Dear Jonathan,my following reply to Steve Weinstein (and indirectly to Charles
Misner) that I mailed to them yesterday may also answer the question
raised in your last sentence:
> Except if there is a wavefunction of the universe, in which case,
>the universe is a pure state and the entropy is zero.
Physical entropy is NEVER calculated from the individual state (that
is, without any coarse-graining) even if such a state is assumed to
"exist" (as in any classical description). This applies to the wave
function of the universe as well, so its physical entropy would NOT
vanish, as you claimed.
In general we may neglect nonlocal correlations, for example, (as
those arising in Boltzmann's collisions). Properties like ergodicity
or mixing are required for the "dynamical autonomy" of a certain
coarse-graining. However, we should be aware that this "physical"
(local) entropy is NOT able to take into account meaningful long
range correlations, such as those which we understand as a
"consistency of documents about the past". It is these correlations
which convinced us that the documents have emerged from a state of
even lower physical entropy rather than being the outcome of an
accidental fluctuation. The latter would be more "probable" if
measured by physical entropy -- as argued long ago by von Weizsaecker
and Laundau.
Best regards, Dieter Zeh
Dear Steve,
I feel that I have discussed most of your questions in my book. Of
course, Paul's book of 1974 contains similar arguments, but he may
not agree with me on every detail.
>I take your point that we can assign a statistical-mechanical
>entropy to a nonequilibrium ensemble. But, leaving aside the
>question of the coarse-graining, *which* ensemble are we to use?
In classical mechanics, you have (conceptually) to coarse-grain that
ONE state in which the system (or the universe) is assumed to exist.
This gives a "representative" ensemble -- for example it may lead to
a canonical one, depending on the course graining. You may instead
start with an ensemble representing your incomplete knowledge, but
this would not define PHYSICAL entropy, but would include "entropy"
of lacking information (defining some "mean entropy"). Both entropies
are dynamically mixed, however. (You will find definitions of various
appropriate Zwanzig projections in my book. For example, see my
Zwanzig projection P_Boltzmann!)
In quantum theory, ensembles are replaced with density matrices,
which can be formally REPRESENTED by ensembles of wave functions
(state vectors). So you have to assume coarse-graining of THE wave
function of the universe (or rather, and very importantly, our
Everett branch, or the corresponding collapse component - there is no
practical difference). But don't ask ME for the interpretation of the
density matrix by those physicists, who regard the wave function
merely as an algorithm for calculating probabilities for classical
quantities! However, because of quantum nonlocality there is an
important difference compared to classical statistics: in quantum
theory, the Zwanzig projection of locality (which is required for
defining an entropy density), leads to a mixed state even when
applied to a pure state. (Locality means that a completely defined
total state defines the states of its parts.)
> The various "canonical" ensembles of Gibbs are stationary
>distributions suited to the describing systems in equilibrium. But
>here we are dealing with a system (the universe) which is arguably
>out of equilibrium.
Of course it IS very much out of equilibrium (that is, unstable).
Didn't you say that you "take my point ..."? Entropy can be defined
by the usual mean log rho (where rho is the course-grained density).
Nonetheless, local parts of the universe are usually in excellent
local equilibrium (and, as I said, locality is the most important
coarse-graining).
> Thus it is not clear what probability distribution to place on the
>phase space (let alone state what the phase space is!).
Of course, we cannot explicitly define statistics of unknown physical
objects (if that's what you mean), but the resulting thermodynical
properties are pretty universal (save gravitating objects).
>I say "arguably" because, on a coarse-grained scale, the universe
>began in a homogeneous state and *continues* to be in a homogeneous
>state. Thus some might argue that it *is* in equilibrium.
It does not continue for long under gravity (after inflation has
ended). And whoever says it is in equilibrium (that is, stable), is
evidently wrong (unless I misunderstood what you mean).
>(This begs the question of how to compare entropy at early and late
>times in an expanding universe.)
What do you mean?
>
>I don't understand what you mean by homogeneity being stronger in
>the quantum than the classical case - perhaps you could expand on
>that.
If you apply a symmetry transformation to a symmetric quantum state,
you reproduce the same state. A transformed classical state is only
macroscopically equivalent, but distinguishable from the original. So
you get a continuum of different states which have all to be counted
statistically. It's the same as with particle permutations, and the
reason for superfluidity, for example. This is why symmetry-breaking
(discussed in my book) is thermodynamically so important -- including
that of the vacuum.
>As for the classical case, I would say that we should only be
>surprised to find homogeneity in a system on which gravity has
>acted. But the initial state is, by definition, not the
>time-development of any other state.
Counting of states has nothing to do with dynamics and history. The
question is how many (possible) states EXIST for each macrosopic
characterization. This requires (microscopic) kinematical concepts to
begin with -- at least in principle.
> Analogously, we would not be surprised to find that a new deck of
>cards is sorted by number and suit. Though we would be surprised if
>a deck which had been shuffled were so ordered.
I feel I have addressed this problem. Statistically we should be
surprised that the world began in a state that now allows us to find
sorted decks of cards (or consistent documents). Most of us are not
surprised because we are used to this fact.
Best regards, Dieter Zeh
P.S.: Charles Misner, in his contribution, mentions the importance of
negative specific heat of gravitating systems. I have discussed this
situation for weakly gravitating systems (stars) at the beginning of
Chapter 5 of my book (since its first edition of 1989, and later in
slightly revised form). Its consequences culminate in the
Bekenstein-Hawking entropy. The intermediate (usually violent) stage
is indeed not well understood, but fortunately not very important in
practice. (The radiation arrow is important here, because of the
no-hair theorem.) In a speculative final section of my Chapter 6, I
have even argued that the interior of black hole horizons may not
exist at all because of combined quantum and thermodynamical effects.
In any case, the emergence of classical properties, including a
classical metric, requires a quantum arrow of time (the irreversible
process of decoherence).
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