Re: [Phys-L] Entropy as a state function...

[John Denker <jsd@av8n.com>]

On 04/13/2014 09:09 PM, rjensen@ualberta.ca wrote:
> I just want to confirm my understanding of entropy. 
>  * The system entropy is a state function.
>  * For any arbitrary process, the universe and surroundings entropy
> are not state functions. 
>  * For a reversible process, the universe and surroundings entropy are
> state functions.

Those are very strange questions.  Before answering about
the entropy, let me answer the corresponding questions about
something simpler and more intuitive, namely the volume.

Suppose we have parcel A which we arbitrarily call "the
system".  Surrounding parcel A we have another parcel B
which we arbitrarily call "the surroundings".  On the
outside of B we have another exceedingly large parcel C
which we arbitrarily call "the universe".

The volume of parcel A is a function of the state of
parcel A ... and not a function of the state of parcel B.

The volume of parcel B is a function of the state of
parcel B ... and not a function of the state of parcel A.

When stated that way, it is blazingly obvious.  Things 
become less obvious when less specific language is used,
for instance when asking about a function of state without
specifying /whose/ state.  This is nontrivial, because
there are conflicting traditions. 
 1) In (say) fluid dynamics, there are lots of parcels,
  and it is traditional to speak of the properties of 
  a given parcel as being functions of state if they 
  depend on the state /of that parcel/.  
 2) In contrast, in introductory thermo books, it is
  common (but not particularly wise) to focus attention
  on capital "The" capital "System" and to assume that
  everything of interest is a function of the state of
  "The System".

Very commonly nothing is a function of state, if you think
too narrowly about what the state-vector is.  For example,
consider "the" temperature of a system where the temperature
is non-uniform.  This is however a fixable problem.  Often 
it is necessary to account for more variables, i.e. redefine 
the state-vector so that it lives in a higher-dimensional space.

===

Asking specifically about "reversible processes" is pointless, 
because nothing about the situation depends on reversibility.

==========

Turning now from volume to entropy, things get messier because
entropy is both subjective and non-extensive.  Suppose I
shuffle a deck of cards, putting them into a random sequence,
and peek to see what the sequence is.  Then I manipulate two 
other decks of cards, putting them into the exact same sequence.

If we pay attention only to the sequence, and not the various
other physical variables, each deck separately has 226 bits
of entropy as far as you are concerned.  I can give you any
one of the three decks, and it will take you 226 yes/no
guesses, on average, to figure out the state.  However, if 
I give you all three decks and tell you that they're all the 
same, the three of them together have only 226 bits.  Let's 
be clear:  any one deck = 226 bits, and all three decks =
226 bits.  The entropy, as far as you are concerned, is very
conspicuously non-extensive, due to correlations.

This shouldn't be very shocking.  The plain old energy of a 
sample of water is not strictly extensive, because of surface
tension and whatnot.

What's worse is that the entropy is subjective.  The entropy
of the card deck is 226 bits as far as you are concerned.
There are 2^226 equally-likely microstates.  However, for me 
the entropy is zero, because I peeked.  I know exactly which 
microstate the deck is in.  This subjectivity is almost never
a problem in practice when dealing with large systems.  However,
the fundamental laws of physics (including thermodynamics) apply
just fine to small systems, in which case we need to be quite
careful about how we define entropy.

Furthermore, here's an even more fundamental issue:  Entropy
is a property of the /ensemble/.  It is a property of the
macrostate.  This stands in contrast to (say) the energy or
the volume, which can perfectly well be defined on a microstate-
by-microstate basis.  So ... things very much depend on what
you mean by "The System".  If you mean the ensemble, that's
one thing.  If you mean a particular instance drawn from the
ensemble, that's something else entirely.


Again, asking specifically about "reversible processes" is 
pointless, because there is no part of this story that depends 
on reversibility.