Re: Entropy, Objectivity, and Timescales

[Leigh Palmer <palmer@sfu.ca>]

I guess I will post this reply to the group. I can't be very tactful
given the first paragraph, but here goes. A correspondent says:

> First let me confess that I haven't followed the entire thread on
> order and disorder, etc.  But the example of shuffling a deck of
> cards brings up an issue that I'd like to comment on.
>
> We usually define entropy as the logarithm of the number of "accessible"
> microstates.  But the word "accessible" cannot possibly be meant to
> imply that the system is ever *actually* in more but a tiny fraction
> of the microstates.  Look at the numbers:  e^(10^25) "accessible"
> microstates for a mole of helium under standard conditions.  Bringing
> up the timescale issue (a good issue, generally) doesn't help here;
> wait as long as you like, billions and billions of years, this system
> will never actually explore the vast majority of the supposedly
> "accessible" states.

No, indeed what you say here is correct. What one can say is that the
interactions of the parts of the system are of such a nature that the
microstate changes during the time period in question in such a manner
that in principle it *could have* reached any of the states which are
considered to be in the set of accessible states. It is not necessary
that the system reach even 10^-10 (or any small fraction) of them.

> It seems to me, therefore, that entropy is fundamentally a *subjective*
> quantity, a measure of our ignorance.

Here you must immediately see the fault in your logic. Mere use of
the word "therefore" does not cover the holes in your argument. Is
there an argument, by the way?

> If I knew the precise microstate
> of the helium now, then I could predict its microstate at all times
> in the future for the next billion years, and even counting every state
> that it explores as "accessible" I'd get an entropy much less than the
> generally agreed upon value.  The agreed upon value, though, includes
> an enormous number of other microstates that the helium will never
> actually explore (though we don't know this).

You don't know (and you can't know) the exact microstate of any
physical system, and it is fundamentally impossible to predict even
its microstate after the next transition, let alone in the distant
future.

> Now back to the deck of cards.  If you shuffle the deck and I don't
> look at the order, why can't I consistently say that its entropy
> is log(52!)?  From my perspective, your shuffling of the deck in the
> past made all states "accessible", in the same sense that all those
> microstates were accessible to the helium gas.  True, the cards won't
> be exploring any more microstates in the future if you're no longer
> shuffling them, so there is a sense in which their entropy is zero,
> but since entropy is subjective anyway, it doesn't bother me that
> entropy might be zero in one sense and nonzero in another.  In cases
> where it matters (i.e., large systems), the difference between
> one person's entropy and another person's entropy will be negligible,
> since nobody will have enough information about the system to cut
> down the entropy significantly.

The entropy of a system is not a subjective quantity. Now that you
understand that you also understand why the entropy associated with
the order of the cards in the deck is zero.

> Well, that's how it seems to me.  But I'm posting this so that those
> of you who disagree will (I hope) explain why.  Can't wait to hear
> from you!

Sorry you had to wait so long.

It seems that this post is an example of a misconception that can
result from the association of an established, well understood
physical concept like entropy with something that is formally similar
(the "Shannon entropy" as Dave Bowman calls it) but physically
unrelated.

Leigh