Re: [Phys-l] equipartition riddle

[John Denker <jsd@av8n.com>]

Here's my analysis of the riddle:

1) There is no such thing as a system with one degree of freedom.

If the /motion/ has one dimension (x), the phase space has two 
dimensions (x and p).  If the motion has three dimensions, the 
phase space has six dimensions.

It is AFAIK impossible to do classical mechanics without looking 
at the phase space, and even more impossible to do any kind of 
thermodynamics without looking at the phase space.  The entropy 
of a system is essentially the logarithm of the number of ways it 
can re-arrange itself _in phase space_.

2) This means that from the point of view of theoretical physics, 
the usual analysis of an ideal gas is a swindle.  The swindle can 
be justified _a posteriori_ by saying that the equations agree with 
experiment, but we can't say that the equations agree with the 
analysis, because the analysis doesn't even agree with itself.  
The analysis is internally inconsistent.

Specifically, there is no scenario for the ideal gas in which the 
position variable can be entirely neglected.  For a particle in a box:
 -- It is unphysical to think that the particle is infinitely small 
  and hard, such that it does not deform when it hits the wall.
 -- It is unphysical to think that the walls of the box are infinitely
  tall, steep, and impenetrable.
 -- Even if you do assume both of the above, it is unphysical to think
  that the floor of the box has no slope at all.  There is always going
  to be some kind of stray field, such as a gravitational field, that
  imparts some slope to the floor.
 -- et cetera

To summarize:  We really need to consider both momentum and position.
The classical Hamiltonian depends on both p and x.  For the square well,
we need to find a theoretically-sound explanation of why the p-related 
term in the Hamiltonian contributes to the heat capacity whereas the 
x-related term does not.

3) Keep in mind that we are working in the classical limit.  That means 
that the temperature is not too high and not too low:
   -- kT must be small compared to mc^2
   -- kT must be large compared to the spacing between energy levels
       at the bottom of the potential well

Philosophical remark: It would be quite wrong to say that the laws of
relativity do not apply, or that the laws of QM do not apply.  Those
laws *always* apply.  The point is that we are working an a regime 
where those laws tell us that the classical approximation is OK.

Also:  We are assuming a canonical ensemble.  That means fixed 
temperature and fixed number of particles.

4) As David Bowman mentioned, the usual pedestrian statement of the 
equipartition theorem says that each _quadratic_ degree of freedom
contributes 1/2 kT to the average energy, in thermal equilibrium.

This leaves us with a gigantic question, namely, what about non-quadratic
degrees of freedom?
 -- What about a DoF that is almost quadratic, such as we find in a
  /slightly/ anharmonic oscillator?
 -- What about a DoF that is nowhere near being quadratic, such as we
  find in a square well, or an almost-square well?
 -- What about the intermediate cases?

We can answer this question using the _generalized_ equipartition theorem
as derived in e.g. Sturge Appendix G:

   〈q ∂H/∂q〉 = kT                          [1]

for each and every canonical variable q that appears in the Hamiltonian, 
and where the brackets 〈...〉 indicate a thermal average.  You can prove 
this starting from the canonical partition function, by integrating by 
parts.

Note that both x and p are canonical variables.  At any given time, the
Hamiltonian depends on both x and p.  Equation [1] applies to each of
them separately.  Take q=p or q=x as appropriate.

5) The non-relativistic KE is p^2 / 2m, so equation [1] tells us that the
KE contributes 1/2 kT to the average energy.  This is what we expected.

6) For the harmonic oscillator, the PE is 1/2 k x^2, so equation [1] tells
us that the PE contributes 1/2 kT to the average energy.  Again, this is
what we expected.

7) More generally, for a power-law potential that goes like |x|^N for some 
positive N, the PE contributes kT/N to the average energy.  This includes 
the harmonic oscillator as a special case (N=2).

8a) For a particle moving in a slightly anharmonic potential, with N very 
near 2, the specific heat capacity is very nearly 1.0 in the appropriate 
units.  This is not particularly remarkable.

8b) For a particle moving in the anharmonic power-law potential with N=10,
the specific heat capacity is 0.6 in the appropriate units, as we see by
combining point (5) with point (7).  This is IMHO a nontrivial result.

We see that there is nothing magical about 1/2 kT.

8c) For large N, the specific heat capacity is 0.5 in the appropriate
units.  This is consistent with observations on ideal gases and the like.
The agreement with observations is unremarkable.  The remarkable point
is that now we have a self-consistent theory that explains the lack of
any PE contribution to the heat capacity.

IMHO it is not obvious that the PE of the ideal gas is negligible.  It
turns out to be true, in the usual classroom situation, but that doesn't 
make it obvious.  And in fact it's not always true!  A tall column of
ideal gas in a gravitational field is easy to analyze.  It does *not* 
obey the so-called ideal gas law, and (in 3D) its specific heat capacity
is *not* 3/2.

9) We can understand the physics of the power-law potential as follows:  
Consider some point x2 in the potential where |x| is greater than 1, 
i.e. some point where the particle is trying to climb the walls of the 
potential well.  Assume thermal equilibrium.  Consider the case of large 
but finite N.  As N becomes larger and larger:
  a) the potential energy at this point x2 gets larger and larger
  b) the amount of time the particle spends at this point get smaller
   and smaller 
  c) effect (b) outweighs effect (a) by a factor of N, so that the thermal
   average of the potential energy scales like 1/N.

In the limit of large N, the PE makes a negligible contribution to the
heat capacity.  The x-variable is still a degree of freedom, but it does
not show up in the heat capacity.

10a) Equation [1] is not quite the same as a virial theorem.  As Sturge 
points out, it looks suspiciously similar, but it's not really the same.  
(I suspect there may nevertheless be some connection, perhaps via path
integrals and analytic continuation to imaginary time, but this is just
a wild guess.)

10b) We have discussed the case of non-quadratic PE.  Special relativity 
would give us a non-quadratic KE.  Equation [1] can handle this case.

10c) There are lots of anharmonic potentials that are not power laws.
Equation [1] can handle this.  In some cases the analysis is easy, 
e.g. the gas column mentioned in item (8c), but in other cases it 
might get messy.

10d) Equation [1] cannot handle the low-temperature limit, where the
energy levels are quantized.  There is a step in the derivation where
the exact sum over states is approximated by an integral over phase
space, and that doesn't work in the low-temperature limit.

On 04/09/2011 10:40 PM, David Bowman wrote:
> > BTW, for a generic dynamical degree of freedom that has a ground
> > state configuration, we typically expect that the minimal energy
> > configuration, if it is a nonsingular point, to be, typically, a
> > quadratic minimum assuming the degree of freedom is a normal
> > sufficiently smooth and differentiable parameter.

That argument evidently doesn't work for the PE of an ideal gas.