Re: [Phys-l] Photon Thermodynamics

[John Denker <jsd@av8n.com>]

On 06/17/2007 10:33 PM, Jeffrey Schnick wrote:
>  ....
> Thanks for your informative response.  I perceive the scaling argument
> to be elegant and powerful.  At present, however, I don't follow it in
> this case.  I assume the equation to which you refer when you write "you
> can't have an extensive variable on one side of the equation and not the
> other" is:
>
> (1)  dU/dV (const T) = 3P + 3V dP/dV (const T)
>
> On the left is the ratio of a change in an extensive variable to a
> change in an extensive variable.  That ratio is an intensive variable.
> The first term on the right is clearly an intensive variable.  The
> second term on the right is an extensive variable times the ratio of a
> change in an intensive variable to a change in an extensive variable.
> The ratio is a reciprocal extensive variable.  The product of the
> extensive variable 3V times that reciprocal extensive variable is thus
> an intensive variable.  As such, every term in equation (1) scales like
> the zeroth power of the size of the system and we can't use the scaling
> argument to establish the fact that the second term on the right must
> vanish.

That argument is a correct non-proof.  It says that a
certain line of reasoning fails to prove that the pressure
is independent of volume, under black-body conditions.

But the door remains open to other lines of reasoning.

In particular, consider a piston between two boxes of black-body
radiation.  Both boxes are at temperature T.  One box is larger
than the other.  My physical intuition tells me that to a very
good approximation (for a wide range of reasonable sizes and
temperatures), the black-body radiation pressure is the same on
both sides of the piston.

There are so many physical arguments supporting this intuition
that I hardly know where to start.  One argument uses the same
apparatus as the Gibbs "paradox" experiment, i.e. a box with
photon gas on both sides of a partition, and then we pull out
the partition.  Neglecting very small effects (Casimir-like
effects etc.) we now have one big box of photons, instead of
two small boxes, and the photon pressure is unchanged.  For me
this is very graphic, very pictorial:  I visualize the photon
modes in the box ... and most of the energy is in modes that
are negligibly affected by the partition.

  Another argument is that I know the black-body radiation brightness
  formula, and it doesn't have a V in it ... but that is slightly
  cheating, because Feynman was starting to /derive/ the blackbody
  formula, and he didn't want to make a circular argument.

Anyway, my advice is don't obsess over equation (1).  Think about
the underlying physics.

(Also, as mentioned in my previous note, beware of negative
transference from your experience with a gas of ordinary
molecules.  Black-body conditions != constant-N conditions.)