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Behaviour of the density of states
Consider an
isolated system in equilibrium whose volume is , and whose energy lies in the
range to .
Let
be the total number of microscopic states which
satisfy these constraints.
It would be useful if we could estimate how
this number typically varies with the macroscopic parameters of the system.
The easiest way to do this is to consider a specific example. For instance,
an ideal gas made up of spinless monatomic particles. This is a particularly
simple example,
because for such a gas the particles possess translational but no
internal (e.g., vibrational, rotational, or spin) degrees of freedom.
By definition, interatomic forces are negligible in an ideal gas. In other
words, the individual particles
move in an approximately uniform potential.
It follows that the energy of the gas is just
the total translational kinetic energy of its constituent particles. Thus,

(104) 
where is the particle mass, the total number of particles,
and the
vector momentum of the th particle.
Consider the system in the
limit in which the energy of the gas is
much greater than the groundstate energy, so that all of the
quantum numbers are large.
The classical version of statistical mechanics, in which we
divide up phasespace into cells of equal volume, is valid in this limit.
The number of
states
lying between the energies and is simply
equal to the number of cells in phasespace contained between these energies.
In other words,
is proportional to the volume of
phasespace between these two energies:

(105) 
Here, the integrand is the element of volume of phasespace, with
where , , and , ,
are the Cartesian coordinates and momentum components of the th particle,
respectively.
The integration is over all coordinates and momenta such that the total energy
of the system lies between and .
For an ideal gas, the total energy does not depend on the positions of the
particles [see Eq. (104)]. This means that the integration over the
position vectors
can be performed immediately. Since each integral over extends over
the volume of the container (the particles are, of course, not allowed to stray
outside the container),
. There
are such integrals, so Eq. (105) reduces to

(108) 
where

(109) 
is a momentum space integral which is independent of the volume.
The energy of the system can be written

(110) 
since
,
denoting the , ,
components by (1, 2, 3), respectively. The above sum contains square terms.
For constant, Eq. (110) describes the locus of a
sphere of radius
in
the dimensional space of the momentum components. Hence, is
proportional to the volume of momentum phasespace contained in the spherical
shell lying between the sphere of radius and that of slightly larger
radius . This volume is
proportional to the area of the inner sphere multiplied by
.
Since the area varies like , and
,
we have

(111) 
Combining this result with
(108) yields

(112) 
where is a constant independent of or , and we have also
made use of . Note that, since the number of degrees of freedom of the
system is , the above relation can be very approximately written

(113) 
In other words, the density of states varies like the extensive
macroscopic parameters of the system
raised to the power of the number of degrees of freedom. An extensive parameter is
one which scales with the size of the system (e.g., the volume).
Since thermodynamic
systems generally possess a very large number of degrees of freedom, this
result implies that the density of states
is an exceptionally rapidly increasing function of
the energy and volume.
This result, which turns out to be quite general, is very useful in statistical
thermodynamics.
Next: Heat and work
Up: Statistical mechanics
Previous: Probability calculations
Richard Fitzpatrick
20060202