Natural Science Forum / Physics / Particle Physics / August 2005

For entropy of fields, start with this paper:

http://xxx.arxiv.org/abs/cond-mat/0209043

You're making the mistake of confusing your intuitive sense of
what is "ordered" with the actual entropy.  There are times when
identifying entropy with "disorder" -- or more clearly, with
"what looks disordered to me" -- is useful, but this is not one
of them.  In a self-gravitating system, or more generally a system
dominated by a purely attractive force, the real, physical entropy
of a uniformly distributed gas is quite low, while the entropy of
a clumpy gas is higher.  

Start with a uniform gas in an otherwise empty region.  If the gas
is big enough or dense enough -- if its volume is larger than roughly
the cube of the Jeans length -- it will begin to collapse into lumps.
This dynamical instability has been known since Jeans' work in 1902.
These days it can be watched quite carefully in many-body computer
simulations.

What happens to entropy during this process?  The modern work on
this apparently began with a 1962 paper by Antonov, but it's in an
obscure Russian journal, and I confess I haven't read it.  The usual
reference is a paper by Lynden-Bell and Wood, Mon. Not. R. Astr. Soc.
138 (1968) 495.  When you do the calculation, you find that the
entropy of each individual clump decreases, but the entropy of the
system as a whole increases.  This is not a contradiction -- it is
possible because during the process of gravitational collapse,
individual particles and small clumps get flung out of the collapsing
regions at very high speeds.  The ``extra'' entropy ends up in a hot,
thin ``interstellar'' gas.

If you think about the energetics, you'll see that this has to happen.
As the gas collapses, it heats up, which stabilizes the collapse.  To
collapse further, it has to get rid of energy; something has to carry
energy away from the lumps.  One way to describe this is to say that a
gravitationally collapsing object has negative specific heat -- as it
radiates energy, its temperature increases.  

This effect can already be seen in simulations with uniform spheres.  
In realistic structure formation, the energy is mainly carried off by
photons, I think, and the entropy mostly goes into a thin photon gas.
One of the standard problems in modeling structure formation is to
understand how, and how fast, collapsing structures can be cooled.

You can, if you like, call this an increase in the entropy of the ``rest
of the Universe.''  But the ``rest of the Universe'' here need not be
some separate system; it can be a portion of the gas you started out
with.  In particular, you don't need to have something else to carry
off the energy, although the clumping will happen faster if you do.
In fact, you can imagine putting the whole system inside a perfectly
reflecting box, so the ``interstellar gas'' can't escape.  If the box is
big enough, you will still get a ``gravothermal catastrophe''---the gas
will segregate into a contracting central core and a surrounding halo.

You might want to look at the beginnning of chapter 5 of Zeh's book
_The Physical Basis of the Direction of Time_ for a short but more
mathematical description.

Steve Carlip