Re: Entropy, Objectivity, and Timescales

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

I had originally planned to sit the rest of this discussion out (after
having helped get it going), but then I came across this plea from Leigh.

> Will someone else please chime in here? A voice of sanity?

I would like to believe that my voice is one of sanity.  (However, I hear
that it is said that the insane always consider themselves quite sane).

My position in the discussion between Leigh and Dan is somewhat
intermediate between each of their's, but I think it is significantly
closer to that of Dan than that of Leigh.  I agree with Dan about the
ultimate subjectivity (but practical objectivity) of thermodynamic entropy.
My reasoning on this matter seems to agree with Dan's.  I agree with Dan
that for large thermodynamic systems the absolute uncertainty in the actual
value of the entropy due to slightly different subjectively specified
macrostates for the same system is a very tiny fraction of the overall
entropy of the system, and thus the entropy is effectively a practically
objective quantity.  When a system has an entropy typically of some
10^24 bits if its precise value is uncertain at a level of a few bits here
and there due to differences in the exact definition of the macroscopic
state caused by differences in the subjective perspectives of two different
macroscopic observers (with somewhat different prior information about the
system's macrostate) then the difference in the value for the entropy for
these observers appears in something like at least the 20th significant
figure.  The entropy for such a system is thus effectively objective to at
least 20 significant figures.  This is much more precision than occurs for
most macroscopic thermodynamic quantities which typically have statistical
fluctuations which show up in the 12th significant figure.  This is because
a system with N degrees of freedom has a standard deviation for its
thermally fluctuating macroscopic properties which scales as 1/N^(1/2).

Dan wrote:

> 2.  Excuse me?  The time-dependent Schrodinger equation is fully
> deterministic and allows one to predict a future state with precision,
> provided that the initial state is also known with precision.

This is true, *but* when discussing a thermodynamic, i.e. a macroscopic,
system the Schroedinger equation ceases to apply (i.e. to precisely and
deterministically evolve the microstate forward in time) after a
microscopic time scale due to tiny distrubances between the system and its
environment (not contained in the system's Hamiltonian operator) which
causes the system to loose any quantum coherence it may have initally had.
Also, it is very unrealistic to suppose that a thermodynamic system's 
microstate could initially be specified with much precision in the first
place.  The state of a thermodynamic system is represented by a very
general density matrix which only contains information about the system's
macroscopic constraints and expectations, and contains essentially no
microscopic information.  After all, it is the information missing from the
system's density matrix necessary to determine the system's exact
microstate, i.e. wave function, which constitute's the system's entropy.

> In any case, the issue of quantum indeterminacy surely is not
> relevant to understanding entropy.  (If you think it is relevant,
> then please elaborate.)

I agree with this.  However, Leigh responded in part with:

>       ....                          Clearly quantum indeterminacy
> is central to microscopic interpretation of thermodynamics.

I believe that this statement greatly overstates the case.  Actually, I
think the converse (inverse, reverse ?) is closer to the case, i.e. that
the irreversibility in the thermodynamics of a microscopic system's
*environment* is central to a microscopic system's quantum indeterminacy
and the irreversibility associated with the "collapse of the wave
function".  Here the macroscopic environment is taken to be in a mixed
state and the unavoidable interactions between the system and its
environment are the source of the measurement-like interactions which lead
to the system's loss of quantum coherence and, consequently, the
indeterminacy associated with quantum measurement-like processes.
[For more details see the article by Wojciech Zurek, "Decoherence and the
Transition from Quantum to Classical", _Physics_Today_, (Oct. 91), and the
book by Roland Omnes, _The_Interpretation_of_Quantum_Mechanics_, Princeton,
(1994).]

After describing a Thermal Physics course (offered I assume at SFU) and its
texts and reference works Leigh says:

> Looks familiar, doesn't it? Would you insert comments about the
> subjective nature of entropy into such a course? I'll wager that
> the texts cited here would not support your view.

Perhaps that is one reason why Dan wrote his own Thermal Physics text?

James McLean later wrote:

> ....
> In order to contribute more substance, maybe I should make a stab at it.
> Take the initial quantum state, and propagate it with the Schrodinger
> eq. until there is a non-zero probability of finding the system in each
> state considered to be accessible.  Does that work for everyone?

Sorry, I don't think this will work.  Recall the quantum evolution operator
which describes the Schroedinger motion of the system in time is *unitary*.
If the initial quantum state is that of a single wave function, i.e. a pure
state, then the Schroedinger evolution of the system keeps the system in a
pure state exactly one state remains accessible for all time.  Even if the
initial state was a mixed-state density matrix when its evolved forward in
time with the Schroedinger equation the time translated density matrix
has exactly the same information that it stated out with and the entropy
associated with the time-evolved density matrix is constant in time and
does *not* increase which would be the case if more microstates were
becoming accessible.  Since the system's entropy actually *does* increase
irreversibly with time (assuming the initial state was not one of
equilibrium) we see that Schroedinger evolution does not evolve the
system's state forward in time for more than the microscopic time required
for the system to irreversbily loose quantum coherence due to unavoidable
interactions between the system and its surroundings.  It should be noted
that the time scale over which quantum coherence is lost is *much* shorter
than the time scale over which thermal equilibrium is established.

David Bowman
dbowman@gtc.georgetown.ky.us