From: kkumer@phy.hr (Kresimir Kumericki)
Subject: Re: Entropy
Date: 30 Aug 2000 23:08:47 GMT
Newsgroups: sci.physics.research
Summary: [missing]

John Baez <baez@galaxy.ucr.edu> wrote:
> Oh... duh!   The assumptions are different: you were requiring that
> the kinetic plus potential energy stay constant, whereas I was letting 
> energy leak off in the form of radiation or "boiling off" or something.  
> The calculations really apply to different regimes: yours applies to
> the "fast regime" when the cloud is getting virialized but (you assume)
> acting like a closed system, whereas mine applies to the "slow regime" 
> when it's already virialized and is slowly losing energy by radiation or 
> boiling off.
> It all makes sense now.  Great!

  Trying to catch up with the news after summer holidays I got interested
in this thread and I tried to do a back-of-the-envelope calculation
to see whether in the "slow regime", where cloud is virialized all the
time, there is enough entropy produced by the radiation/evaporation to make 
up for the reduced entropy of the cloud as it shrinks. I know it should
come out right (in fact, I think the calculation below is in effect
tautologic) but this gravity-entropy business can be tricky so I
wanted to make an explicit check. So, if anyone's still interested,
here it is:

  We already know that the entropy of the cloud is

S_cloud = kN [(1/2) ln N + (1/2) ln V + C] ,

and this means that for a small change dR of cloud radius we have
entropy change

dS_cloud = (3/2) kN dR/R    (*) .

On the other hand, the assumption that cloud is virialized means
that 
  
dP = -2 dK = -2 dU ,

where U is the energy of radiation/vapour. 
If we take that energy U was radiated in a reversible process
we will get the lower bound on the produced entropy:

dS_rad >= dU/T = dK/T .

But we know that potential energy P is proportional to -1/R and this
means, again by virial theorem, that K ~ 1/2R, or that for ideal
monoatomic cloud

dK = -K dR/R = -(3/2) kNT dR/R .

This finally means that radiated entropy is

dS_rad >= -(3/2) kN dR/R ,

which exactly cancels (*):

dS_total = dS_cloud + dS_rad >= 0 ,

and saves the Second Law.
 As an aside, note that for short-range forces V ~ -1/r^n, n>2, it is
impossible to virialize the cloud and still have it shrinking because
this would mean |dK|>|dV| which would violate energy conservation.


-- 
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Kresimir Kumericki  kkumer@phy.hr  http://www.phy.hr/~kkumer/
Theoretical Physics Department, University of Zagreb, Croatia
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