Is Classical Statistical Mechanics dead?

Vonny N.

With the (oft-quoted) tremendous success of Quantum Mechanics (QM)
predicting all types of phenomena, is there still any application for
the subject of Classical Statistical Mechanics (CSM)? The two subjects
do not represent limits of each other in any rigorous sense, so CSM
cannot be justified as a convenient approximation to QM in the same way
that Newtonian Mechanics can for Special Relativity Theory.

So are there any natural phenomena that find a prediction/explanation
only in CSM? I say 'natural' because obviously we could invent one, by
building a macroscopic object out of a large ensemble of ball-bearings,
say, if we wanted to, but this would surely be a bit self-serving.

Vonny N.

Douglas Natelson

Vonny N. wrote:
> [snip] is there still any application for
> the subject of Classical Statistical Mechanics (CSM)? The two subjects
> do not represent limits of each other in any rigorous sense, so CSM
> cannot be justified as a convenient approximation to QM in the same way
> that Newtonian Mechanics can for Special Relativity Theory.

Classical statistical mechanics is a perfectly reasonable limit
of quantum statistical mechanics, though perhaps not at the level
of rigor you're considering.

For example, the Maxwell-Boltzmann distribution function is a
reasonable approximation to both the Fermi-Dirac and Bose-Einstein
distributions, in the appropriate limits (kT much larger than the
intrinsic level spacings of the system).

> So are there any natural phenomena that find a prediction/explanation
> only in CSM? I say 'natural' because obviously we could invent one, by
> building a macroscopic object out of a large ensemble of ball-bearings,
> say, if we wanted to, but this would surely be a bit self-serving.

Are there any natural phenomena that find a prediction/explanation
*only* in Newtonian mechanics?

The kinetic theory of gases is pretty well described by classical
statistical mechanics. I should also point out that there is
active research still going on in what you might consider
analogous to your ensemble of cannon balls: granular media
(like sand piles). Amazingly, there are still many unanswered
questions about basic things like mixing and force distributions
in ensembles of classical grains, even though this seems like the
sort of thing that should have been solved 150 years ago by
a dead French mathematician whose name probably starts with "L".

ebunn@lfa221051.richmond.edu

In article <1138560039.263237.85710@g14g2000cwa.googlegroups.com>,
Vonny N. <vonnyn@hotmail.com> wrote:
>With the (oft-quoted) tremendous success of Quantum Mechanics (QM)
>predicting all types of phenomena, is there still any application for
>the subject of Classical Statistical Mechanics (CSM)? The two subjects
>do not represent limits of each other in any rigorous sense, so CSM
>cannot be justified as a convenient approximation to QM in the same way
>that Newtonian Mechanics can for Special Relativity Theory.
>
>So are there any natural phenomena that find a prediction/explanation
>only in CSM? I say 'natural' because obviously we could invent one, by
>building a macroscopic object out of a large ensemble of ball-bearings,
>say, if we wanted to, but this would surely be a bit self-serving.

Maybe I'm missing your point, but isn't this precisely what the
kinetic theory of gases is? We model a gas as a collection of many
ball-bearings undergoing elastic collisions with each other, and
pretend the particles are classical rather than quantum. Kinetic
theory's still useful, even though everyone knows that the exact
theory is quantum rather than classical.

-Ted

--
[E-mail me at name@domain.edu, as opposed to name@machine.domain.edu.]

Arnold Neumaier

Vonny N. wrote:

> With the (oft-quoted) tremendous success of Quantum Mechanics (QM)
> predicting all types of phenomena, is there still any application for
> the subject of Classical Statistical Mechanics (CSM)? The two subjects
> do not represent limits of each other in any rigorous sense

Neither classical nor quantum statistical mechanics is applicable
to real phenomena 'in any rigorous sense'; hence this argument does
not prove classical statistical mechanics dead.

Indeed, most of chemical engineering is done on the basis of classical
statistical mechanics, quantum methods being only used selectively where
absolutely essential.

Arnold Neumaier

Ilja Schmelzer

"Vonny N." <vonnyn@hotmail.com> schrieb
> With the (oft-quoted) tremendous success of Quantum Mechanics (QM)
> predicting all types of phenomena, is there still any application for
> the subject of Classical Statistical Mechanics (CSM)? The two subjects
> do not represent limits of each other in any rigorous sense, so CSM
> cannot be justified as a convenient approximation to QM in the same way
> that Newtonian Mechanics can for Special Relativity Theory.

It can. If there are some loopholes in the justification (I don't want
to argue about their existence) there are similar loopholes also in the
justification of this limit from classical Hamiltonian physics.

Some people argue that the justification of the limit is even easier
with QM as the starting point. Last not least, we already start with
probability distributions.

> So are there any natural phenomena that find a prediction/explanation
> only in CSM?

Everything in its domain of application too complicate to compute it in
full quantum beauty.

Ilja

Lou Pecora

In article <1138560039.263237.85710@g14g2000cwa.googlegroups.com>,
"Vonny N." <vonnyn@hotmail.com> wrote:

> With the (oft-quoted) tremendous success of Quantum Mechanics (QM)
> predicting all types of phenomena, is there still any application for
> the subject of Classical Statistical Mechanics (CSM)? The two subjects
> do not represent limits of each other in any rigorous sense, so CSM
> cannot be justified as a convenient approximation to QM in the same way
> that Newtonian Mechanics can for Special Relativity Theory.
>
> So are there any natural phenomena that find a prediction/explanation
> only in CSM? I say 'natural' because obviously we could invent one, by
> building a macroscopic object out of a large ensemble of ball-bearings,
> say, if we wanted to, but this would surely be a bit self-serving.
>
> Vonny N.
>

This is not exactly my field, but I know there has been a lot of effort
to link classical statistical mechanics to dynamics, especially
(conservative) chaotic dynamics. This work develops relationships
between a system's lyapunov exponents and stat mech quantities, among
other things. Check out the name Dorfman (at University of Maryland,
USA) for more info. I'm pretty sure there are others workin on this,
too.

-- Lou Pecora (my views are my own) REMOVE THIS to email me.

Vonny N.

> Maybe I'm missing your point, but isn't this precisely what the
> kinetic theory of gases is? We model a gas as a collection of many
> ball-bearings undergoing elastic collisions with each other, and
> pretend the particles are classical rather than quantum. Kinetic
> theory's still useful, even though everyone knows that the exact
> theory is quantum rather than classical.

My question, rather than my point, is: What exactly do people mean when
they say, as you do, things like "Kinetic theory's still useful"? We
know for sure that the predictions of the classical theory are
incorrect. We know for sure that the predictions of the quantum theory
fair much much better. We also now know that the quantum theory does
not support a picture of reality that is anything at all like the
'elastic ball-bearing' models - or even a picture of microscopic
reality at all for that matter. Given these assertions (which, of
course, I open for debate), I am wondering where classical statistical
mechanics still gets used productively.

Of course, if a system 'really is' composed of an ensemble of
macroscopic bodies, one would expect the classical theory to work, and
I would be interested in practical, rather than pedagogical, examples
of such systems. But in modeling gases, and matter in general, as
ensembles, the classical picture is so completely wrong that I just
can't imagine it being used by real physicists and engineers to solve
real problems. So again, if this reasoning is incorrect, I would be
very interested in (counter?)-examples.

Vonny

Igor Khavkine

Vonny N. wrote:
> With the (oft-quoted) tremendous success of Quantum Mechanics (QM)
> predicting all types of phenomena, is there still any application for
> the subject of Classical Statistical Mechanics (CSM)? The two subjects
> do not represent limits of each other in any rigorous sense, so CSM
> cannot be justified as a convenient approximation to QM in the same way
> that Newtonian Mechanics can for Special Relativity Theory.
>
> So are there any natural phenomena that find a prediction/explanation
> only in CSM? I say 'natural' because obviously we could invent one, by
> building a macroscopic object out of a large ensemble of ball-bearings,
> say, if we wanted to, but this would surely be a bit self-serving.

Classical statistical mechanics is not dead, it is very much alive.
However, it has merged with field theory. That may be a reason why you
wouldn't hear much about it.

The Euclidean version of the path integral formulation of quantum
mechanics (specifically quantum field theory, the QM of fields) is
equivalent to a classical statistical mechanical system with one extra
space dimension. So 3 (space) + 1 (time) dimensions in field theory
goes into 4 (space) dimensions in statistical mechanics. There is no
time considered in statistical mechanics since we assume everything is
in equilibrium. The identification between the two theories puts in
correspondance the Wick rotated phase factor with the Boltzman factor,
while the Wick rotated classical action corresponds to the Hamiltonian.
With the same identification, hbar corresponds to inverse temperature.

In short, classical statistical mechanics is as alive today as is field
theory.

Igor

Vonny N.

Douglas Natelson wrote:

> Classical statistical mechanics is a perfectly reasonable limit
> of quantum statistical mechanics, though perhaps not at the level
> of rigor you're considering.
>
> For example, the Maxwell-Boltzmann distribution function is a
> reasonable approximation to both the Fermi-Dirac and Bose-Einstein
> distributions, in the appropriate limits (kT much larger than the
> intrinsic level spacings of the system).

Here is a problem I have with your answer. If these distributions have
already been established using quantum theory, then there is no need to
re-derive them using classical theory. The fact that the latter might
be simpler, for example, is irrelevant since we already have the
theorem, so the work is done. Nor can we claim that the classical
derivation gives some sort of physical insight into what is going on
because we already know that classical statistical mechanics is founded
on a microscopic picture of reality which is completely and utterly
incorrect and untrustworthy.

If we take a result from classical statistical mechanics, whose quantum
counterpart has not yet been established, would anybody trust it
without doing the quantum calculations first? I would imagine not. In
this case, we once again have a result which has been established
quantum mechanically, and I can't see how we benefit from an
untrustworthy re-derivation using flawed classical methods.

>
> > So are there any natural phenomena that find a prediction/explanation
> > only in CSM? I say 'natural' because obviously we could invent one, by
> > building a macroscopic object out of a large ensemble of ball-bearings,
> > say, if we wanted to, but this would surely be a bit self-serving.
>
> Are there any natural phenomena that find a prediction/explanation
> *only* in Newtonian mechanics?

I take your point :-) In retrospect I shouldn't have inserted 'only'
since that really wasn't my point.

> The kinetic theory of gases is pretty well described by classical
> statistical mechanics.

This is just a tautology. The kinetic theory of gases is a theory
established within a classical framework. My question relates precisely
to the empirical usefulness of such theories.

I should also point out that there is
> active research still going on in what you might consider
> analogous to your ensemble of cannon balls: granular media
> (like sand piles). Amazingly, there are still many unanswered
> questions about basic things like mixing and force distributions
> in ensembles of classical grains, even though this seems like the
> sort of thing that should have been solved 150 years ago by
> a dead French mathematician whose name probably starts with "L".

I would certainly be interested in finding out a bit about this kind of
research. One would expect classical statistical mechanics to work in
dealing with ensembles of 'real' macroscopic objects. On the other
hand, I remain unconvinced that this theory is useful in dealing with
matter, such as gases, assumed to be made of such ensembles.

Vonny N.

Douglas Natelson

Vonny N. wrote:
> My question, rather than my point, is: What exactly do people mean when
> they say, as you do, things like "Kinetic theory's still useful"? We
> know for sure that the predictions of the classical theory are
> incorrect. We know for sure that the predictions of the quantum theory
> fair much much better. We also now know that the quantum theory does
> not support a picture of reality that is anything at all like the
> 'elastic ball-bearing' models - or even a picture of microscopic
> reality at all for that matter. Given these assertions (which, of
> course, I open for debate), I am wondering where classical statistical
> mechanics still gets used productively.

I could make analogous statements about Newtonian gravity
and general relativity: everyone knows that GR is more accurate
at calculating things like deflection of starlight by the
Sun's gravity, and the precession of the perihelion of Mercury.
In some philosophy of science sense that you're using, this makes
Newtonian gravity "incorrect". It is nonetheless extremely
useful for calulating, say, the period of Jupiter's orbit around
the sun, to a high (though not arbitrary) accuracy.

> But in modeling gases, and matter in general, as
> ensembles, the classical picture is so completely wrong that I just
> can't imagine it being used by real physicists and engineers to solve
> real problems.

Why do you say this? Kinetic theory lets one derive the
"ideal gas law". In the regime where kinetic theory's approximations
are reasonable, the ideal gas law works very very well.
Real physicists and engineers use it all the time to do very
useful things.

Just because a theory doesn't describe every aspect of a
system flawlessly under every circumstance doesn't render
the theory absolutely "incorrect" and somehow useless.
It seems like you're confusing a "theory" with some platonic
ideal of understanding....

Hendrik van Hees

Vonny N. wrote:

> Here is a problem I have with your answer. If these distributions have
> already been established using quantum theory, then there is no need
> to re-derive them using classical theory. The fact that the latter
> might be simpler, for example, is irrelevant since we already have the
> theorem, so the work is done.

I doubt that classical statistics is simpler than quantum statistics.
There is so much trouble in classical statistics which is avoided in
quantum statistics (and then taking the classical limit when
appropriate to make practical calculations simpler), e.g., Gibbs's
paradoxon (about the mixing entropy of, e.g., identical parts of an
ideal gas). I don't know whether there is a simple "classical solution"
for this paradox, while it is simply not there in quantum statistics.
There in many-body theory we know from quite basic principles that
there are only bosons and fermions (if the space dimension ist >=3),
and this solves the riddle from the very beginning.

Another example is the derivation of Boltzmann's transport equation
which is quite straight-forwardly given in the Schwinger-Keldysh
time-contour formalism (see vol. 10 of Landau and Lifshits's textbook
on theoretical physics).

Why should we bother ourselves (and students!) with classical
statistics, when we have quantum statistics? We can take the classical
limit anyway, if it is applicable for a given system in question and
this helps to solve the problem.

--
Hendrik van Hees Texas A&M University
Phone: +1 979/845-1411 Cyclotron Institute, MS-3366
Fax: +1 979/845-1899 College Station, TX 77843-3366
http://theory.gsi.de/~vanhees/ mailto:hees@comp.tamu.edu

Vonny N.

Douglas Natelson wrote:

> I could make analogous statements about Newtonian gravity
> and general relativity: everyone knows that GR is more accurate
> at calculating things like deflection of starlight by the
> Sun's gravity, and the precession of the perihelion of Mercury.
> In some philosophy of science sense that you're using, this makes
> Newtonian gravity "incorrect". It is nonetheless extremely
> useful for calulating, say, the period of Jupiter's orbit around
> the sun, to a high (though not arbitrary) accuracy.

I disagree, because in the case of gravity the relationship between the
Newtonian and General Relativistic accounts is well understood. That
is, one can derive the Newtonian version from the General Relativistic
version and see where and why the former will provide perfectly
adequate, and much simpler, approximations in many cases. My argument
is that the same cannot be said in statistical mechanics because there
is no connection between the classical and quantum versions, with
respect to which each can be seen as a limiting case of the other. Of
course, we are all presented with heuristic arguments in text books
that such a limit exists, but as physicists and philosophers working on
the foundations of quantum theory are well aware, there are serious
holes in these reductionistic claims.

The radical, and poorly understood, distinction between the quantum and
classical theories surely makes it dubious to solve a problem in
statistical mechanics using the classical theory, arguing that it will
give an approximation to the quantum version. Who says so? There are
endless examples of statistical predictions which the classical theory
gets hopelessly wrong, and there is no prescribable set of conditions
one can use to decide that classical statistical mechanics will be
reasonably accurate. In Newtonian mechanics this is not the case. We
know what conditions on velocity and mass are necessary for this theory
to give good answers. The limiting process between the quantum and
classical statistical theories, on the other hand, is at best a
mystery, and at worst, non-existent.

> Just because a theory doesn't describe every aspect of a
> system flawlessly under every circumstance doesn't render
> the theory absolutely "incorrect" and somehow useless.
> It seems like you're confusing a "theory" with some platonic
> ideal of understanding....

I have no such platonic view, and am perfectly comfortable in the
knowledge that no theory is 'correct' or 'incorrect' in any objective
sense. Hopefully the above explanation clarifies my real issue here.

Vonny N.

Alex

Vonny N. wrote:
> > Maybe I'm missing your point, but isn't this precisely what the
> > kinetic theory of gases is? We model a gas as a collection of many
> > ball-bearings undergoing elastic collisions with each other, and
> > pretend the particles are classical rather than quantum. Kinetic
> > theory's still useful, even though everyone knows that the exact
> > theory is quantum rather than classical.
>
> My question, rather than my point, is: What exactly do people mean when
> they say, as you do, things like "Kinetic theory's still useful"? We
> know for sure that the predictions of the classical theory are
> incorrect. We know for sure that the predictions of the quantum theory
> fair much much better. We also now know that the quantum theory does
> not support a picture of reality that is anything at all like the
> 'elastic ball-bearing' models - or even a picture of microscopic
> reality at all for that matter. Given these assertions (which, of
> course, I open for debate), I am wondering where classical statistical
> mechanics still gets used productively.

Rethinking of space and time in Special Relativity changes the basis of
classical mechanics. Nothing is more fundamental to mechanics than its
spatial framework. It is natural to talk about succession here.

The situation is different in statistical mechanics. Of course, one can
say:

>>>For example, the Maxwell-Boltzmann distribution function is a
>>>reasonable approximation to both the Fermi-Dirac and Bose-Einstein
>>>distributions, in the appropriate limits (kT much larger than the
>>>Intrinsic level spacings of the system).

But it is not about fundamentals of classical or quantum statistics.

Probability distributions in classical statistical mechanics can be
obtained from simple assumptions of underlying kinetic theory ("a large
ensemble of ball-bearings"). The same kind of basis is not present in
quantum statistical theory (I would be happy to hear that it is wrong).

If someone could provide "quantum kinetic theory" and get statistical
distributions from it, we could talk about succession in the same sense
as in SR. A new understanding of kinetic theory could be significant in
attempt to look beyond QM.

Alex

Douglas Natelson

Hendrik van Hees wrote:
> Why should we bother ourselves (and students!) with classical
> statistics, when we have quantum statistics? We can take the classical
> limit anyway, if it is applicable for a given system in question and
> this helps to solve the problem.

This reminds me of a discussion several of us had back in
grad school. We were all TA-ing different sections of introductory
mechanics (for pre-meds, for engineers, and for would-be physicists).
One of the TAs was advocating starting off with Lagrangians,
since teaching people Newton's laws and forces and such was
silly and hid the beauty of things like Noether's theorem.
While his point was well taken, the rest of us all made the case
that that would be a pedagogically unwise approach, particularly
for undergrads who had never seen variational calculus before and
who would undoubtedly ask, "But *why* is the action extremized?"
Of course, our colleague really would've preferred starting off
with Feynman-Hibbs path integrals and explaining taking the
classical limit, but that, too, was not considered useful for
teaching freshmen.

acl

> If we take a result from classical statistical mechanics, whose quantum
> counterpart has not yet been established, would anybody trust it
> without doing the quantum calculations first? I would imagine not.

One example that springs to mind is liquid state theory (see eg Hansen
and McDonald, "Liquid state theory"). This subject is highly nontrivial
(eg how to compute things near the critical point? A partial answer is
given by the so-called "Hierarchical reference theory" of Parola,
Reatto et al). Understanding the structure of the liquid state is also
nontrivial, because it's not close to a nice, well-understood system
(unlike solids, which are close to a periodic lattice, or gases, which
are close to ideal gases). Most of the results in this area are
obtained using purely classical statistical mechanics, and yes,
specialists in the field believe them.
There are other examples as well, that have already been mentioned; for
example, granular matter is tentatively being explored. Spin glasses,
and, in general, disordered systems, are much harder to understand than
they look at first glance. And so on.

It's just that subjects such as these are not well known, because they
don't sound as exciting as (say) particle physics.

Alex

Vonny N. wrote:
> I disagree, because in the case of gravity the relationship between the
> Newtonian and General Relativistic accounts is well understood. That
> is, one can derive the Newtonian version from the General Relativistic
> version and see where and why the former will provide perfectly
> adequate, and much simpler, approximations in many cases. My argument
> is that the same cannot be said in statistical mechanics because there
> is no connection between the classical and quantum versions, with
> respect to which each can be seen as a limiting case of the other. Of
> course, we are all presented with heuristic arguments in text books
> that such a limit exists, but as physicists and philosophers working on
> the foundations of quantum theory are well aware, there are serious
> holes in these reductionistic claims.
>
> The radical, and poorly understood, distinction between the quantum and
> classical theories surely makes it dubious to solve a problem in
> statistical mechanics using the classical theory, arguing that it will
> give an approximation to the quantum version. Who says so? There are
> endless examples of statistical predictions which the classical theory
> gets hopelessly wrong, and there is no prescribable set of conditions
> one can use to decide that classical statistical mechanics will be
> reasonably accurate. In Newtonian mechanics this is not the case. We
> know what conditions on velocity and mass are necessary for this theory
> to give good answers. The limiting process between the quantum and
> classical statistical theories, on the other hand, is at best a
> mystery, and at worst, non-existent.

Rethinking space and time in Special Relativity changes the basis of
classical mechanics. Nothing is more fundamental in mechanics than its
spatial framework. It is natural to talk about succession, when
fundamentals are "improved".

The situation is different in statistical mechanics. Of course, one can
say:

>>For example, the Maxwell-Boltzmann distribution function is a
>>reasonable approximation to both the Fermi-Dirac and Bose-Einstein
>>distributions, in the appropriate limits (kT much larger than the
>>Intrinsic level spacings of the system).

But it is not about fundamentals of classical or quantum statistics.

Probability distributions in classical statistical mechanics can be
obtained from simple assumptions of underlying deterministic kinetic
theory ("a large ensemble of ball-bearings"). The same kind of basis is
not present in quantum statistical theory.

However, technically it is possible to change the assumptions of
kinetic theory. As a result, related statistical outcome
(distributions) will vary. It would be nice, if one could provide
"quantum kinetic theory" and get statistical distributions from it ...
then we could talk about succession in the same sense as in SR and
Newtonian Mechanics, for example. At least such exercise would help to
understand better the origins of quantum statistics.

Alex

DRLunsford

Vonny N. wrote:
> With the (oft-quoted) tremendous success of Quantum Mechanics (QM)
> predicting all types of phenomena, is there still any application for
> the subject of Classical Statistical Mechanics (CSM)? The two subjects
> do not represent limits of each other in any rigorous sense, so CSM
> cannot be justified as a convenient approximation to QM in the same way
> that Newtonian Mechanics can for Special Relativity Theory.
>
> So are there any natural phenomena that find a prediction/explanation
> only in CSM? I say 'natural' because obviously we could invent one, by
> building a macroscopic object out of a large ensemble of ball-bearings,
> say, if we wanted to, but this would surely be a bit self-serving.
>
> Vonny N.

Certainly not - whenever the Maxwell-Boltzmann statistics apply
(distinguishable particles) you're in the classical realm.

-drl