points in phase space, each point tracing out its own (independent) trajectory with
time. Note that as each system is isolated in a microcanonical ensemble, this ensures
its independence.
The role of the ‘equal a priori probability’ postulate is to require there to be a
representative phase point in phase space for each and every set of coordinates and
conjugate momenta that is consistent with the fixed values of N, V , and U . For
a microcanonical ensemble, this requirement reduces to a uniform density over a
constant energy surface in phase space. The cloud of phase points is thus very dense,
which allows us to define a number density f (N ) (p, q, t) which has the property that
the number of systems in the ensemble that have phase points in a volume element
dpdq about the point (p, q) at time t is given by f (N ) (p, q, t) dpdq. We need to
impose the constraint that the number of members of our ensemble (i.e., the number
of N-particle systems) be given by
N =
f
(N ) (p, q, t) dpdq .
(7.5.2)
As we have seen in Chap. 3, we may define the ensemble average of a quantity
M(p, q) as
M(t) ≡
1
N
M(p, q)f
(N ) (p, q, t) dpdq .
(7.5.3)
We may also postulate, as was done originally by Gibbs, that the ensemble average
is the same as the corresponding thermodynamic function. To proceed further, we
note that as the equations of motion determine the trajectories of the phase points in
phase space, they must also determine the density f (N ) (p, q, t) of the phase points
at any time if the dependence of f (N ) (p, q, t) on p, q is known at some initial time
t 0 .
Let us consider a volume element δpδq ≡ δp 1 · · · δp sN δq 1 · · · δq sN about (p, q):
the number of phase points δN inside this phase-space volume element will be
δN = f
(N ) (p, q, t) δpδq .
(7.5.4)
Now, let us determine the difference between the numbers of phase points entering
and leaving this volume element in phase space via a three-step process. In step 1 we
shall consider the net change obtained for one variable, in step 2 we shall extend the
argument utilized in step 1 to the full set of sN conjugate variables and determine
383
determined by the location of the phase point at some initial time t = t 0 . Of course,
this cannot actually be accomplished in practice.
We shall now introduce a classical microcanonical phase space ensemble to aid
us in developing the distribution function concept. Consider a large number N of
isolated systems, each having the same values of the macroscopic variables N , V ,
and E ≡ U . As each N-particle system in this ensemble will have a representative
phase point in the same phase space, the entire ensemble will make up a cloud of
points in phase space, each point tracing out its own (independent) trajectory with
time. Note that as each system is isolated in a microcanonical ensemble, this ensures
its independence.
The role of the ‘equal a priori probability’ postulate is to require there to be a
representative phase point in phase space for each and every set of coordinates and
conjugate momenta that is consistent with the fixed values of N, V , and U . For
a microcanonical ensemble, this requirement reduces to a uniform density over a
constant energy surface in phase space. The cloud of phase points is thus very dense,
which allows us to define a number density f (N ) (p, q, t) which has the property that
the number of systems in the ensemble that have phase points in a volume element
dpdq about the point (p, q) at time t is given by f (N ) (p, q, t) dpdq. We need to
impose the constraint that the number of members of our ensemble (i.e., the number
of N-particle systems) be given by
N =
f
(N ) (p, q, t) dpdq .
(7.5.2)
As we have seen in Chap. 3, we may define the ensemble average of a quantity
M(p, q) as
M(t) ≡
1
N
M(p, q)f
(N ) (p, q, t) dpdq .
(7.5.3)
We may also postulate, as was done originally by Gibbs, that the ensemble average
is the same as the corresponding thermodynamic function. To proceed further, we
note that as the equations of motion determine the trajectories of the phase points in
phase space, they must also determine the density f (N ) (p, q, t) of the phase points
at any time if the dependence of f (N ) (p, q, t) on p, q is known at some initial time
t 0 .
Let us consider a volume element δpδq ≡ δp 1 · · · δp sN δq 1 · · · δq sN about (p, q):
the number of phase points δN inside this phase-space volume element will be
δN = f
(N ) (p, q, t) δpδq .
(7.5.4)
Now, let us determine the difference between the numbers of phase points entering
and leaving this volume element in phase space via a three-step process. In step 1 we
shall consider the net change obtained for one variable, in step 2 we shall extend the
argument utilized in step 1 to the full set of sN conjugate variables and determine