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Is it possible that prefect reversibility is a mathematical ideal that
does not apply exactly to any system found the the real world of
nature?

Did Poincare already discover this during the 1892-1899 period when
modern chaos theory was founded in his "New Methods of Celestial
Mechanics"?

Are the examples of revesibility that physicists frequently cite
actually either artificial idealizations, or refer to systems
maintained briefly in periodic states, but whose full, and
unmanipulated, behavior would include the much more extensive behavior
of nonlinear dynamical systems?

What are the best examples of candidates for truly and ideally
reversible systems?

Robert L. Oldershaw

[[Mod. note -- Hmm, elastic scattering of uncharged subatomic particles
or atoms certainly comes very close.  "Uncharged" means there shouldn't
be any electromagnetic radiation emitted, although there will still be
(very very *very*) tiny amounts of gravitational radiation emitted.
-- jt]]

Consider fluxional tunneling between the two equivalent structures of
semibullvalene (in vacuum) around 300 K.  

http://pubs.acs.org/doi/abs/10.1021/ja00816a037
 Low temp solution kinetics (two interconverting minima)
http://pubs.acs.org/doi/abs/10.1021/ja00544a056
 Low temp solid state freeze-out (two minima)
<http://www3.interscience.wiley.com/journal/106588467/abstract>
 Stabilized transition state (single minimum)

If you like larger numbers, parent bullvalene has 10!/3 = 1,209,600
fluxional structures - though it must be warmed to about 400 K.

http://en.wikipedia.org/wiki/Bullvalene

--
Uncle Al
http://www.mazepath.com/uncleal/
 (Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz4.htm

On Oct 30, 3:28 pm, Uncle Al <Uncle...@hate.spam.net> wrote:

What about the quantum harmonic oscillator, or quantum tunneling in a
symmetric atom like ammonia?

On Oct 30, 12:08 pm, "Robert L. Oldershaw" <rlolders...@amherst.edu>
wrote:

Interesting.  Can you do an actual experiment with neutrons or photons
wherein the particles interact and subsequently are made to retrace
their steps exactly and end up exactly in their original starting
places, states, etc.?

Or can this only be done in the Platonic world of mathematics?

RLO
www.amherst.edu/~rloldershaw

Heating and cooling a piece of metal within a moderate range of
temperatures is also generally regarded as a reversible change
of the metal.

Superconductivity is a more truly reversible quantum phenomenon.

But of course, all physical laws are mathematical ideals, so you
may be hunting for the impossible.

Arnold Neumaier

Ammonia inversion is reversible, but is it symmetric?  It is a general
question pertinent to any two-well "symmetric" oscillator (e.g.,
timekeeping).

Classically, the ammonia umbrella has the same energy before and after
being turned inside out.  In QM this is only true to first order.
Even and odd states corresponding to the electronic groundstate of the
NH3 molecule have energies differing by micro-eV, corresponding to a
frequency in the microwave range,

23.6944955 GHz
23.6893348 GHz

A low-loss cavity filled with the antisymmetric form spontaneously
oscillates (ammonia maser).  The almost equal populations at room
temperature in a molecular beam can be separated by travel through a
hexapole cylindrical electrostatic field that scatters "gerade"
(quantum state with a negative Stark effect) and focuses "ungerade"
(quantum state with a positive Stark effect) molecules.

<http://ticc.mines.edu/csm/wiki/index.php/The_ammonia_Maser>
Z. Phys. D  37 333 (1996)
<http://www.opus-bayern.de/uni-regensburg/volltexte/2002/57/pdf/diss.pdf>

"Gerade" and "ungerade" re Hund's Paradox are not identical to
geometric chirality, though there is significant overlap,

http://www.ir.ethz.ch/research.htm
 "6. Theory of fundamental symmetry principles in chemical reactions
and of parity violation in polyatomic (chiral) molecules"

The two ground states, superositions of "gerade" and "ungerade"
contributors with off-diagonal elements, are then subject to more
rigorous and subtle analysis of their moving positions on an SU(2)
sphere.

================= Moderator's note ================================

Hund's paradox concerning the enantiomeric state of chiral molecules
has been solved only quite recently:

J. Trost, K. Hornberger, Hund's Paradox and the Collisional Stabilization
of Chiral Molecules, PRL *103*, 023202 (2009)

HvH.

--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic  URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz4.htm

On Oct 31, 3:58 am, Arnold Neumaier <Arnold.Neuma...@univie.ac.at>
wrote:

Yes!  That is exactly what I am hunting for: an admission of the
possibility that every system in nature, if studied with unlimited
precision and accuracy, would be found to be a nonlinear dynamical
system that is not ideally reversible or integrable, although the
system could asymptotically approach such an ideal, or be wildly
nonintegrable.

Please Note: I am not trying to convince anyone that nature is built
this way, and certainly I am not saying that I have the required
evidence to prove it.  What I hope the reader will take home from this
thread is the idea that nature might be this way.  At least until
someone demonstrates that nature could not be this way.

In subsequent discussion it is very important to distinguish among:
perfectly reversible/integrable; approximately reversible/integrable;
mildly irreversible/nonintegrable; strongly irreversible/
nonintegrable.

RLO
www.amherst.edu/~rloldershaw

On Oct 31, 1:42 pm, "Robert L. Oldershaw" <rlolders...@amherst.edu>
wrote:

Refining the general question of whether exact reversibility/
integrability is an idealization or is actually realized in nature,
one could narrow the discussion as follows. Are atoms correctly
characterized by linearity, reversibility and integrability or is this
characterization a good but limited approximation to a more
sophisticated characterization of atoms as nonlinear dynamical
systems.

When chaos theory [aka NLDS theory] was first acknowledged as being
fundamental to modeling much of natural phenomena, it was thought that
its application was limited to the macroscopic domain.

Then one began to see the first papers arguing that period-doubling
and other chaotic phenomena could be observed in the atomic domain, if
one looked hard enough.

In the last decade the application of NLDS modeling to atomic scale
phenomena has been steadily accelerating, especially in regard to
atoms in highly excited Rydberg states.

Now, in the 10/8/09 issue of Nature, we see a potentially paradigm-
changing paper by Chaudhury et al which may herald the advent of a new
era in the modeling of atoms. In this paper the nuclear and electronic
interactions of a single are shown to display a quantum version of
classical chaotic behavior: the kicked top phenomena.

The authors also state: "We ... present experimental evidence for
dynamical entanglement as a signature of chaos.

So it is not unreasonable to ask: are atoms nonlinear dynamical
systems?

RLO
www.amherst.edu/~rloldershaw

The answer is clearly: nonlinear. After all, at high enough excitation
energies (few eV) atoms ionize, which is not linear at all! And at much
higher energies (MeV), the atomic nuclei transmute into other nuclei,
particle pairs are produced, and a host of highly nonlinear phenomena
occur. When one gets above TeV energies, we simply don't know what
happens....

Bottom line: theoretical concepts like reversibility apply to our
various THEORIES, not to the world we inhabit. For every theory we have,
there is a boundary beyond which it is not applicable, or beyond which
clearly nonlinear phenomena occur.

Tom Roberts

Atoms are quantum objects, hence their states (density matrices) satisfy
(to the approximation that the atomic picture of a point charge nucleus
with point charge electrons is valid) the linear quantum Liouville
equations. (And pure states - a further idealization - satisfy the
Schroedinger equation.)

Of course, the nucleus/electron picture is an idealization.

Moreover, linearity only holds for the dynamics of the density matrix,
but not for any reduced dynamics of  system of actually observable
quantities. The latter is highly nonlinear, and - no suprise - may
therefore be chaotic.

http://www.nature.com/nature/journal/v461/n7265/full/nature08396.html

I see there nothing indicating a new era in the modeling of atoms.
The Caesium atoms involved are assumed to satisfy the standard linear
quantum laws.

And the experiments are reported to have a 5% error, so they imply
nothing about exact reversibility/integrability.

I also don't understand why you use reversibility/integrability
in this combination as if these were essentially synonymous.

Every (classical or quantum) Hamiltonian system is reversible,
while only very simple or very idealized systems are integrable.

Arnold Neumaier

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Actually, it follows from the assumption that the universe as a whole
is reversible that asny subsystem of it (in particular anything we
cannot observe) is not reversible, since it depends on interaction
with the remainder of the universe.

So the only perfectly reversible system (if any) is the universe as a
whole (or a set of perfectly noninteractiung universes - of which we can
of course know only the single one we are in).

The mainstream belief is that, indeed, the universe as a whole is
reversible. But various alternatives have also been suggested, and of
course, perfect reversibility is not experimentally testable.

On the other hand, many small systems that we can observe can be
taken routinely as approximately reversible, with good success.

Arnold Neumaier

Yes, actual observable quantities are highly nonlinear
but the idealized nucleus/electron picture
represents an elastic (reversible) state
for the majority of the universe mass.

Consider the carbons atoms in your body.

They have maintained
their carbon (nuclear) elastic (reversible) identity
since their super nova creation.

So goes most of the universe
existing in an internally elastic (reversible) state
for billions of years interrupted by infrequent irreversible transitions
to some other time predominate elastic (reversible) condition.

Richard D. Saam

On Nov 1, 9:36 am, Arnold Neumaier <Arnold.Neuma...@univie.ac.at>
wrote:

[[Mod. note -- 31 excessively-quoted lines snipped here.  -- jt]]

Does a hard boiled egg count? (rhetorical).
One heats it, it goes from liquid to solid and there is no way
to get it back to liquid, are we discussing the arrow of time?
Ken

On 1 Nov., 18:36, Arnold Neumaier <Arnold.Neuma...@univie.ac.at>
wrote:
...

It is correct, that the quantum Liouville (von Neumann) equation is
linear in the density matrix. But the interaction enters
parametrically (as in the Schrödinger equation), hence, non-linear.

...

Striking!

Best wishes,
Peter

Robert L. Oldershaw wrote on Fri, 30 Oct 2009 12:08:59 -0400:

(...)

*Any* law of physics is a mathematical ideal. One would not confound
reality with our models of her.

For certain systems the production of entropy is so low that cannot be
differentiated from zero and are explained using reversible models.

Poincaré showed that not all mechanical systems are integrable due to
presence of resonances.

The so-called Brussels school tries to build models of irreversibility and
to solve the problem of the arrow of time using Poincare theorems.

They think that irreversible systems are LPS (Large Poincaré Systems).

They also think that resonances introduce an arrow of time.

Their work is well explain to broad audiences in the best-seller book

http://www.amazon.com/End-Certainty-Ilya-Prigogine/dp/0684837056

where he explain how to extend reversible theories from particle physics
to general relativity for accounting for irreversible phenomena, including
a resolution of the measurement problem in quantum mechanics as a bonus.

Whereas I agree on motivations, I disagree on the details of their theory.
In my opinion resonances are not the origin of the time arrow.

It depends. A reversible model of Moon motion is an excellent idealization
and the time-reversible mechanical equations work fine. A reversible model
of dissipation in a fluid would be artificial.

The second law says: reversible systems are those for which production of
entropy is zero.

In thermodynamics we compute the production of entropy (using the
well-known product of forces and fluxes) for checking irreversibility.

Microscopically we have the dissipative quantum equation

d(rho)/dt = L rho + D

when D is zero the production of entropy is also zero and the resulting
dynamics is reversible and described by the Liouville equation

d(rho)/dt = L rho

When the state can be approximated by a pure state

rho = |Psi><Psi|

then the Liouville equation reduces to the Schrödinger equation

d|Psi>/dt = H |Psi>

Therefore one computes D and it if it is zero or close to zero, the
dynamics is reversible.

The big question is what is the new term D? Nobody knows for sure.

Each School propose a diferent D. Some people has proposed
phenomenological terms in wait for a theory of irreversibility.

Zubarev School proposes D = epsilon (rho - rho_R)

where epsilon is a positive infinitesimal and rho_R an auxiliary state
postulated according to certain kernels and phenomenology.

Lindbald proposes another D

http://en.wikipedia.org/wiki/Lindblad_equation

assuming some mathematical properties.

Prigogine School proposes another

http://www.amazon.com/End-Certainty-Ilya-Prigogine/dp/0684837056

where the new term is explained in terms of collision operators
that contain resonances among degrees of freedom.

Byung Chan Eu proposed other based in a generalization of
Boltzmann kinetic theory and the observation of behavior of
hundred of physicochemical systems he studied

http://www.canonicalscience.org/en/researchzone/time.html

Etc.

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Arnold Neumaier wrote on Mon, 02 Nov 2009 11:51:17 -0500:

(...)

Untrue. It is not possible to derive irreversibility from reversibility.
As Van Kampen brilliantly noted "One cannot escape from this fact by any
amount of mathematical funambulism".

The open-system approach is totally inconsistent. The subdynamics of a
reversible system is of course reversible. The so-called derivations of
irreversibility are mathematical and physically invalid.

Those "perfectly noninteractiung universes" that we cannot know belong
to the world of fantasy not to physics.

(...)

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As an approximation, there is nothing inconsistent.

All of physics is valid only approximately anyway; so approximations
are legitimate. In particular, one conventionally approximates the
dynamics of a part of a larger system (whether or not the latter is
assumed to be reversible) successfully as that of an irreversible
system.

This approximation process is well understood - see, e.g.,
    H Grabert,
    Projection Operator Techniques in Nonequilibrium
    Statistical Mechanics,
    Springer Tracts in Modern Physics, 1982.
It is often applicable with much success.

In all serious applications of physics, one reduces a system description
to something manageable by replacing its interaction with the unmodelled
environment, using some approximation that accounts for its influence
without having to model it. This makes the system open, but amenable to
a stochastic description. Or, with further approximation, even to a
deterministic description.

If one does not allow for that, one cannot do any physics at all.

We cannot even know whether they are fantasy or physics.
They might exist, and still we could never find out. But of course,
one can ignore them completely without losing anything of
predictive value. This is why I put the statement in parentheses.

Arnold Neumaier

On Nov 4, 6:39 am, "Juan R." González-Álvarez

I am also troubled by AN's comment that: "it follows from the
assumption that the universe as a whole is reversible..."

(1) There is considerable confusion over what the term "universe as a
whole" actually means. In fact, the phrase is scientifically undefined
at this point.

(2) Assuming this undefined thing is "reversible" just adds insult to
injury. Who says it must be so?  Where is the evidence?

I realize that AN was just speaking in the vernacular, but woe be to
science when assumptions are treated as facts and and used as such in
reasoning.

RLO
www.amherst.edu/~rloldershaw

Arnold Neumaier wrote on Sat, 07 Nov 2009 09:26:11 +0000:

Evidently both Van Kampen (one of most respected physicists
in the field)

http://www.amazon.com/Views-Physicist-Selected-Papers-Kampen/dp/
981024357X

and myself (not at his level of course) are aware of the importance
of approximations. You missed the whole point

I agree with him on that the claimed 'derivations' of irreversibility
from reversibility are based in some "amount of mathematical
funambulism".

His remark is totally general and also applies to the claimed
'derivations' using PO techniques.

PO techniques introduced in NESM in early 60s are rather useful [#].
But its lack of usefulness beyond the weak limit (more exactly in
regimes where the reduced kinetic equation is not closed) is also
well-known.

Moreover, PO techniques are only a clever and *fast* technique to
decompose the so-named "relevant" and "irrelevant" subspaces.
PO techniques do not provide a foundation for NESM neither solve
the problem of the arrow of time.

A more modern and rigorous discussion of those issues was given in a
recent Solvay conference devoted to the problem. Contributions were
published in the next volume

http://www.amazon.com/Resonances-Instability-Irreversibility-Advances-
Chemical/dp/0471165263

I agree on their motivations and welcome their attempt to substitute
"mathematical funambulism" by a more rigorous and axiomatic approach.
However, I want to remark that I disagree with all the theories
presented there.

That in your own words "set of perfectly noninteractiung universes -
of which we can of course know only the single one we are in" do not
belong to physics.

[#] I want to reproduce here an interesting episode. It is often
    acknowledged in NESM literature that PO techniques were introduced
    by Nakajima, Zwanzig, and Mori. However, in a personal
    communication with Prigogine coworker, Gonzalo Ordonez, he said me
    that Prigogine had introduced PO techniques during a talk he gave
    and Zwanzig attended. Some time after Zwanzig published his
    foundational paper on the PO method. Gonzalo said me that Zwanzig
    gave a more elegant formulation but the original idea was from
    Prigogine!

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Arnold Neumaier wrote on Sat, 07 Nov 2009 09:26:11 +0000:

Evidently both Van Kampen (one of most respected physicists
in the field)

http://www.amazon.com/Views-Physicist-Selected-Papers-Kampen/dp/
981024357X

and myself (not at his level of course) are aware of the importance
of approximations. You missed the whole point

I agree with him on that the claimed 'derivations' of irreversibility
from reversibility are based in some "amount of mathematical
funambulism".

His remark is totally general and also applies to the claimed
'derivations' using PO techniques.

PO techniques introduced in NESM in early 60s are rather useful [#].
But its lack of usefulness beyond the weak limit (more exactly in
regimes where the reduced kinetic equation is not closed) is also
well-known.

Moreover, PO techniques are only a clever and *fast* technique to
decompose the so-named "relevant" and "irrelevant" subspaces.
PO techniques do not provide a foundation for NESM neither solve
the problem of the arrow of time.

A more modern and rigorous discussion of those issues was given in a
recent Solvay conference devoted to the problem. Contributions were
published in the next volume

http://www.amazon.com/Resonances-Instability-Irreversibility-Advances-
Chemical/dp/0471165263

I agree on their motivations and welcome their attempt to substitute
"mathematical funambulism" by a more rigorous and axiomatic approach.
However, I want to remark that I disagree with all the theories
presented there.

That in your own words "set of perfectly noninteractiung universes -
of which we can of course know only the single one we are in" do not
belong to physics.

[#] I want to reproduce here an interesting episode. It is often
    acknowledged in NESM literature that PO techniques were introduced
    by Nakajima, Zwanzig, and Mori. However, in a personal
    communication with Prigogine coworker, Gonzalo Ordonez, he said me
    that Prigogine had introduced PO techniques during a talk he gave
    and Zwanzig attended. Some time after Zwanzig published his
    foundational paper on the PO method. Gonzalo said me that Zwanzig
    gave a more elegant formulation but the original idea was from
    Prigogine!

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