Re: [Phys-l] Gibbs paradox

[John Denker <jsd@av8n.com>]

On 03/31/2011 09:14 AM, Carl Mungan wrote:
> I note that Jaynes "fixes" the problem by imposing extensivity on
> entropy. Swendsen on the other hand denies that entropy has to be
> extensive. (He says it has to be additive, but there in no intrinsic
> reason entropy need be extensive.)


There is nothing about this thread that makes sense to me.

Let's do a simple numerical example, using playing cards
instead of atoms.  The cards are all distinguishable.

Sample S consists of all 13 spades.  These are shuffled, which
gives a configurational entropy of log(13!) or about 32.5 bits.

Sample H consists of all 13 hearts.  This sample, by itself,
has 32.5 bits of configurational entropy, for the same reason.

If you pile the two samples on top of each other, S on top of
H, the total entropy is 65.1 bits.

If you pile them on top of each other and don't tell me which
one is on top, it will cost me a bit to find out, so the total
is then 66.1 bits.

Now ... if you pile them on top of each other and shuffle them,
the entropy jumps up to log(26!) or about 88.4 bits.

===

To summarize:  It appears that one of the central claims is
that when we take two samples of distinguishable particles 
and mix them, the entropy is additive.  This claim cannot
withstand even the slightest scrutiny.

If somebody has cooked up a partition function that supports
this claim, the partition function is wrong.

============

While we're in the neighborhood, let's keep in mind that there
are a lot of different Stirling approximations, some of which
are accurate down to very small N ... and for small N you don't
need to make an approximation at all.

This is relevant because if you rederive the Sackur-Tetrode
equation and keep the next term(s) in the Stirling expansion,
you discover that the entropy is non-extensive even in the
simplest case, i.e. an ideal gas of identical particles.  See
  http://www.av8n.com/physics/thermo/z-particles.html#eq-sackur-non-intensive
and the surrounding derivations and discussion.

This is not a big deal.  The energy is not quite extensive, 
because of surface-tension and surface-reconstruction effects, 
and this has not caused the world to end.  Discovering that
the entropy is not quite extensive will also not cause the
world to end.  For large systems, the correction terms will
be insignificant ... but they exist in principle and they
are significant for small systems.

The /principles/ of thermodynamics apply to all systems, not
just large systems.  Some of the /secondary/ results involve 
simplifications that are only valid for large systems, but we
are allowed to use the non-simplified results if we want.