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Re: entropy



James McLean asks a question of Leigh that I wish to jump in and answer.

Am I correct that the difference between this disorder you are refering to
and the kind appropriate to entropy is this:
A single pack of cards in a particular sequence can be "disordered",
while to talk about entropy you need an ensemble of sequences.

Yes.

I'm not sure I've said that well. What I'm trying to suggest is that
entropy (one kind of disorder) applies when a system is changing between
many microstates.

This is essentially what I have been trying to say. However I called this
case entropy not disorder. In this instance, the system (as defined
macroscopically) is sampling many different microstates (nearly all of which
are indistinguishable at the macroscopic level), and the entropy is the
average (expectation over the distribution of the microscopic possibilities)
of the information necessary to determine which of the microstates the system
is in. When the available microscopic possibilities are all equally likely
this information is just the logarithm of the number of possibilities
available. When they are not all equally likely then this information for
each microstate is the logarithm of the reciprocal (or equivalently, the
negative of the logarithm) of the probability for that particular microstate
and the entropy is the average of these logarithms over the non-uniform
distribution of possibilities. In contrast, I suggested disorder as the
minimal information necessary to *characterize* a given microstate. A
disordered state is one that requires a lot of information to describe
sufficiently to exactly reproduce it. This concept is close (but not
identical) to the information that is averaged in evaluating the entropy. In
the entropy case the information averaged is only the information necessary to
distinguish/determine/identify a given microstate from the space of all
available competing other microstates (i.e. it is a counting or labelling
kind of information) while the information that measures the disorder of a
microstate is the information needed to reconstruct that state (i.e. it is a
descriptive or characterizing kind of information). My definition here of
disorder was thrown out only as an example of my original point that the
notion of disorder is usually quite fuzzy in people's minds and there are many
*different* ways to make a fuzzy notion precise each with inequivalent
definitions. My definition of disorder was only thrown out as a counter-
example to the definition given in Doug C.'s web page of disorder being a
measure of the difference between a given state and a fiducial "ordered" state
which is produced by a given "ordering rule". The notion of disorder, being
somewhat fuzzy, can be defined in multiple inequivalent ways, some of which
are related to the (statistical mechanically) fixed definition of entropy and
some of them cannot be so related.

A second kind of disorder is when a system is in a
single, unorganized microstate. Are you are concerned about students
making a distinction between these two kinds of disorder?

I'm not too overly concerned about the students making the subtle distinctions
I made above. I do wish them to come away from a unit on entropy knowing that
a thermodynamic system has MANY more disordered microstates (however that idea
is defined) than it has ordered ones and that as an isolated system
equilibrates by making all of the accessible microstates equally likely, the
disordered ones, being so much more numerous, dominate the possibilites
sampled by the system's dynamics so that if the system started in a somewhat
ordered state and the ensemble of disordered states is made available to the
system, that after a while, the system will, almost certainly, be found in a
disordered state. IOW, I do want the students to understand the mechanism of
the 2nd law.

Or is there a third (, fourth...) kind of disorder that I'm missing and
that muddies the waters further?

Yes there are. For instance, in the stat mech of phase transitions there is
the idea of an "order parameter" that macroscopically and quantitatively
distinguishs a condensed low symmetry phase from an uncondensed high symmetry
phase. I do not suggest we give beginning students the full litany of all
the various kinds of order and disorder there may be.

David Bowman
dbowman@gtc.georgetown.ky.us