Quantum mechanics appeared as the description of discrete atomic states. But there are already discrete states in non-linear classical mechanics. They are Poincare limiting cycles. Why can not they be quantum states? Who has considered this possibility?
We should distinguish Poincare recurrence, which indeed occurs in all bounded Hamiltonian systems, from limit cycles, which are typical of dissipative systems. Poincare recurrence refers to the fact that any initial condition with probability one returns infinitely often to an arbitrarily small but fixed neighbourhood of the initial condition. In typical (chaotic) systems, this does not lead to periodic, or even quasiperiodic behaviour, because, once the orbit has nearly returned to the initial condition, the precision is not sufficient to guarantee that it will come back a second time the same way. So no periodicity is involved. Also, the times for this phenomenon are large: they go approximately as the volume of all phase space divided by the volume of the initial neighbourhood
Poincare recurrence does have a well known quantum equivalent: when a system is bounded, the spectrum is discrete, so that the amplitude of returning to an arbitrary initial state is a quasi periodic function, and since it starts at one, comes infinitely often back arbitrarily close to one.
I think it is necessary to consider from the historical perspective. The development of quantum mechanics greatly influenced by the tools of statistical physics. Determinism against spontaneity which came from experimental data processing.
But who has considered, Vasiliy? Discussions about interpretation of quantum mechanics continue.
The development of quantum mechanics greatly influenced by the tools of statistical physics - QM is not a statistical physics You can show that by algebras with you get from statistics an from quantum mechanics.
I think that At the begining try prof that Poincare limiting cycles form Hilbert space - and what kind of states they should describe
I think such ideas can be classified as hidden variable models of Quantum Mechanics, which inevitably run into problems by failing to violate the Bell inequalities. Maybe they could work if formulated as some kind of nonlinear dynamics in Hilbert space instead of physical space. But there are quite good experimental tests of the superposition principle, and the Born probability interpretation, which therefore cannot be violated too much.
But a relativistic invariant dynamical model replacing the collapse of the wave function by something less subjective would be welcome...
We should distinguish Poincare recurrence, which indeed occurs in all bounded Hamiltonian systems, from limit cycles, which are typical of dissipative systems. Poincare recurrence refers to the fact that any initial condition with probability one returns infinitely often to an arbitrarily small but fixed neighbourhood of the initial condition. In typical (chaotic) systems, this does not lead to periodic, or even quasiperiodic behaviour, because, once the orbit has nearly returned to the initial condition, the precision is not sufficient to guarantee that it will come back a second time the same way. So no periodicity is involved. Also, the times for this phenomenon are large: they go approximately as the volume of all phase space divided by the volume of the initial neighbourhood
Poincare recurrence does have a well known quantum equivalent: when a system is bounded, the spectrum is discrete, so that the amplitude of returning to an arbitrary initial state is a quasi periodic function, and since it starts at one, comes infinitely often back arbitrarily close to one.
Dear colleagues!
Hilbert space, superposition principle and all quantum ideology is linear. Linearity is the simplest way to extrapolate and describe. But experimental evidence is discreteness of atomic states (spectral lines) for which mathematical instruments exist in non-linear classical mechanics. Simply, they were not explored so far because of their difficulty. Bogolyubov's consideration appeared only in the second half of twentieth century and till now many purely mathematical problems exist.
Let us start from the fact that at the heart of quantum mechanics is the Schrödinger equation, which is a consequence of the formalism of classical mechanics, and the principle of additive solutions. Then we come to the conclusion that the quantum mechanics, as well as classical mechanics, is not capable describe rigorously nonlinear processes of evolution for which the second law of thermodynamics is the basis
Vyacheslav,
non-linearity is mathematics. Don't mix it with physics (thermodynamics) and philosophy (evolution). Poincare and Bogolyubov could rigorously describe nonlinear processes not only in classical mechanics. Other people are also working.
To say that quantum mechanics is unable to handle nonlinear processes because the Schrödinger equation is linear, is like saying that classical mechanics is unable to handle nonlinear processes because the Liouville equation is linear. In both cases any kind of Hamiltonian dynamics, linear or nonlinear, is reformulated in terms of a linear equation.
Dear Eugene.
It is clear that all evolutionary processes are non-linear, but it is obvious that not all non-linear processes are evolutionary. So here the word "non-linearity" is not the key. The main thing is that the canonical formalism of classical mechanics does not describe the evolution. Hence the well-known problem of irreversibility.
We don't consider now evolution, Vyacheslav. The question is: Can descrete spectral lines be described by non-linear classical mechanics and electrodynamics or not?
Kare,
of course, concrete QM problems, for example Hartry-Fock, are non-linear also. But I was speaking about postulates of QM, coming from linearity.
I should perhaps stress that I am not hostile to such ideas, since I have spent considerable time thinking in these directions (some of my colleagues even more, cf. the linked preprint). To me the unsurmountable obstacle (last time I thought) was to construct a model which violates the Bell inequalities.
Article Collapse of the Quantum Wavefunction
Vyascheslav> the canonical formalism of classical mechanics does not describe the evolution. Hence the well-known problem of irreversibility.
Nevertheless, molecular dynamical simulations of thermal systems is an important practical approach, for which the canonical formalism of classical mechanics is essential (to generate the right statistics). I believe few of those practicing molecular dynamics consider irreversibility a problem.
Sorry Eugene, for following up this out-of-topic (for this thread) post.
About evolution, Vyascheslav,
in non-linear classical mechanics determenistic chaos exists.
I have read your article, Kare. I also think, that in linear approximation nothing happens in Universe, but I don't understand, how You managed to make quantum mechanics determenistic. May be in some special sense? Also I think, that neglecting phase angle in your equation (9) is unjustified.
Regards,
Eugene.
Our equations are deterministic, just like the Schrödinger equation (or any other set of classical dynamical equations) is deterministic before imposing the collapse postulate, which we don't do. However, it is nonlinear, due to its coupling with the reservoir/measurement apparatus, and this nonlinearity forces the system towards one of several (in this case two) attractors (limiting cycles in your lingo), where each attractor has a basin of attraction of size in (approximate) agreement with the Born interpretation.
We don't justify neglecting the phase angle in equation (9); we are saying that its value will not modify the relative sizes of the basins of attraction in any essential way. It was simply a matter of available computer time: Scanning over (say) twenty phase angles for each of the other variables would have increased the simulation time by a factor twenty, beyond the thesis deadline of Håkon Brox.
The history of this work is that I had a project student a few years earlier, who investigated a simplified version of this model. In that model the phase angle was the only unobserved (i.e., hidden) variable to vary over; it lead to similar results than in the paper. Independent of this, later Håkon started a master thesis project with Kiet Anh as advisor along similar lines, but with a much more sophisticated model for the reservoir/measurement apparatus. The collaboration occurred when we learned about each others work; but I was a fairly passive member.
It is a quite clever model, if I must say myself. Quite in agreement with your idea of limiting cycles. But it does belong to the category of hidden variable models; hence it cannot get around the Bell equalities. Which is the main reason why it was never published beyond the preprint.
Dear Eugene,
Please allow a lengthy response to your question using a general mathematical argument. Until we develop working physical models of the sub-atomic particles, radiation, and the fundamental forces that are all compatible with each other and with experiment, quantum mechanics is required, but will remain dissatisfying to many.
Per Richard Feynman, "I think I can safely say that nobody understands quantum mechanics." Per Steven Weinberg, "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics." These expressions of uncertainty still exist today because QM was empirically derived based on the observations and measurements of the behavior of particles and forces - NOT on the underlying mechanisms of the particles and forces that give such observations and measurements. Since QM was empirically derived using quantum logic, substituting part of a QM concept with a classical mechanical concept, based on classical logic, would be incompatible with the overall QM approach used to describe the overall subject process and system.
Once, working models of particles and forces are developed, then we will be able to conceptually explain how and why the observations and measurements of them occur as they do, without having to invoke QM. Why discrete spectral lines occur, could then be understood, and described using classical mechanics. Due to the randomness of particles, a probabilistic mathematical platform will still be required in some cases, but we will understand why we get the measurements and observations that we do.
Following are a few examples that offer physical explanations of some phenomena that only QM can currently explain. The physical explanations are based on derived models of sub-atomic particles, photons, and forces in a book, from which the attachment is excerpted.
1. If a photon were an oscillating fiber that oscillates and twirls in the same planes perpendicular to its propagation, then it would trace a waveform during translation, but would not be a wave unto itself. Please remember that Maxwell’s differential wave equations only indicate that radiation translates in a waveform and is not a wave itself. Although photons are not spherical particles or waves, they would still exhibit wave-particle duality in an interferometer.
2. Due to the photon fiber’s length, it is capable of interacting with another photon fiber that appears to be separated from it. Thus, what appears to be quantum entanglement is only an interaction between the two photons.
3. Due to the photon’s twirling motion as it translates, it can appear to be in two states at the same time – vertical and horizontal polarization, a QM phenomenon.
4. If each sub-atomic particle were a finite circumferential/cylindrical-shaped field, not “particle”, then it would be capable if interacting with other “particles” that appear to be separated from it. Thus, what appears to be quantum entanglement is only an interaction between the two “particles”.
5. If the construction of an electron (and atoms) were known, then it would not be necessary to consider electrons as an uncertain, non-deterministic, smeared, probabilistic wave–particle orbital about the nucleus; and by extension it would not be necessary to invoke QM to understand why bulk matter doesn’t collapse under the electric force alone.
6. If the construction of an electron (or other sub-atomic particle) were known, then it would not be necessary to invoke QM to understand how and why it exhibits wave-particle duality in an interferometer, without being a wave or particle itself.
7. If the construction of an electron (and atoms) were known, then it would be possible to know the position of the electron. If its position is deterministic, then its momentum would be also; thus, it would not be necessary to invoke QM to understand atomic behavior.
8. If the construction of sub-atomic particles were known, then it would be possible to understand how and why high-impact particle collisions can result in numerous uncommon hadrons and lepton particles that are mostly unstable; and why multiple transitions to more than one type of particle may occur before a stable particle is obtained or why it disintegrates completely, resulting in radiation.
If a conceptual understanding of the underlying mechanisms for particles and forces is not available, then it can be difficult to interpret the results of experimental research; this then, further compounds the difficulty of understanding physics phenomena, such as those described by QM. Hence, as a result, basic simple phenomena such as that, which gives discrete spectral lines and which cannot be described plainly using classical mechanics, often arise. A derivation of how discrete spectral lines are created is given in Sections 6 & 7 of the referenced book. I apologize for the lengthy comments on QM vs. CM, but it seems necessary as a step towards gaining a physical understanding of physics phenomena.
Regards,
Dan Correnti
Article UNVEILING of the ELECTRON (Post #1) [A Proposed Electron Structure]
How many pictures do you need to explain the Bell-viololating experiments?
http://www.nature.com/nature/journal/v497/n7448/full/nature12012.html
So, dear colleagues, I, personally, don't see principle limitations for classical mechanics to describe descrete spectral lines.
Thanks to all.
Dear Eugene F Kislyakov
I think you will be interested in our results of calculation of the classical oscillator passing through a potential barrier. According to these results in classical mechanics are not excluded discrete lines. Pleas see in attachmants.
Your work is very interesting, Vyacheslav, giving new insight into the whole problem of tunneling. Also, tunneling particle (electron) is usually considered as structureless in quantum mechanics, everybody knows, that static potential barrier is fiction, not existing in nature. Moreover, there is no any potential at all in many body quantum mechanical problem.
To be published, english of your paper should be improved. Also, You can consider parabolic potential barrier, where exact quantum mechanical solution exists, and compare it with your results.
Regards,
Eugene.
Eugene,
For reference, it should be noted that the Bell’s inequality equation was derived utilizing “local hidden variables” that would only have linear effects on the correlations determined at different angles of measurement of a spin ½ particle or photon. For example, the correlations would vary linearly with the angle of measurement.
The electron and the photon have axes along which, their dynamic elements translate and/or oscillate as shown in the Figures 1 and 2 of the link previously given. Measurements and/or affects on the properties of either the electron or photon vary in accordance with the cosine of the angle between the axis of measurement and the axis of either the electron or photon. As you know, this wouldn’t be a linear relationship.
If the construction of a spin ½ electron or a photon were known at the time of Bell’s work, it would have been realized that the correlation/angle relationship would not be linear, but instead would be related to the cosine function as reproduced in experiments and would not need to be attributed to quantum mechanics. In all cases, the Bell inequality equation would be violated.
If a conceptual understanding of the underlying mechanisms for particles and forces is not available, then it can be difficult to interpret theory and the results of experimental research; this then, further compounds the difficulty of understanding physics phenomena, such as those described by QM. Hence, as a result, basic simple phenomena such as that related to Bell’s work, which cannot be described plainly, often arise. I know this isn’t in the direction you are going with your work, but it seems necessary to put out the above comments as a step towards gaining a physical understanding of physics phenomena.
Regards,
Dan Correnti
Dear Eugene F Kislyakov
commenting on your phrase; "I, personally, don't see principle limitations for classical mechanics to describe descrete spectral lines."
I agree with you, but the only one thing that is needed to be considered in classical mechanics is the scaling effects. The scaling effects may lead to change the form of real equations into complex ones. So instead of observing the real particle we just can observe its projections.
Dan> ...it should be noted that the Bell’s inequality equation was derived utilizing “local hidden variables” that would only have linear effects...
I don't know where the above misconception comes from. But, for reference, it should be noted out that it is just that.
Thank You for interesting comment, Dan.
I, personally, divide all this problem into description and understanding (interpretation). To my mind the problem is that QM gives good description, but good interpretation is lacking and may be impossible. Contrary, classical mechanics satisfies people (they understand it), but possible non-linear description of atomic phenomena is very difficult and actually lacking till now.
Regards,
Eugene.
P.S. The solution may be Poincare limiting cycles as a bridge between quantum and classical descriptions.
Regards to all.
Kare & Eugene,
Thanks for taking the time to reply and making clarifying points.
The “local hidden variables” concept was only mentioned because Bell set out to disprove their existence when he derived the inequality equation. He assumed they would only affect the system linearly. However, if they affect the system per the cosine function, then his inequality equation is violated. For the case given in the earlier post, the “local hidden variable” is that the axis of either the electron or photon can occur at different angles to the measuring device.
It is hoped that the earlier posts and references will be helpful in developing mathematical description of the factors giving discrete energy quanta emanating from atoms based on classical mechanics. Hoping not to belabor the points in the first post and as the above example shows, it is better to derive appropriate mathematical formulations based on working physical models of particles and forces, not just on observations and measurements; otherwise the theory would be empirical and abstract, as QM is. For the model that gives a physical description and view of how, why, and when a quantum of energy is emitted from the hydrogen molecule, please refer to the previous references. Both of you receiving this is appreciated.
Regards,
Dan Correnti
Dan,
emition of the quant of energy must include interaction of the field with itself and it is the non-linear process. Only discretization of the space-time can stop this self-interection loop and eliminate infinities.
Regards,
Eugene.
Dear Eugene F Kislyakov
It is hardly possible to answer your question without determining the area limitation of the formalism of classical mechanics. One of the key limitations connected with the use of the hypothesis of holonomic constraints in the derivation of the equation LaGrange
Charles,
see this historically. The only problem of foundators of QM was discreteness. But at that times they did not know, that classical mechanics can give discreteness as well. The works of Poincare were poorly known and understood, and Bogolyubov was not yet born.
Regards,
Eugene.
Excuse, F. Leyvraz, but limiting cycles and recurrence are different things and limiting cycles don't need dissipation.
Regards,
Eugene.
I understand your view point, Charles, but the problem of understanding (interpretation) of QM remains. From where are quantum logic and so on?
Dear Charles
Space time is the origin of all the problem of hardly understanding of quantum mechanics.
As I told you before space time is not so simple like what is described by Lorentz transformations. When the matter is related to elemantary particles it will be complex. This can give a logic way to derive quantum mechanics using classical logic. You don't need to add hoc the i factor to the wave functions to get the right solution. Differentiating between quantum mechanics and space time behavior is the cause of most mystries in physics.
@Eugene:
``Excuse, F. Leyvraz, but limiting cycles and recurrence are different things and limiting cycles don't need dissipation.''
Liouville's theorem makes limit cycles impossible for mechanical systems. What you can have---for very anomalous systems, such as the harmonic oscillator---is the fact that each iinitial condition lies on a periodic orbit. But this is exceedingly rare, and quite incompatible with chaos. My point was that Poincaré recurrence does not involve cycles (limit or otherwise, because it only involves approximate return. And even when there are periodic orbits in chaotic systems, they are unstable, thus very far from being limit cycles in any sense).
Regards
Francois
Harmonic oscillator is linear system. In two dimensional systems chaos is impossible. What Liouville's theorem do You mean, Francois?
Regards,
Eugene.
Francois> Liouville's theorem makes limit cycles impossible for mechanical systems.
If you have a microscopic system (like an atom) coupled to a macroscopic one, both Hamiltonian, things are more subtle. In a semiclassical (non-linear) approximation, an atomic state which is a superposition of two or more eigenstates will generate a current which leads to a radiation of energy. This selects the atomic eigenstates to be approximate limiting cycles of the atomic system. But the total universe do not approach a limiting cycle, only well decoupled small subsystems. I stress that this is a semiclassical viewpoint.
Total universe is surely not quantum system, Kare. It has no observer.
Eugene> Total universe is surely not quantum system
This is almost my point also (but not including the word surely, and replacing the observer with something less subjective --- f.i. macroscopic systems).
@Kare: Fair enough. I was replying to Eugene's claim that dissipation is not needed for limit cycles. It is. In your example, the macroscopic external system can provide it
@Eugene: Liouville's theorem stating that in a Hamiltonian system phase space volume is conserved in time. If I understand a limit cycle to be a periodic orbit which atracts a domain of finite volume around it, then such limit cycles cannot exist in Hamiltonian systems. As I said, it can happen, both in linear and nonlinear systems, that all orbits are periodic. However, as you were repeatedly talking about chaos, I pointed out that this cannot happen in chaotic systems. Rather, in chaotic systems, there are periodic orbits, but they are unstable.
Additionally, there does exist some kind of correspondence between these periodic orbits and quantum levels, but it is quite complicated and many-to-one. Thus a single orbit corresponds to an oscillation in the density of eigenvalues over a large number of eigenvalues. See the Gutzwiller formula for details.
Dear Eugene
the term i can have many physical meanings, so it's not a matter of just adding it or not. The most important is why we need it in quantum mechanics? This question hasn't being answered physically yet, while it's mathematically well being understood. The dynamic space time offers such answer physically and mathematically.
The periodicity can give such justification, but there will be a problem that if you apply such periodicity on Lorentz transformations you will get a real function, becasue what is periodic for elementary particles it does't require to be periodic for classical objects so you need complex transformations to achieve the physical meaning. while the complex one will give you a complex function, which avoid falling on such paradox.
Thank You for the reminiscence about Gutzwiller, Francois. But I strongly recommend You Bogolyubov, Andronov and original Poincare's works. They did not know much about chaos but solved concrete non-linear problems. It is differential equations, Francois. Who has proved that limiting cycles are not possible without dissipation? And what does it mean dissipation for arbitrary differential equation?
Theorems are good, Francois, but any working generator is the model of limiting cycle.
From mathematical (topological) point of view it is the topic of one of the Hilbert's problems.
Regards,
Eugene.
Уважаемый Евгений ! Спасибо за приглашение. У меня когда-то были аналогичные размышления на тему о физической сути квантования. Возможно, мои соображения окажутся сходны с Вашими соображениям в связи с Poincare limiting cycles, которыми я, признаюсь сразу, профессионально не занимался. Вместо того, чтобы углубляться в суть квантовой механики, я предложил бы Вам остановиться на поразительном сходстве двух наших конкретных статей: Ваша Electromechanical generator: going from micro to nano size и моя Uppsalator's acceleration. Мой Уппсалатор - это тоже генератор (колебаний), и тоже электромеханический: в нем электроосмос сопряжен с вязким течением. Самое интересное, что в нем основной процесс тоже разыгрывается на масштабах от микро- до нанометровых. Последняя статья вышла у меня в 2007 в Electrophoresis. Особенность этой работы в том, что мне удалось подкорректировать теорию автоколебаний в этой системе, предложенной Т.Теореллом в университете г. Уппсала 65 лет назад. Теоретическая поправка позволила предсказать режим автоосциллятора, который в течение более чем полувека считался невозможным в рамках старой теории. Меня занимает простой вопрос: есть ли в истории науки аналогичный случай, когда предсказание возможности явления, более чем полвека считавшегося невозможным, оставалось длительное время без экспериментальной проверки? Не кажется Вам эта головоломка гораздо ближе к основам науки, чем квантование времени? Обе мои статьи на эту тему, 2007 и 1997, можно скопировать на моей странице. Думаю, при Вашем уровне владения физикой у Вас не может быть сложностей с этими статьями, но на любые вопросы отвечу без задержки. Можно сразу на [email protected]
Спасибо за электронный адрес, Василий. Единственная проблема у меня - это со временем (моим личным). Его не хватает. Но рано или поздно я обязательно займусь сравнением Вашей работы с моей. Парадокс заключается в том, что закончив аспирантуру на кафедре Боголюбова, я только пару лет тому назад узнал о его деятельности в нелинейной науке, когда столкнулся с конкретной нелинейной проблемой в своей прикладной области. Математики Беларуси ни чем кроме некоторых ссылок мне помочь не смогли, а с помощью метода Боголюбова мне удалось решить эту задачу.
То, что в классической механике электрон падает на атом, - это сказки для детей. Ответ на мой вопрос к научной общественности о том, кто решал эту задачу с учетом действия поля на себя, Вы видели в этой дискуссии. Никто. Ближе всех к этой проблеме подошел Гутцвиллер, но и он не включал в свое рассмотрение излучение.
Мой электронный адрес: [email protected]
Надеюсь на плодотворное сотрудничество с Вами.
С уважением,
E.F.
``Who has proved that limiting cycles are not possible without dissipation? And what does it mean dissipation for arbitrary differential equation?''
The two questions have related answers: I say an equation is non-dissipative if it can be derived from a Hamiltonian, and if the motion in the phase space of this Hamiltonian remains bounded. This means, of course, both with respect to position and momenta.
Who has proved that there are no limit cycles in such a case? It is a trivial consequence of Liouville's theorem in mechanics, which states that if you start with a set of initial conditions which fill out a volume V0, then for later times, the propagated set will also fill out a volume V0. Here, of course, the volume is simply the element of volume
dp_1 dp_2...dp_N . dq_1 dq_2 .... dq_N
where the phase space has dimension 2N. This excludes limit cycles, since, if you start out with a small but non-zero volume of initial conditions near the supposed limit cycle, these initial conditions should all tend to the periodic orbit. But since the orbit is bounded by hypothesis, this means that he volume tends to zero, which contradicts Liouville's theorem.
You might, of course, disagree with my definition of non-dissipative. However, since you seem to be interested in quantum mechanics, it would certainly seem relevant to use concepts that apply to Hamiltonian mechanics.
No, Francois, I don't limit myself by Hamiltonian mechanics. Nevertheless, I accept your explanation in the frames of Hamiltonian mechanics. Thank You.
Leon Brillouin deals with the theorem of Liouville and shows that the statement that the area is conserved with time only is possible in the case of a constant restoring force with time - SHM. With inherent non-linearity in the restoring force, the Liouville shape and area is not conserved and thus the response of the oscillator in both phase and amplitude to perturbation is dependent on the time of application in the phase space. And if we assume an error in determining the frequency of the oscillator and measure the frequency over many cycles, again the Liouville area is not conserved.Thus a Hamitonian system can only be considered conservative under strict conditions - perfect linearity and complete isolation. As a conservative oscillator has no means to dispose of changes in its energy, other than the imposition of an exactly contrary force, it must take up a closed integral curve exactly representing that energy condition. after perturbation - it must follow the noise. And with non-linearity in either restoring force or mass, the degree of perturbation increases with time and with it the Liouville area. Brillouin calls the harmonic oscillator - 'an anomaly' and in physical terms, the assumption of a steady state is extremely doubtful.Limit cycles only apply to systems where both energy input and output are controlled by a non-linear relationship between the variables. However, such a condition is conceivable within an atom, but you need an energy source with which energy can be taken and returned - the weight driving the clock. The evidence of effects such as quantum entanglement suggest that such a source exists but we are a long way from understanding it.But when the solution is found it will assuredly be a non-linear one.
Mervyn: I like your conclusion "you need an energy source with which energy can be taken and returned ". This is why by mentioning Poincare limiting cycles I wanted to attract the attention of the participants to the problem of Uppsalator, a real oscillator. demonstrating limit cycles in a dissipative system and therefore having a direct relation to measuring and properties of time. My question would be: is it possible to understand, why people prefer to discuss artificially constructed problems and neglect real ones? This is a frequent case in politics, but in science it produces an extremely strange impression.
@Mervyn: your claim that the Liouville theorem holds only for the harmonic oscillator is simply incorrect For a proof that the volume is conserved (the shape is indeed not conserved) see any textbook on classical mechanics, say Goldstein, or Arnold. In fact, the theorem even holds iin the case fo time-dependent Hamiltonians, though that was not essential to the point I was making.
Do you have the reference to Leon Brillouin?
Francois,
I took down my copy of Classical Mechanics, Goldstein and compared the conclusion from that excellent book and those of Brillouin, ( 'Scientific Uncertainty and Information' , Academic Press 1964, Chapter VIII) and I have to accept that Brillouin's conclusions are correct, The classical view of Liouville can only apply to a completely isolated system, devoid of any other interaction. Brillouin gives examples of both linear and non-linear oscillating systems and shows how the Liouville area is transformed and the volume sometimes conserved. Only in the case of the conservative linear oscillator is the statement that the error in the initial conditions can be recovered and as required dD/dt = 0. Thus the conservative oscillating system and its steady state is a mathematical fantasy which collapses immediately we see perturbation. Brillouin goes on to show how a linear oscillator, subject to internal fluctuation, steadily accumulates an error in phase with time - the Liouville area is not conserved, the experience of every clockmaker and oscillator designer for the last 350 years. Only in very specific circumstances of fluctuation power spectral density can we minimise the accumulated error with time.
It is possible to conceive of an oscillator, non-linear, that to the casual observer, appears to comply with Liouvilles theorem, but is dissipative and exhibits Poincare recurrence very closely. This is the Harrison clock tested at the RGO Greenwich which has a seconds pendulum swinging a large arc at atmosheric pressure and had an accumulated error at the end of 440 days of less than 1 second.
In one sense it answers Eugene's excellent question - it is possible for a non-linear oscillator to exhibit limit cycle conditions and Poincare recurrence if it is dissipative - in the case of the Harrison regulator, the weight driving the clock. But the weight can only supply the necessary force and therefore energy because it is supended in a gravitational field generated by the mass of the earth. As the weight falls, how much difference would you see in the overall gravitational potential or following Newton's third law, in the position of the Earth? I could easily be forgiven, finding an immearsurable change, for concluding that a Harrison clock is completely conservative....
We are lacking Paul Ehrenfest, dear colleagues, to solve this problem.
Eugene,
Following Born (Atomic Physics, pp 108 - 112) Ehrenfest's qualification of adiabatic invariance for the harmonic oscillator as an explanation of quantized states, at least following Born's mathematics, is no proof at all and indeed rules out the possibility of a linear oscillator as the basis of an explanation. As is well understood in non-linear oscillator theory, an oscillator may jump to a new frequency within half a cycle, as long as the energy before and after the jump is conserved. This is the total energy in the system, not just the energy in an assumed resonator. Thus Born's statement using the mathematics of the linear oscillator contains no mechanism to explain quantum jumps or to justify them. If we moved to a Mathieu's equation where the mass or stiffness factor is modulated in a periodic manner, either harmonic or subharmonic, then we have a mechanism for a change both in frequency and in the resultant energy - many oscillators have been built on this principle. So did Ehrenfest originally say something quite different to that expressed by Born? The implication from Born is that anything less than the energy difference between states cannot initiate a 'jump' - as is observed - both the existance of assumed limit cycles - tightly controlled energy and frequency and jumps between states can only really be explained by a non-linear oscillator, as required by your question.
Totally agree, Mervyn. I see, that You have thought about this much. Now , I am reading last Brillouin's books.
Regards,
Eugene.
Eugene,
Thankyou for your kind support - my interest in non-linear oscillator theory dates back to the late 1970s when I was puzzling over John Harrison's stated ideas and could find no support for them in linear theory. I then found a copy of Andonov and Chaikin's 'Theory of Oscillations' ( 1949) and Rayleigh's 'Theory of Sound' on a rubbish dump where I worked! As Goethe remarked, 'Truth may be found in the strangest places, at the bottom of a well or in a drawer in an old oak dresser,' Or, in this case on a rubbish dump. It is an established myth that the 1927 Solvay conference generated a concensus. The majority of the participants went away deeply concerned and only published their real concerns much later. Both Brillouin and de Broglie were very sceptical of the view that a measure of finality had been reached. However, Born pushed through the 'wave function plus statistics' model which is now known as the Copengagen concensus. Brillouin, because of his background in information theory was a firm believer in the statistical foundation but not the Hamilton-Jacobi model as a final solution. De Broglie published a very interesting book on non-linear wave mechanics - there are now a number of groups actively investigating that possibility.
However, my feeling, based on the discoveries made of the ideas of John Harrison, who definitely developed a non-linear model of an oscillator in the 18th C, is that any model based on a linear oscillator lacks physical rigor - because it is impossible to prove the dynamic stability of a linear oscillator - the analyst can only assume stability by saying that energy conditions are 'bounded' without a means to rigorously prove such a contention.
Of course, linear oscillator is non-existing idealization, Mervyn.
I have graduated Bogolubov's chair in Moscow state university, but began to work with non-linearities only three years ago because of my problems in applied science for what I am payed. Now, last russian eddition (1981) of Андронов, Витт and Хайкин (it is right russian spelling) is my table book. I have faild to receive the works of Frederix about non-linear problems. If You can , please, help me. He was in Geotingen simultaneously with von Neuman, but less succeeded, though, he maid works, parallel Neuman and was better educated in physics, than Neuman.
Хайкин was professor in Moscow state university when I was studing there. He has many textbooks in mechanics.
Bитт was murdered in 1930-th.
Excuse for my misspelling, Mervyn. I mean:
K.O. Friedrichs, On nonlinear Vibrations of Third Order, Studies in Nonlinear Vibration Theory, Institute for Mathematics and Mechanics; New York University, 1946.
I can not find this work. If You can, please, help me.
Thank You in advance,
Eugene.
Eugene F Kislyakov wrote “Now, last russian eddition (1981) of Андронов, Витт and Хайкин (it is right russian spelling) is my table book.”
This is a classical handbook on auto oscillations. Therefore my question: Does this mean that you are generally interested in nonlinear oscillators ? If yes, why are you then indifferent to the prediction of oscillations in Uppsalator, which is a typical nonlinear autooscillator? Under condition of constant current, the oscillations in this system were known starting from the end of 40th years of previous century, but at constant voltage they were considered as impossible from the very beginning. A more detaled physical analysis has shown that it should oscillate at constant voltage too. In this way “a prediction of an impossible phenomenon” has appeared. This is sufficient reason to get interested and verify the prediction experimentally. Do you, as a high level specialist in oscillators, see any alternative to experimental verification of the prediction?
I am interested in all this, Vassili, but I am simply lacking time. Excuse.
One of the analagous to your cases was described by Andronow. In the begining of his work with auto oscillations he had the case, where he could prove, that there is no Poincare limit cycle in the system, but generator was working. Conversations with Mandelstamm helped him to understand situation. Now it is the one of the few classical solved problems in auto oscillations. The point was, that the solution was not continuous.
Excuse once more, may be I'll have time for your Uppsalator after some weeks.
Regards,
Eugene.
Dear Eugen. My situation is completely inverted with respect to the case of Andronov, because he had to adjust the theory to already existing experimental data, whereas I have predicted the autooscillations at constant voltage, in spite of the fact, that the existing at that time equations showed in one step the impossibility of this phenomenon.
The similarity exists, but, in a sense, Andronov jumped down from the mountain, whereas I have climbed to its top over a vertical wall. Important is, however, the difference between us: in his case the criterion of truth, experiment did already exist, and in my case it is difficult to convince the scientists in principal importance of such a criterion.
As to UPPSALATOR papers, both texts can be downloaded from my page here in RG: "Uppsalator's acceleration", Electrophoresis 2007, and "Teorell's membrane system oscillates at constant voltage", Bioelectrochemistry and bioenergetics, 1997.
There are many such examples in Nature, Vassili. I am, personally, interested how human's heart works. Where is it's generator?
High Eugene! You say "There are many such examples in Nature".
Please tell me only one example out of many similar situations known to you. The similarity should be based on at least two conditions :
1) Theory predicts possibility of a real physical phenomenon, which was considered by specialists in different countries as impossible for more than half a century.
2) Nevertheless, no experimental verification has been made during at least two decades, as in my case
(my first paper has appeared in 1997). If you find such an example, I shall explain you with my pleasure the physico-chemical principles of the autooscillator which controls the frequency and amplitude of heart beating.
Several similar problems can be found in our book published in Series Nonequilibrium Problems in the Physical Sciences & Biology: V.S.Markin, V.F.Pastushenko, Y.A.Chizmadzhev, Theory of Excitable Media, 316 pp, Wiley 1987, the year of defense of my doctoral thesis in Moscow State University.
I believe this kind of science should be now widely progressed because of its high practical importance for the humanity.
Good question, Eugene! It has no straightforward answer!
I have defended my cand.chem.sci. thesis in 1969, Ac.Sci. USSR, worked in the Institute of Electrochemistry of Ac. Sci. from 1966 until 1991, defended dr.phys-math.sci. thesis in Biophysics, 1987, in MSU, biological faculty, and in 1991-2008 worked in teaching (German, English) and research, Inst. Biophysics, J.Kepler University of Linz, Austria, where my skills were estimated as Dr. tech.
Now I live about 40 km from J.Kepler University, and my main scientific concern is the wish to finish the unbelievable adventures of the Uppsalator, which goal can be achieved only by a kind of "experimentum crucis"
Vassili,
I have read your paper about Uppsalator. I think, that numerically nothing can be proved because as usual results are precision dependent. Of course, last word is always experiment and numerical calculations are useful hint.
Standard procedure in Poincare and Bogolubov analysis of nonlinear problem is to reduce it to the system of first order differential equations. It is always possible for any differential (and I think integro-differential) equations. Then some qualitative results can be proved by topological methods. Mathematically it is not simple problem. Many mathematicians are engaged in it since Lyapunov.
Good luck.
Thanks Eugene! Apparently you have read the first paper 1997, otherwise you would see that in my case a high-precision (rel err 1e-6) solution can be achieved by solving a system of 1000 ODE, 8 years ago with CPU Intel 3 within several tens of seconds, and now probably within seconds or faster. This high precision solution is exactly the basis to announce the prediction of o regime which has been considered as impossible within the frames of the old theory.
I still remain awaiting from you any single example of a similar situation: for a long time remaining considered as impossible, a phenomenon is predicted due to a new theory as well as precise and complicated calculations, but the experimental science remains calm and ignorant...
Best wishes!
I would like to point out the following book, on the link, who provides some insight within this theme
My sincere apologies, Î happened not to post it!
The book is "Basics and Highlights in Fundamental Physics", by Antonino Zichichi.
Page 408 is specially relevant in providing some insight into this problem.
https://books.google.pt/books?id=uiRYU19QhMQC&lpg=PA408&ots=tyGoMJCo2w&dq=poincare%20limiting%20cycles%20quantum%20states&hl=pt-PT&pg=PA408#v=onepage&q=poincare%20limiting%20cycles%20quantum%20states&f=false
Alcides Simão : "It focuses on the basic unity of fundamental physics at both the theoretical and the experimental level".
Sounds very reasonable. If a science produces many different theories, but does not care about any experimental verification, it is then not really a science. Almost 20 years ago the existence of a new type limit cycle was predicted in Teorell's electroosmotical system, but any experimental verification is still missing. I am interested whether any of participants of the 37th Course of the International School of Subnuclear Physics in Erice might be really interested in such a situation.
Dear Vasili,
Is this the theory you were mentioning?
DOI: 10.1002/bbpc.19950990211
Alcides,
send me your e-mail and I'll send You requested by You my papers.
Sorry, Eugene, my auto correct decided to change 'Spasibo' in the mail title to other word...
Alcides Simão
Is this the theory you were mentioning?
DOI: 10.1002/bbpc.19950990211
Dear Alkides,
This was a conference report. Main papers are
Article: Uppsalator's acceleration
Vassili Ph Pastushenko ·
[Show abstract]
Electrophoresis 02/2007; 28(4):683-90. DOI:10.1002/elps.200600214 · 3.
and
Teorell's membrane system oscillates at constant voltage
V.Ph. Pastushenko ·
[Show abstract]
Bioelectrochemistry and Bioenergetics 06/1997; 43(1):143-150. DOI:10.1016/S0302-4598(96)05168-9
If you google "UPPSALATOR", you can additionally find and download my synonymous Matlab program, which calculates and illustrates the Uppsalator's behavior for any values of concentrations and voltage. Only measurements are missing because I am a theoretician.
Regards
Thanks, Vassili, for sharing that info!
UPPSALATOR is quite a nod to Uppsala, indeed! Also, I've seen that ethymology seems to be a hobby of yours, and indeed, you are correct - Alkides is indeed the greek root of my name, or even Heracles/Herakles. When the Romans adopted the greek gods, they translated Herakles to Hercules.
I appreciate this kind of intelligent sense of humor - one can learn from it, while being amused.
Dear Alkides! I should do my best in order to correspond to your appreciation of my level in ethymology. In fact I was too slow in this case, because Brusselator developed in Brussels and Oregonator in Oregon were known much earlier. We in the Institute of Electrochemistry Ac Sci USSR in Moscow started to work theoretically with Teorell's system in late sixtieth. However, at that time it seemed to be little important, whether the dynamical equations are correct or not.
Already in 1964 Kobatake and Fujita discussed disagreement of the dynamical equations with fundamental Smoluchowski theory. I was nevertheless little interested to look for correct description, because at those times the computers were still missing in our work. Only in Linz, Austria in early 90-th I became really interested in more correct dynamics, because I had the Idea to propagate the new measure of nonequilibrium towards Brusselator (done successfully) and similar systems. To this end I needed a more detailed physical description instead of the available phenomenology.
I was lucky to write down more correct equations. However, calculations in paper 1997 were still approximate, because of numerical complexities: nonlinear integral boundary value problem with partial derivatives. No appropriate software in Matlab was available at that time. Only 10 years later I have managed these difficulties in paper 2007.
The main result is: correction of equations has enabled the regime of oscilations at constant voltage. The complicated and slow developments of the progress with Uppsalator, with exciting results at the end, urgently need some attention, in order to put an experimental point in decades of theoretical research. I am convinced, it is not necessary to be as powerful in science as Heracles (or Hercules) in ancient Greece, in order to check out the theory experimentally. The only heroic effort needed here is just to have a true love to truth in science.
Please, specify, Alcides, which paper in your link do You mean explicitly?
Dear Vasili,
I mentioned previously that in the book 'Basics and Highlights in Fundamental Physics: Proceedings of the International School of Subnuclear Physics', there was an interesting conclusion in page 408, which stated the following:
"'In the above models, what we call 'quantum states', coincides with Poincaré limit cycles of the universe. We repeat, just because our model universes are so small, we were able to identify these. When we glue tiny universes together to obtain larger and hence more interesting models, we get much longer Poincaré cycles, but also much more of them. Eventually, in practise, sooner or later, one has to abandon the hope of describing complete Poincaré cycles, and replace them by the more practical definition of equivalence classes. At that point, when one combines mututally weakly interacting universes, the effective quantum states are just multiplied into product Hilbert spaces''
Afterwards, there was some mention to Teorell's Electroosmotic theory - which was completely unknown to me - and after some googling, I found the following article: "Electroosmotic and Hydrodynamic Oscillations in the Teorell Membrane System", DOI: 10.1002/bbpc.19950990211 , at which point I asked if Vasilli was indeed refering to this particular theory.
I hope I didn't have misunderstood your question!
Sorry, Eugene, I didn't meant t'Hooft's paper, of which I am even unaware of its existence!
Dear Alkides,
I am sorry, at first glance at your reference 1995 I thought it were my abstract for some conference, and I did not even try to find the authors name. I was in contact with Dr Herbert R. L. Drouin before 1997, and I have invited him to become a co author in that paper. I knew that he had developed earlier a detailed realization of Teorell's electroosmotical cell. For this reason I expected he might be interested in cooperation, because next step should have been experimental one.
However, he was sure that my analysis is not correct, although no reason was declared. Well, with regrets I had to go on alone.
To be honest, I was myself not completely sure that the prediction 1997 was correct, because for numerical illustrations the approximate relaxation method of Teorell was used. Only the paper in 2007, where the calculations were done strictly, has eliminated my own doubts, whatever small they were. I did not try to address Dr Herbert R. L. Drouin any more: once rejected, it is not pleasant to be judged as impolite. However, if he or his coworkers are able to do the measurements, it could be a solution to the problem of Uppsalator.
Many thanks for taking part,
Vassili
Spasibo, Vasilli, for clarifying this point. I think it is rather unethical to dismiss someone's work without justification. It reminds me of the stories between Bartlett and Bauer over electron diffraction... I prefer a weak excuse to none!
Dear Eugene, the paper I refered before was "Electroosmotic and Hydrodynamic Oscillations in the Teorell Membrane System", DOI: 10.1002/bbpc.19950990211
Alcides Simão
it is rather unethical
Dear Alkides,
I think this is first of all the problem of religious thinking which is usual not only for some people from very hot countries, but for many scientists too. Dr Herbert R. L. Drouin is not any theoretician, otherwise he would be positively impressed by my results. Therefore most probable is that he believed not me, an unknown to him person, but some other person. However, I do not have any idea, who might pretend to know my potentials and simultaneously understand anything in the theory of Teorell's sysnem. I mean this kind of situation is frequently observed in political events, when the first stupid assumptions are immediately believed as a final truth which does not need any proof. We do not need here to go deeper into this stuff. Simply, science is not free of political aspects, and in this sense is not 100% pure science.
Just to add a point: My country (Portugal) is, since its inception, a member of NATO, and of course, it was on the 'other side' of the Iron Curtain (specially since it had a right-wing dictatorship). But despite this, there was an incredible respect for soviet science - so much that my University Chemistry Library has an rather extensive collection of Soviet Journals and their translations, some of them complete! I'm always amazed - not to say incredibly satisfied - to see that, despite political differences, someone stood up and made the difference - Science is above politics - it is indeed Humanity's greatest triumph! And it must be preserved as such - no matter what!
One says "good intentions lead to hell". Science can not decide. It can only advice something. Decisions are made by politicians, who are in average less clever and knowlegeable than scientists. Such a situation brakes the development and creates sometimes difficulties. One of examjples is the calm neglection of the Uppsalator story, although this story has very little analogies, if any at all.
Back to the main topic, I have a question: What would be the main consequences if Poincaré limiting cycles were indeed quantum states?
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