"Juan R." González-Álvarez
2010-05-21 17:00:40 UTC
##########################################################
THE QUEST FOR THE ULTIMATE THEORY OF TIME, AN INTRODUCTION
TO IRREVERSIBILITY
##########################################################
I do not believe that we, scientists, can develop any "ultimate theory"
and I do not claim to have one. I have chosen the above title for this
event, in direct relation to the last book by Sean Carroll —The quest for
the ultimate theory of time— because I wanted to be a bit ironic here.
I smile each occasion that arrogant string theorists talk me about their
"theory of everything", specially because their approach is built over
several approximations and outdated formalisms. Somehow related is the
recent perplexity of mathematical physicists as John Baez who has reported
his shocking finding of "a curious generalization of Hamiltonian mechanics
that involves entropy" without being aware that more fundamental and
elegant formalisms, as the canonical one, have been known to physical
chemists and chemical physicists for more than thirty years now!
My comments on entropic generalizations of mechanics are available in This
Week's Finds in Mathematical Physics (Week 295). For a basic review of the
outdated aspects of both string and M theory see Canonical science: its
history, goals, and future.
I applaud Carroll valiant attempt to consider irreversibility seriously,
but his work is far from the usual standards in specialized literature. In
fact, whereas several of the authors cited below have advanced our
understanding of irreversibility, and some of them even provided us some
irreversible equation that explains a determined range of phenomena, there
is not such stuff as a Carroll equation that was used in the laboratories
by us. Carroll's scientific contributions to the theory of irreversible
thermodynamics, nonequilibrium statistical mechanics, or kinetic theory,
are easily summarized: zero. Moreover, he is misguided on many points. I
have done some comments and corrections in Carroll's blog, for instance in
Chapter Eight, Chapter Eleven, Chapter Eleven (b), Chapter Twelve, Chapter
Fourteen, Chapter Fourteen (b), and Chapter Fourteen (c) (external
hyperlinks), but the list is far from exhaustive; what is more, I have
decided to ignore both his unscientific speculations about the multiverse
and its reflections on philosophy of science.
The problem of the "arrow of time" is one of the oldest fundamental
problems of physics. The world, as we observe around us, is irreversible:
ice cubes melt and the water gets slightly cooler, broken eggs cannot be
recovered, paper ages, people passes away, etc. All those observations
finally led to the celebrated second law of thermodynamics, for the which
no violation is known [1,2]. However, the fundamental laws of mechanics
are time-reversible. There exist three attitudes toward this confrontation
between mechanics and observations: the coarse-grained, the pragmatic, and
the fine-grained.
*The coarse-grained attitude*
The authors who share this attitude affirm that, at some fundamental
level, Universe is time-reversible and that irreversibility is only an
illusion arising from our ignorance. This misguided attitude was initiated
by Ludwig Boltzmann, who was the first who tried to obtain the second law
of thermodynamics from mechanics. A short but beautiful history of the
early objections that his work has raised is given in the book by Laidler,
Meiser & Sanctuary [3]. For instance, the authors report how Boltzmann
paper of 1877 used the probability argument, contradicting his 1872
article, and add:
Much confusion resulted from the fact that for many years Boltzmann
was inconsistent in his statements about the matter, sometimes using
the probability argument and sometimes saying that it was not
necessary to invoke probability theory. When pressed, Boltzmann
invoked the probability argument to justify his arguments but
sometimes insisted that he had arrived at his function H on the basis
of pure dynamics.
I agree with the authors that Boltzmann never understood that he had made
some hidden assumptions that left out the terms that would lead to the
wrong conclusions and included only the terms that led to the second law.
Rather than giving the second law a purely mechanical origin, he used
extra-mechanical assumptions for breaking the time-symmetry of mechanics,
forcing it to match with the observations predicted by the law.
As Nico G. van Kampen —considered one of the "most outstanding theoretical
physicists of the second half of the 20th century" [4]— has noticed:
"Obviously it is a logical impossibility to deduce from reversible
equations an irreversible consequence" and emphasized that "One cannot
escape from this fact by any amount of mathematical funambulism". Many
important physicists, chemists, and mathematicians have stated similar
views. For instance, Landau & Lifshitz speculated, in their celebrated
course in theoretical physics [5], that the second law of thermodynamics
would be the macroscopic form of some fundamental law of irreversibility
associated to the measurement problem in quantum mechanics —recent
research partially confirms their suspicion but going on the line of
solving the measurement problem of quantum mechanics by using a
microscopic version of the second law!—. Truesdell's criticism is much
more acid and, in his classic monograph Rational thermodynamics, writes
about "microscopic reversibility" in the next terms [6]:
In fact, it requires no great mathematician to see that the
reversibility theorem and Poincaré's recurrence theorem makes
irreversible behavior impossible for dynamical systems in the
classical sense. Something must be added to the dynamics of
conservative systems, something not consistent with it, in order to
get irreversibility at all. Everyone competent in mechanics has known
this for a long time.
Unfortunately, some modern physicists —especially those "who would never
dream of trying to go through the mathematics and check the "proofs"",
again in the acid words of Truesdell— continue to repeat both the
ambiguities and subtle mistakes done by Boltzmann, and even add some
recent ones. I recall that in one occasion, I was so perplexed after
reading Science of Chaos or Chaos in Science? that I asked to the Nobel
laureate Ilya Prigogine about his opinion; he just confirmed my findings
with his concise, "the work of Bricmont is completely wrong".
Unfortunately, J. Bricmont's work has been indeed influential, especially
among non-experts [7].
Moreover, the amounts of "mathematical funambulism" do not seem enough and
some authors are utilizing metaphysical and religious arguments to sustain
their credo on coarse-graining. In a recent monograph [8], Robert Zwanzig
affirms that an ice cube, melted in water, will eventually reappear, but
"we never see it happen". Nevertheless, even when Zwanzig is ignoring the
experimental basis of the scientific method, he is still obligated to
conclude that the eventual reappearing would violate the second law of
thermodynamics. It is at this point when Zwanzig reveals us the reach of
his faith: "what we know about time irreversibility is obtained by
experiments on a human time scale". For authors as him, the paradox of
time is gone, the second law of thermodynamics, the laws of chemical
kinetics, or the evolution law of biology are only illusions obtained by
humans, mere mortals, whereas the time-reversible and deterministic laws
of mechanics have a divine origin and thus atemporal, being valid for
scales of time beyond our experience [9]!
It was not my objective to give here to my readers a detailed and complete
criticism of the coarse-grained attitude nor of the epistemological
fallacies traditionally associated to it. I plan to write a report about
all this.
*The pragmatic attitude*
This starts from the failure of the coarse-grained attitude. Irreversible
equations of motion are postulated, which are then justified a posteriori
sought in comparison with experimental data measured in the laboratory or
observed in nature. This is the attitude taken by Byung Chan Eu on his
interesting formulation of nonequilibrium statistical mechanics [10]:
We take the viewpoint that dynamical systems consisting of 10^23
particles require dynamical equations of broken time reversal symmetry
which are not derivable from the time reversal invariant Newtonian or
quantum equations of motion for the systems.
Another example is Zubarev's theory. He added a small infinitesimal source
term to the Liouville equation, breaking its time-symmetry.
The equations proposed have limited validity —for instance, the Eu
equation uses a Markovian approximation— and the question of the origin of
irreversibility is not even asked in this attitude.
This is also the case in Lindblad's semigroup approach [11]: "Thus the
present formalism has nothing to say about the origin of time's arrow".
Lindblad's theory has been criticized in the basis of its original
obtaining from purely mathematical assumptions and also by its detected
unphysical behaviors. Recent fine-grained —see below— developments confirm
the affirmations of Ulrich Weiss: "one should not attribute fundamental
significance to the Lindblad master equation". In the fine grained
attitude we can amend the equation with physical terms, study more general
dynamical regimes, and correct Lindblad qualitative assertions about
irreversibility.
*The fine-grained attitude*
This is the more physical, modern, and sophisticated attitude of the
three. Its goal is to find the microscopic source of the irreversibility
and to give a rigorous derivation of the macroscopic irreversible
equations, providing us with their microscopic generalizations too. As a
bonus, this attitude tries to solve other fundamental problems of physics
and of philosophy, such as the problem of free-will or the problem of
measurement in quantum mechanics. This attitude and the recent research
advances done at this Center will be discussed in the second part:
Trajectory branching in Liouville space as the source of irreversibility.
*References and notes*
[1] Regularly new articles are published which pretend to provide a real
violation, but each time it is showed that the violation is only in the
title or abstract of the articles. This was also the case for the last
assertions of violations of the second law done around the year 2002. See
the next reference for details.
[2] Has been thermodynamics violated? 2003: CPS:physchem/0309002.
González—Álvarez, Juan R.
[3] The Approach to Equilibrium In Physical Chemistry 2003: Houghton
Mifflin Company, Boston; 4th Ed. Laidler, Keith J.; Meiser, John H.;
Sanctuary, Bryan C.
[4] Nico van Kampen: charlatans beware! 2001: Physics World, January, 46.
ter Haar, D.
[5] Pag. 32 In Course of Theoretical Physics Volume 5, Statistical Physics
Part 1 1980: Pergamon Press, Oxford; 3rd Ed; Lifshitz, E. M.; Pitaevskii,
L.P. (Revision and Enlargement); Sykes, J. B.; Kearsley, M. J.
(Translation from Russian). Landau, L. D.; Lifshitz, E. M.
[6] Pag. 121 In Rational Thermodynamics 1968: MacGraw-Hill Book Company,
New York. Truesdell, C.
[7] It comes now to the mind the names Cosma Rohilla Shalizi and Carlos
Zuppa, basically because both are the last authors who I read. It is
rather amazing to read such misguided claims and unfair accusations
against Prigogine, specially when the authority of important —fictitious—
people as "Osanger [sic]", "Feymann [sic]", and "Keiser [sic]" is invoked
as argument, for instance by Zuppa; which gives a good idea of the
terribly ill-informed that he is.
[8] The Paradoxes of Irreversibility In Nonequilibrium Statistical
Mechanics 2001: Oxford Uniersity Press, Oxford. Zwanzig, Robert.
[9] In their work Entropy: A dialogue, Joel L. Lebowitz & Christian Maes
reproduce a conversation between a physicist and an... angel who has a
perfect knowledge of the past and present state of a perfectly isolated
macroscopic system of particles moving according to Hamiltonian dynamics.
It is worth asking if this is the same angel who convinced Zwanzig of that
the second law was obtained from experiments performed by humans and,
therefore, would not be trusted.
[10] Pag. 55 In Nonequilibrium Statistical Mechanics, Ensemble Method
1998: Kluwer Academic Publishers, Dordrecht. Chan Eu, Byung.
[11] Pag. 19 In Non-equilibrium Entropy and Irreversibility 1983: D.
Reidel Publishing Company, Dordrecht. Lindblad, Göran.
##############
NEWS AND BLOG:
##############
http://www.canonicalscience.org/publications/canonicalsciencetoday/canonicalsciencetoday.html
http://www.canonicalscience.org/publications/canonicalsciencetoday/20100521.html
THE QUEST FOR THE ULTIMATE THEORY OF TIME, AN INTRODUCTION
TO IRREVERSIBILITY
##########################################################
I do not believe that we, scientists, can develop any "ultimate theory"
and I do not claim to have one. I have chosen the above title for this
event, in direct relation to the last book by Sean Carroll —The quest for
the ultimate theory of time— because I wanted to be a bit ironic here.
I smile each occasion that arrogant string theorists talk me about their
"theory of everything", specially because their approach is built over
several approximations and outdated formalisms. Somehow related is the
recent perplexity of mathematical physicists as John Baez who has reported
his shocking finding of "a curious generalization of Hamiltonian mechanics
that involves entropy" without being aware that more fundamental and
elegant formalisms, as the canonical one, have been known to physical
chemists and chemical physicists for more than thirty years now!
My comments on entropic generalizations of mechanics are available in This
Week's Finds in Mathematical Physics (Week 295). For a basic review of the
outdated aspects of both string and M theory see Canonical science: its
history, goals, and future.
I applaud Carroll valiant attempt to consider irreversibility seriously,
but his work is far from the usual standards in specialized literature. In
fact, whereas several of the authors cited below have advanced our
understanding of irreversibility, and some of them even provided us some
irreversible equation that explains a determined range of phenomena, there
is not such stuff as a Carroll equation that was used in the laboratories
by us. Carroll's scientific contributions to the theory of irreversible
thermodynamics, nonequilibrium statistical mechanics, or kinetic theory,
are easily summarized: zero. Moreover, he is misguided on many points. I
have done some comments and corrections in Carroll's blog, for instance in
Chapter Eight, Chapter Eleven, Chapter Eleven (b), Chapter Twelve, Chapter
Fourteen, Chapter Fourteen (b), and Chapter Fourteen (c) (external
hyperlinks), but the list is far from exhaustive; what is more, I have
decided to ignore both his unscientific speculations about the multiverse
and its reflections on philosophy of science.
The problem of the "arrow of time" is one of the oldest fundamental
problems of physics. The world, as we observe around us, is irreversible:
ice cubes melt and the water gets slightly cooler, broken eggs cannot be
recovered, paper ages, people passes away, etc. All those observations
finally led to the celebrated second law of thermodynamics, for the which
no violation is known [1,2]. However, the fundamental laws of mechanics
are time-reversible. There exist three attitudes toward this confrontation
between mechanics and observations: the coarse-grained, the pragmatic, and
the fine-grained.
*The coarse-grained attitude*
The authors who share this attitude affirm that, at some fundamental
level, Universe is time-reversible and that irreversibility is only an
illusion arising from our ignorance. This misguided attitude was initiated
by Ludwig Boltzmann, who was the first who tried to obtain the second law
of thermodynamics from mechanics. A short but beautiful history of the
early objections that his work has raised is given in the book by Laidler,
Meiser & Sanctuary [3]. For instance, the authors report how Boltzmann
paper of 1877 used the probability argument, contradicting his 1872
article, and add:
Much confusion resulted from the fact that for many years Boltzmann
was inconsistent in his statements about the matter, sometimes using
the probability argument and sometimes saying that it was not
necessary to invoke probability theory. When pressed, Boltzmann
invoked the probability argument to justify his arguments but
sometimes insisted that he had arrived at his function H on the basis
of pure dynamics.
I agree with the authors that Boltzmann never understood that he had made
some hidden assumptions that left out the terms that would lead to the
wrong conclusions and included only the terms that led to the second law.
Rather than giving the second law a purely mechanical origin, he used
extra-mechanical assumptions for breaking the time-symmetry of mechanics,
forcing it to match with the observations predicted by the law.
As Nico G. van Kampen —considered one of the "most outstanding theoretical
physicists of the second half of the 20th century" [4]— has noticed:
"Obviously it is a logical impossibility to deduce from reversible
equations an irreversible consequence" and emphasized that "One cannot
escape from this fact by any amount of mathematical funambulism". Many
important physicists, chemists, and mathematicians have stated similar
views. For instance, Landau & Lifshitz speculated, in their celebrated
course in theoretical physics [5], that the second law of thermodynamics
would be the macroscopic form of some fundamental law of irreversibility
associated to the measurement problem in quantum mechanics —recent
research partially confirms their suspicion but going on the line of
solving the measurement problem of quantum mechanics by using a
microscopic version of the second law!—. Truesdell's criticism is much
more acid and, in his classic monograph Rational thermodynamics, writes
about "microscopic reversibility" in the next terms [6]:
In fact, it requires no great mathematician to see that the
reversibility theorem and Poincaré's recurrence theorem makes
irreversible behavior impossible for dynamical systems in the
classical sense. Something must be added to the dynamics of
conservative systems, something not consistent with it, in order to
get irreversibility at all. Everyone competent in mechanics has known
this for a long time.
Unfortunately, some modern physicists —especially those "who would never
dream of trying to go through the mathematics and check the "proofs"",
again in the acid words of Truesdell— continue to repeat both the
ambiguities and subtle mistakes done by Boltzmann, and even add some
recent ones. I recall that in one occasion, I was so perplexed after
reading Science of Chaos or Chaos in Science? that I asked to the Nobel
laureate Ilya Prigogine about his opinion; he just confirmed my findings
with his concise, "the work of Bricmont is completely wrong".
Unfortunately, J. Bricmont's work has been indeed influential, especially
among non-experts [7].
Moreover, the amounts of "mathematical funambulism" do not seem enough and
some authors are utilizing metaphysical and religious arguments to sustain
their credo on coarse-graining. In a recent monograph [8], Robert Zwanzig
affirms that an ice cube, melted in water, will eventually reappear, but
"we never see it happen". Nevertheless, even when Zwanzig is ignoring the
experimental basis of the scientific method, he is still obligated to
conclude that the eventual reappearing would violate the second law of
thermodynamics. It is at this point when Zwanzig reveals us the reach of
his faith: "what we know about time irreversibility is obtained by
experiments on a human time scale". For authors as him, the paradox of
time is gone, the second law of thermodynamics, the laws of chemical
kinetics, or the evolution law of biology are only illusions obtained by
humans, mere mortals, whereas the time-reversible and deterministic laws
of mechanics have a divine origin and thus atemporal, being valid for
scales of time beyond our experience [9]!
It was not my objective to give here to my readers a detailed and complete
criticism of the coarse-grained attitude nor of the epistemological
fallacies traditionally associated to it. I plan to write a report about
all this.
*The pragmatic attitude*
This starts from the failure of the coarse-grained attitude. Irreversible
equations of motion are postulated, which are then justified a posteriori
sought in comparison with experimental data measured in the laboratory or
observed in nature. This is the attitude taken by Byung Chan Eu on his
interesting formulation of nonequilibrium statistical mechanics [10]:
We take the viewpoint that dynamical systems consisting of 10^23
particles require dynamical equations of broken time reversal symmetry
which are not derivable from the time reversal invariant Newtonian or
quantum equations of motion for the systems.
Another example is Zubarev's theory. He added a small infinitesimal source
term to the Liouville equation, breaking its time-symmetry.
The equations proposed have limited validity —for instance, the Eu
equation uses a Markovian approximation— and the question of the origin of
irreversibility is not even asked in this attitude.
This is also the case in Lindblad's semigroup approach [11]: "Thus the
present formalism has nothing to say about the origin of time's arrow".
Lindblad's theory has been criticized in the basis of its original
obtaining from purely mathematical assumptions and also by its detected
unphysical behaviors. Recent fine-grained —see below— developments confirm
the affirmations of Ulrich Weiss: "one should not attribute fundamental
significance to the Lindblad master equation". In the fine grained
attitude we can amend the equation with physical terms, study more general
dynamical regimes, and correct Lindblad qualitative assertions about
irreversibility.
*The fine-grained attitude*
This is the more physical, modern, and sophisticated attitude of the
three. Its goal is to find the microscopic source of the irreversibility
and to give a rigorous derivation of the macroscopic irreversible
equations, providing us with their microscopic generalizations too. As a
bonus, this attitude tries to solve other fundamental problems of physics
and of philosophy, such as the problem of free-will or the problem of
measurement in quantum mechanics. This attitude and the recent research
advances done at this Center will be discussed in the second part:
Trajectory branching in Liouville space as the source of irreversibility.
*References and notes*
[1] Regularly new articles are published which pretend to provide a real
violation, but each time it is showed that the violation is only in the
title or abstract of the articles. This was also the case for the last
assertions of violations of the second law done around the year 2002. See
the next reference for details.
[2] Has been thermodynamics violated? 2003: CPS:physchem/0309002.
González—Álvarez, Juan R.
[3] The Approach to Equilibrium In Physical Chemistry 2003: Houghton
Mifflin Company, Boston; 4th Ed. Laidler, Keith J.; Meiser, John H.;
Sanctuary, Bryan C.
[4] Nico van Kampen: charlatans beware! 2001: Physics World, January, 46.
ter Haar, D.
[5] Pag. 32 In Course of Theoretical Physics Volume 5, Statistical Physics
Part 1 1980: Pergamon Press, Oxford; 3rd Ed; Lifshitz, E. M.; Pitaevskii,
L.P. (Revision and Enlargement); Sykes, J. B.; Kearsley, M. J.
(Translation from Russian). Landau, L. D.; Lifshitz, E. M.
[6] Pag. 121 In Rational Thermodynamics 1968: MacGraw-Hill Book Company,
New York. Truesdell, C.
[7] It comes now to the mind the names Cosma Rohilla Shalizi and Carlos
Zuppa, basically because both are the last authors who I read. It is
rather amazing to read such misguided claims and unfair accusations
against Prigogine, specially when the authority of important —fictitious—
people as "Osanger [sic]", "Feymann [sic]", and "Keiser [sic]" is invoked
as argument, for instance by Zuppa; which gives a good idea of the
terribly ill-informed that he is.
[8] The Paradoxes of Irreversibility In Nonequilibrium Statistical
Mechanics 2001: Oxford Uniersity Press, Oxford. Zwanzig, Robert.
[9] In their work Entropy: A dialogue, Joel L. Lebowitz & Christian Maes
reproduce a conversation between a physicist and an... angel who has a
perfect knowledge of the past and present state of a perfectly isolated
macroscopic system of particles moving according to Hamiltonian dynamics.
It is worth asking if this is the same angel who convinced Zwanzig of that
the second law was obtained from experiments performed by humans and,
therefore, would not be trusted.
[10] Pag. 55 In Nonequilibrium Statistical Mechanics, Ensemble Method
1998: Kluwer Academic Publishers, Dordrecht. Chan Eu, Byung.
[11] Pag. 19 In Non-equilibrium Entropy and Irreversibility 1983: D.
Reidel Publishing Company, Dordrecht. Lindblad, Göran.
##############
NEWS AND BLOG:
##############
http://www.canonicalscience.org/publications/canonicalsciencetoday/canonicalsciencetoday.html
http://www.canonicalscience.org/publications/canonicalsciencetoday/20100521.html
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