From: burchard@geom.umn.edu (Paul Burchard)
Newsgroups: bionet.info-theory
Subject: Re:  A Mathematical Fireside Chat at 300K about 0K
Date: 17 Nov 92 15:59:36 GMT
Distribution: bionet
Organization: University of Minnesota

In article <9211152208.AA19972@eureka.buc.edu.au>
kjm@buc.edu.au (Kevin Moore) writes:
> 
> In article <9211140822.AA01880@net.bio.net>
> burchard@horizon.math.utah.edu (Paul Burchard) writes:
> 
>> Our conclusion?  Temperature implies an upper bound on (physical)
>> entropy, and under physically correct assumptions, this upper 
>> bound goes to zero as the temperature goes to zero.  Therefore,
>> zero temperature implies zero entropy.
> 
> I'll go along with this.

Well, that is the basic argument I wanted to make, so let's
work backwards from there and see where I may have erred
along the way.

What seems to have gotten people the most upset was my talk
of absolute energy scales.  As Kevin points out, you can of
course shift the entire theory to any energy "origin" and
it will work fine.  (For exponentials, translation and
scaling are the same thing, hence the Z.)

So to make my point in a less inflamatory way, what I'm claiming
is that we must be given an absolute energy minimum as prior
information.  Without this, there is no consistent way to
define a heat bath, whose only property is temperature.  For
simplicity, we can then shift our energy scale to put that
absolute minimum at zero.  (I remain skeptical, however, that
this absolute minimum would always be set to the ground state
energy in the quantum case.)

The other thing that proved to be kindling was my definition
of temperature as average energy per state.  Here the disagreement
is more interesting:

>> Note that T also shows up as the exponential decay rate of p(E).  
>> This suggests an alternative definition of temperature, which 
>> allows the possibility of "negative absolute temperature".  We 
>> ignore this definition.
> 
> Why? That T corresponds to the integrating factor in
> dQ=dS/T. It therefore corresponds to the thermodynamic
> definition. 

That's an interesting point.  Could you explain this a bit further?
The correspondence isn't immediately obvious to me...

> As I said earlier, the average energy definition is a BAD
> one. In fact it only works for classical particles, with
> no exchange interaction. All the particles in the ground
> state is 0K. ($\beta=1/kT$ is infinite) 

My definition was E per state, not per particle.

In any case, the REAL test of which temperature definition is
correct must be settled by checking which one gives the correct
direction of heat flow when two systems of different temperature
are "weakly coupled".  I would think that the average energy per
state would equalize between the two---are you claiming that the
exponential decay rates of the energy distribution equalize
instead, in the case when these two definitions disagree?

>> 	2. In order to make sense of the limiting maximum entropy
>> 		as T goes to zero, we had to assume discreteness
>> 		of energy levels.  Classically, it's trouble again.
>> In fact, I'm skeptical that we can make classical sense of
>> entropy at 0K.  The entropy bound's divergence to minus infinity 
>> reflects the infinite information difference between a continuous 
>> and a discrete probability distribution.  It might still be 
>> possible to renormalize these calculations somehow, but that's a 
>> slippery business in itself.
> 
> Yep. You use $\int \rho\log\frac{\rho}{m} d\tau$,
> where $m$ is the Lebesgue measure on phase space $\tau$.
> The $m$ is important, because it makes the information
> entropy invariant under choice of parametrisation of
> phase space. Klir, the fuzzy set person, (unforgivably)
> sets up a straw man by leaving out the measure when
> criticising the maximum entropy procedure. 

Wait!  I think max entropy is great, and in fact I'm already using
your definition for continuous distributions.

My point is that we get in trouble defining the *limiting* entropy
in the continuous case.  Measured continuously, the limit is minus
infinity; however, that corresponds to the fact that the continuous
distribution is becoming discrete---somehow we need to "subtract an
infinite constant" to make sense of the limit.  As long as the energy 
minimum is non-degenerate, we could hope that this minus infinity 
"renormalizes" to zero, because in that case, the distribution is
approaching a single-point discrete distribution.

>> P.S.  Please criticize my *definitions* first, before
>> flaming my conclusions!
> 
> I did, and I hope you're satisfied. :-) 

Thank you for your interesting comments!

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Paul Burchard   <burchard@geom.umn.edu>
``I'm still learning how to count backwards from infinity...''
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