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Natural Science Forum / Physics / Research / October 2007The least action principle as a consequence of Maxent
I have discovered a simple and amazing result, a surprising consequence of the maximum entropy principle, and would like to know if it is true or not. Here it is: Mathematically, from the set of constraints (p_i reads p with indice i, sums are over all values of i): I = sum (-p_i ln p_i) maximum x = sum (p_i * x_i) t = sum (p_i * t_i) one can deduce, using Lagrange multipliers, that: L = dI/dt satisfies d°L/d°x = (d/dt)d°L/d°(x_dot) where d° means partial differential and x_dot==dx/dt. L has the property of the Lagrangian in physics. Up to a multiplicative constant of dimension Action, L is the Lagrangian, I the action, the Lagrange multiplier of the (generalised) coordinates are the (generalised) momenta, and the Lagrange multiplier of the time is the opposite of the Hamiltonian. All Lagrangian mechanics can be deduced from this set of constraints. A detailed demonstration can be found at http://hal.archives-ouvertes.fr/hal-00123252/en/ It means that all lagrangian mechanics can be deduced from: 1/ the hypothesis (more natural, I think, that the least action principle) that all we know is an average path in spacetime coordinates. 2/ the use of Maxent, which is justified by the repeatability of the experiment. Could anyone confirm (or infirm) this fact? Jean-Bernard Brissaud wrote: > I have discovered a simple and amazing result, a surprising > consequence of the maximum entropy principle, and would like to know [quoted text clipped - 15 lines] > is the opposite of the Hamiltonian. All Lagrangian mechanics can be > deduced from this set of constraints. A nice observation, which looks correct, though presented somewhat sloppy. You should keep lambda since the multipliers will also need a lambda-dependence, and you must integrate over lambda. Also, you need to take into account boundary conditions in the integral to get the correct form of the stationary action principle. (You don't prove minimality of the action, although perhaps you could with more refined arguments or assumptions. Note that in some applications, the action is maximized rather than minimized!) > A detailed demonstration can be found at http://hal.archives-ouvertes.fr/hal-00123252/en/ and arXiv:physics/0701127. What you call Cauchy-Riemann equations are generally called Euler-Lagrange equations. Also it is strange that you say after (1) that you maximize information and after (13) call your principle the least information principle. You'd check your productions for consistency internally and with the literature before you make them public! Rather than introducing the constant K after the derivation of the equations, you'd put it into the definition of I. One can give the entropy arbitrary units without changing anything; this corresponds choosing an arbitrary basis for the logarithm. (For example, in computer science, one takes the basis 2 instead of e.) This also makes your remark on K in Section 3 obsolete. (Anyway, it should have said raises instead of rests.) > It means that all lagrangian mechanics can be deduced from: > 1/ the hypothesis (more natural, I think, that the least action > principle) that all we know is an average path in spacetime > coordinates. > 2/ the use of Maxent, which is justified by the repeatability of the > experiment. This is just an opinion, not a fact. Rewriting existing physics in another form is relevant only if it produces visible results, such as shorter derivations of consequences, better numerical methods, new applications, etc.. What you found is a connection between the two settings, not a more natural foundation. Regarding the status of the maximum entropy principle as a fundamental principle of Nature, you may wish to read the strong counterarguments given in Appendix B of my paper A. Neumaier On the foundations of thermodynamics Manuscript (2007) arXiv:0705.3790 You might wish to relate your results to the laws of thermodynamics as presented there. Your basic equation (2) is just an expression of the first law of thermodynamics as derived in Theorem 10.2 of my paper. But I derive the extremal principle and the first law from assumed Gibbs states, instead of basing it on the questionable max entropy principle. (If you assume the relation in your equation before (20) and write S_i for -log p_i, you are in the setting of my Definition 4.2 for a Gibbs state.) This will also give the right framework for a quantum version of your result. There has been previous work relating mechanics and max entropy. Your work reminds me of BR Frieden Physics from Fisher Information: A Unification Cambridge Univ. Press 1998, although I don't remember the details. (A nice observation without further consequences, as in your case, but then inflated into a series of papers and a book.) A review of the book is in Amer J Physics 68 (2000), 1064-1065. There is also a second edition, now called (more grandiose) 'Science from Fisher Information' which I haven't read. Arnold Neumaier > I have discovered a simple and amazing result, a surprising > consequence of the maximum entropy principle, and would like to know [quoted text clipped - 26 lines] > > Could anyone confirm (or infirm) this fact? Oscillating chemical reactions (e.g., Belousov-Zhabotinsky) do not smoothly go asymptotic to maximum entropy over time. Energetic systems far from equilibrium and with positive feedback spotaneously walk all over the map with strange attractors and repeated catastrophic transitions - Ilya Prigogine's nonequilibrium thermodynamics. thermodynamics + Bekenstein bound = metric gravitation Is my preceding paragraph then pertinent to your assertions about maximizing entropy to trace optimally efficient paths? Entropy is a weak arrow of time. The Second Law is strictly probabilistic. Angular momentum is an absolute arrow of time, e.g., Feynman's sprinkler and fluid diodes with no internal parts. (Pump water orthogonally into the center of a motion picture film can and tangentially out its edge and it flows freely. Pump water tangentially into the film can's edge with a center exit port and... no flow. It is not symmetric to time inversion.)  SignatureUncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/lajos.htm#a2 > (Pump water orthogonally into the center of a motion picture film can > and tangentially out its edge and it flows freely. Pump water > tangentially into the film can's edge with a center exit port and... > no flow. It is not symmetric to time inversion.) Based on your picture above, you are wrong if the can spins. Starting from a given external potential one can feed water orthogonally into the center of a motion picture film can. If the can is spinning at the right speed water will go out not tangentially, but orthogonal. So you can feed the same spinning can externally and make water come out the other way around. Otherwise we can build a perpetual motion machine simply assuming a spinning can. > > (Pump water orthogonally into the center of a motion picture film can > > and tangentially out its edge and it flows freely. Pump water [quoted text clipped - 15 lines] > Otherwise we can build a perpetual motion machine > simply assuming a spinning can. If the can spins it is not an inertial frame of reference. Spin axis is presumed to be parallel to and concentric with the center feed pipe orthogonal to the flat plane of the can. If water enters/exits edge-orthogonal, why spin the can at all? That isn't the example. If there is tangential *entry* and you spin the can at a *necessary* speed, how do you connect to the tangential port without leaks? Its bore is not orthogonal to the edge. Centripetal force doesn't time reverse. The original analysis holds: entropy is a statistical and weak arrow of time; angular momentum can be a strong absolute arrow of time. After all, there is nothing wrong with the Earth spinning the other way. The film can fluid diode is something different. SignatureUncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/lajos.htm#a2 > I have discovered a simple and amazing result, a surprising > consequence of the maximum entropy principle, and would like to know [quoted text clipped - 26 lines] > > Could anyone confirm (or infirm) this fact? I have had a look at your paper, and have a couple of initial observations: 1. The result does not apppear to apply to ALL Lagrangian mechanics, as the Lagrangian you obtain is always linear in the velocities. Such Lagrangians are rare - the only physical one I can think of is for fields rather than particles - for the Schrodinger equation in fact, which looks something like L(psi, psi*, dpsi/dt, dpsi*/dt) = int dx [ i hbar/m Im{ psi* dpsi/ dt} + V psi psi* ]. So, perhaps the Schrodinger equation can be interpreted as a MaxEnt evolution over an ensemble of field evolutions. 2. I am puzzled by a couple of aspects of your derivation. (a) First, you can only obtain your symmetry conditions \partial alpha_k / \partial A_l = \partial alpha_l / \partial A_k if you assume that (quite reasonably) the average values A_k are all chosen independently of one other. However, you then (identifying your beta with alpha_0 and your t with A_0), assume an explicit dependence of the A_k on A_0. How is this consistent with the derivation of the symmetry conditions? (b) Second, given that the A_k are all logically independent, you must be making some sort of added assumption when you restrict them all depend on A_0 (i.e., t). That is, you appear to be restricting consideration to some subset of possible average values, where this subset is parameterised by A_0=t. This consideration is put in by hand, and I'm not sure what the physical meaning is: I guess you are picking out one of the average values to be a "time" variable, and asserting that all the other values are to be chosen in a way that is consistent with MaxEnt for each possible "time" t. I guess your result shows that this "simultaneous consistency" restriction corresponds to Lagrangian evolution for the remaining A_k, with a Lagrangian linear in dA_k/dt (providing point 2(a) above is not a problem). > I have discovered a simple and amazing result, a surprising > consequence of the maximum entropy principle, and would like to know [quoted text clipped - 26 lines] > > Could anyone confirm (or infirm) this fact? Yes, overall mechanics can be derived. Quantum mechanics (e.g. the Schrödinger equation) can be also derived from a corresponding quantum thermodynamics. This is not new. A more general expression for a Lagrangian was proven by Joel Keizer time ago L(q, {DOT q}) = {{DOT q} - Hq}^T {1/GAMMA} {{DOT q} - Hq} T, H, and GAMMA defined in [1]. With asociated Lagrange equations d/ds { {PARTIAL L} / {PARTIAL {DOT q}} } = { {PARTIAL L} / {PARTIAL {DOT q}} } determining the extremal, and only the extremal, path. Those equations are useful for drawing an analogy with mechanics but of little applicability in real life laboratories because two main reasons: 1) In laboratory one prefer Hamiltonian mechanics (first order equations) because of mathematical and computational difficulties with second order Lagrangian dynamics. 2) Above Lagrangian equations determine the average evolution and not the observed deviations. For instance, you cannot study Brownian motion or biochirality using Lagrangian equations because the observed path in laboratory is *not* the extremal one. Some remarks about your paper: i) {BLOCKQUOTE Thermodynamics, on the other hand, can be entirely deduced from the maximum entropy principle (Maxent), and introduces a so-called "arrow of time". } MaxEnt methods are not a foundation basis for the whole of thermodynamics because extremum principles cannot model the fluctuation spectra in the linear regime, because cannot predict the average path in the far from equilibrium regime, because does not give kinetics equations for the non-conserved variables... Ilya Prigogine (Nobel Prize winner by dissipative structures) showed that extremum principles do not apply to fully developed non-linear regimes. Notice that Prigogine developed the main extremum principle in the linear regime (the theorem cited in the reference 1 you reference in your own paper). The equations of motion for a dissipative structure we use in laboratory do *not* follow from minimizing (or maximizing) some action. You need other tools like bifurcation theory. It is well-known that, near a bifurcation point, entropy is not longer a maximum. The system is unstable at the bifurcation point and MaxEnt hypotesis breaks down. About arrows of time, like you know thermodynamics introduces a fundamental (non-statistical) arrow of time asociated to production of entropy. However, MaxEnt methods cannot explain that fundamental arrow just because is formulated over that assumption. ii) {BLOCKQUOTE Note that in this problem, there are no assumptions about the nature of the (Ak). } No really, you are asumming many things in an implicit way. Between others: a) Ak are differentiable everywhere (you are ruling out bifurcation and singular points in state space). b) I assume that (18) is defined for some t, then you are not considering memory effects, which are considered in other formulations, e.g. Truesdell thermomechanics for materials with memory. c) Does not the constraint SUM p_i = 1 imply you are ruling out open systems? d) Are not you asumming a classical theory of measurement when doing the informational interpretation? Also the information function is not that in quantum thermodynamics e) You state are not assuming the Ak to be extensive quantities and that extensivity is "a thermodynamical concept not necessary for Maxent". However, the whole method is based in an extensive informational function I = - {SUM p_i {LN p_I}} Probably the most known and studied non-extensive function was Tsallis [2] I = {1 - {SUM p_i^q }} / {q - 1} q parameter measures non-extensivity. In the limit q --> 1 Tsallis reduces to the classical Gibsian function used in MaxEnt I = - {SUM p_i {LN p_I}} iii) {BLOCKQUOTE Maxent as a fundamental physical principle certainly has epistemological implications. } As stated it is not a fundamental physical principle and has limited applicability even in the linear regime. In my opinion, the epistemological implications derived from the limitations of variational principles are much more interesting because open a broad scientific vision of Nature and because link with fundamental epistemological questions like cosmological origins, human free-will. A discussion of epistemological implications is brilliantly presented in [3]. [1] J. Chem. Phys. 1975, 63(1), 398. [2] Journal of Physics A: Math. Gen. 1991, 24, L69. [3] Prigogine, Ilya. The End of Certainty. Free Press Ed edition, 1997. On 10 sep, 12:05, "Juan R." <juanrgonzal...@canonicalscience.com> wrote: > > I have discovered a simple and amazing result, a surprising > > consequence of the maximumentropyprinciple, and would like to know [quoted text clipped - 162 lines] > > [3] Prigogine, Ilya. The End of Certainty. Free Press Ed edition, 1997. Thank you all for these pertinents answers. As I understand it, this result is already known (and hence true). My faults concern thermodynamics, not the maths. M. Neumaier, I agree with you when you say that this needs boundary conditions. However, the entropy is maximal according to the probability distribution, and minimal (well, extremal) according to a variation in the path. No contradiction or paradox here. Note that this entropy is a priori different from thermodynamical entropy. About the use of Maxent, the debate is closed for me, your arguments included (yes, we have to choose the correct variables and constraints, the variables having an uniform distribution as previous knowledge). Well, this subject seems almost a religious one, so, tolerance. I found your paper very interesting, but I don't think we really talk about the same subject. Juan R., I'd like to know the generalisation to quantum mechanics. I can't find the references (my institution didn't pay for) and Joel Keiser is unknown to google (scholar or not). Could you please give me a clue. Thanks a lot for your answer, and all the time you spend on these forums. > On 10 sep, 12:05, "Juan R." <juanrgonzal...@canonicalscience.com> > wrote: > Juan R., I'd like to know the generalisation to quantum mechanics. You worked at the classical Lagrangian level of mechanics. However a quantum Lagrangian version is not desired by a number of reasons, including that quantum Lagrangian Path integral methods can give wrong S-matrices for important kind of systems. See Weinberg [6] for detailed discussion about that. A Hamiltonian approach is more interesting for quantum purposes. There exists several quantum approaches by different authors. For instance, in "The Liouville Space Extension of Quantum Mechanics" [1] you can find one sophisticated and rather complete theory. The senior author won a Nobel Prize by extension of thermodynamics, and with his group formulated one of the two main approaches to non-equilibrium statistical mechanics. The equation fundamental in [1] is (taking i = 1) {PARTIAL |RHO_B>> / PARTIAL t} = THETA_B |RHO_B>> where |RHO_B>> represents the quantum state (a generalization of quantum wavefunctions) and THETA_B is the evolutor (generalizates the Hamiltonian operator). The expressions are rather elaborated, e.g. THETA _B = LAMBDA_B {SUM_v SUM_alpha |F_alpha^(v)>> Z_alpha^(v) <<TILDE F_alpha^(v)|} {LAMBDA_B}^-1 Where (see [1] for the details): - LAMBDA_B is a non-unitary (super)operator built over collision and destruction operators. I avoid to write the math here by lack of time (see [1] for details). - Z_alpha^(v) are eigenvalues associated to a novel complex spectral representation - The | >> and << | denote a generalization of Dirac bra and kets. As you know the Schrödinger equation of quantum mechanics is {PARTIAL |PHI> / PARTIAL t} = H |PHI> This equation is recovered like a special case of the above fundamental equation [1]. (See also appendix E: dynamical groups as approximations of dynamical semigroups on [1]). In a recent work I. Prigogine writes: {BLOCKQUOTE We see therefore that kinetic theory and thermodynamics are well- defined extensions of classical or quantum dynamics. } Whereas the derivation of quantum dinamics from thermodynamics [*] is interesting for unification issues, the more interesting physics arises when you consider thermodynamical corrections to Schrödinger. Then, thanks to non-Schrodinger terms, you can built explicit 'thermomechanical' models for the collapse of wavefunctions. (See appendix J: collapse of wave functions on [1]). It is generally thought that collapse is a genuine irreversible process; it is not so well-known that collapse generates entropy in agreement with the second law of thermodynamics. If you are interested, you would contact them [2] for details on recent Brushles-Austin theory. I am working a more general theory inspired in early Keizer theory [4]. The physical ideas behind the derivation of quantum mechanics are rather close to [1] but technical details are different (e.g. here Poincaré resonances play no fundamental role). An important aspect is that theories and equations derived from different groups arise here as a special kind of this general theory [*]. The basic view is as follow (I avoid most of technical details and do not use special notation for vectors, operators, and superoperators because lacking time): The fundamental equation now is {PARTIAL RHO / PARTIAL t} = S + D + f where RHO represents the quantum state. S is streaming term, D introduces dissipation, and f possible random contributions (e.g. due to quantum structure of space). If both f and D terms can be ignored the equation reduces to the standard Liouville & von-Neumann equation {PARTIAL RHO / PARTIAL t} = L RHO where L is the quantum Liouvillian. If the state of the quantum system can be approximated by a pure state RHO = |PHI><PHI| Then the Liouville & von-Neumann reduces to Schrödinger equation {PARTIAL PHI / PARTIAL t} = H PHI Note that the Schrödinger equation of quantum mechanics has been here derived after making three approximations with clear physical meaning: no random, no dissipation, pure states. By relaxing the approximations in different ways then you get different generalizations to Schrödinger quantum mechanics. For instance, if you maintain the random term f you get a so-called Ito-Schrödinger equation {PARTIAL PHI / PARTIAL t} = H PHI + f If you ignore f but approximate D in the Markovian limit (physically means no memory effects), second order in potential (usually called the lambda^2 t aprox.), Abel kernel... then you get a Zubarev NESOM equation {PARTIAL RHO / PARTIAL t} = {L RHO} + EPSILON {RHO - {BAR RHO}} used in MaxEnt-NESOM [5]. Etcetera. > I can't find the references (my institution didn't pay for) and Joel > Keiser is unknown to google (scholar or not). It is Keizer. Their contributions to physics, chemistry, and biology are listed in both Google engines, SciFinder, and mayor standard literature resources. He was a very brilliant scientist (I consider a genious) but unfortunately got cancer and passed away some few years ago. An obituary is here [3]. > Could you please give me > a clue. Thanks a lot for your answer, and all the time you spend on > these forums. Due to flaming in sci.physics.relativity both my email is disabled. Therefore, I am not acessible now. For October you would be able to get my new academic e-mail. Please check my blog http://canonicalscience.blogspot.com/ periodically and contact with me then. I will try to help you if possible. [1] Adv. Chem. Phys. 1997, 99, 1. Petrosky, Tomio; Prigogine, Ilya. [2] http://order.ph.utexas.edu/Prigogine.htm [3] http://hector.ucdavis.edu/pbk/Obituaries/jkeizer.html [4] E.g. search in Scholar by Keizer canonical theory. [5] http://arxiv.org/abs/cond-mat/9909160 [6] The Quantum Theory of Fields (Vol. 1). Cambridge University Press; 2000. Weinberg, Steven. [*] As stated by the author there thermodynamics is equated to certain extension of dynamics. Unfortunately, most of physicists still think thermodynamics applies only to macroscopic systems at equilibrium. Maybe it is better to leave thermodynamics in his usual meaning in practice, introducing the word "thermomechanics" for a thermal generalization of mechanics. With mechanics (classical or quantum) recovered like a special case. Some authors are already using thermomechanics in this sense. What do you think? [**] It seems that a Brushels-Austin like equation {PARTIAL |RHO_B>> / PARTIAL t} = THETA_B |RHO_B>> can be derived like a special case from {PARTIAL RHO / PARTIAL t} = S + D + f But this is still under active research. Juan R. schrieb: [Moderator's note: Quoted text trimmed. -P.H.] > Whereas the derivation of quantum dinamics from thermodynamics [*] is > interesting for unification issues, the more interesting physics [quoted text clipped - 10 lines] > If you are interested, you would contact them [2] for details on > recent Brushles-Austin theory. Brushles Austin has no useful hits in google. It is also not mentioned in > [2] http://order.ph.utexas.edu/Prigogine.htm Please give more details. What is it about? A reference? Arnold Neumaier On Sep 28, 12:49 pm, Arnold Neumaier <Arnold.Neuma...@univie.ac.at> wrote: > Juan R. schrieb: > > If you are interested, you would contact them [2] for details on > > recent Brushles-Austin theory. > > Brushles Austin has no useful hits in google. It is also not > mentioned in Sorry by typo; it is Brussels-Austin. The Brussels-Austin group is the group was leaded by Prigogine. We informally call the theory the "Brussels-Austin theory". Still a Google search will return results. The best way is to follow directly the authors' literature [2]. > Please give more details. What is it about? The idea fundamental is that Poincare resonances split models into two large classes: integrable and non-integrable. See [1, 2, 3] and previous message for details on that follows. Non-integrable systems need of an extension of dynamics and that extension deal with phenomena like irreversibility, dissipation, and so called non-deterministic chaos. Examples of non-integrable class (called LPS [*] in the theory) are thermodynamics systems, measurement apparatus in quantum mechanics, instable particles (fields)... Regarding collapse, this theory *dinamically* differentiates simple quantum systems (following Schrödinger equation) from measurement apparatus (Schrödinger does not apply). The difference finds in the spectra. Spectral decomposition for an usual quantum system (e.g. atom): H = SUM_j |Phi_j> E_j <Phi_j| Spectral decomposition for a measurement apparatus: L = SUM_v SUM_alpha |F_alpha^(v)>> Z_alpha^(v) <<TILDE F_alpha^(v)| Equation of motion for an usual quantum system (e.g. atom): {PARTIAL |PHI> / PARTIAL t} = H |PHI> Equation of motion for a measurement apparatus: {PARTIAL |RHO_B>> / PARTIAL t} = THETA_B |RHO_B>> whith THETA _B = { LAMBDA_B L {LAMBDA_B}^-1 } This dynamical distinction is a fundamental point you do not find in other theories dealing with quantum measurement and collapse. For example, decoherence approaches _a la_ Zurek, or the histories formalism of Gell-Mann and Hartle are based in _ad hoc_ splitting into quantum system and measurement apparatus. Therefore, the development of realistic models may be guided by experience instead pure theory. Penrose also tries to dinamically explain collapse. Penrose speculates that quantum gravity effects in 'large' systems collapse wave functions. He even offers an estimation of size and find a large enough system would be of order of average molecules. This is right since we know that average molecules are not completely quantum [**] but in the border between quantum and classical worlds. Prigogine and me agree in several fundamental points but both disagree on the role of Poincare resonances. I do not think Poincare resonances are fundamental but product in the large limit (LPS). I think a nice research program would be so find the roots for LPS because that would solve some weak points on the Brussels-Austin approach. As stated in a previous message, it seems that a Brussels-Austin like equation {PARTIAL RHO_B / PARTIAL t} = THETA_B RHO_B can be derived like a special case from a Keizer-like one {PARTIAL RHO / PARTIAL t} = S + D + f One can formally obtain the equation in the large limit (LPS), with a Theta superoperator verifying (6.7) in [1]: THETA _B = { LAMBDA_B L {LAMBDA_B}^-1 } --> L_0 Therefore there exists a clear link between both theories. But this is still under active research. > A reference? In [2] you can see list of recent published references including best- seller books like _The End of Certainty, Time, Chaos and the New Laws of Nature_ for general audiences. In that book Prigogine presents the novel theory in a broad context (cosmology, evolutionary biology, nonlinear chemistry, geometry and general relativity) and discusses philosophical implications. A concise self-contained presentation of the extended quantum formulation including appendices to specific topics (wave collapse, Friedrichs model, instable particles, quantum Lorentz gas, anharmonic lattices...) and further references on detailed models is [1]. Extension of classical mechanics is presented on [3]. > Arnold Neumaier Due to flaming in sci.physics.relativity my email is disabled. Therefore, I am not acessible now. [*] Large Poincare Systems. [**] Its electronic structure is, of course, purely quantum but the nuclear framework is not, any quantum chemist know. [1] Adv. Chem. Phys. 1997, 99, 1. Petrosky, Tomio; Prigogine, Ilya. [2] http://order.ph.utexas.edu/Prigogine.htm [3] Cha. Sol. & Fract. 1996, 7(4), 441-497. Petrosky, Tomio; Prigogine, Ilya. [was: The least action principle as a consequence of Maxent] Juan R. schrieb: > On Sep 28, 12:49 pm, Arnold Neumaier <Arnold.Neuma...@univie.ac.at> > wrote: [quoted text clipped - 3 lines] > informally call the theory the "Brussels-Austin theory". Still a > Google search will return results. One of the Google results (which compared to your references has the advantage of being online) is R.C. Bishop Brussels-Austin Nonequilibrium Statistical Mechanics: Large Poincare Systems and Rigged Hilbert Space, http://www.igpp.de/english/tda/pdf/BrusselsAustin.pdf Does it contain the essence? > The best way is to follow directly > the authors' literature [2]. This is a very long list. Surely not all of them are equally relevant. Which references there gives the key to this theory? >> Please give more details. What is it about? > [quoted text clipped - 34 lines] > > THETA _B = { LAMBDA_B L {LAMBDA_B}^-1 } All this looks to me to be the same, apart from a change in notation. > This dynamical distinction is a fundamental point you do not find in > other theories dealing with quantum measurement and collapse. >> A reference? > > In [2] you can see list of recent published references including best- > seller books like _The End of Certainty, Time, Chaos and the New Laws > of Nature_ for general audiences. I am interested in a (recent) survey for experts, not in a laymen's book. Theory is easier to understand if not enclouded in philosophy for everyone... Arnold Neumaier On Oct 5, 12:41 pm, Arnold Neumaier <Arnold.Neuma...@univie.ac.at> wrote: > [was: The least action principle as a consequence of Maxent] > [quoted text clipped - 18 lines] > > Does it contain the essence? this is a very nice summary often you will see the "brussels" school or "brussels-austin" approach discussed in fragmentary expositions tucked in proceedings on foundations of physics or on chaos or dissipative structures they always seemed to have a hard time fitting in but this paper doesn't go into some of the important issues regarding the operationalist foundations these approaches require the issues with time reversibility seem to rely on dynamical extensions that occur in an operationalist language and the rigged hilbert space approach it is interesting that the "brussels school" nowadays is still very operationalist as exemplified by the work of bob coecke and others but that the newer formalism interpreted as a galois adjunction from a hilbert space category to a topos interpretation is a much more coherent approach which many important computational descriptions of time irreversibility -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar On Oct 5, 9:41 pm, Arnold Neumaier <Arnold.Neuma...@univie.ac.at> wrote: > [was: The least action principle as a consequence of Maxent] > [quoted text clipped - 16 lines] > > Does it contain the essence? It contains the essence of LPS and RHS. And discussed the relatinoship with dissipation and irreversibility. However, it contains not the essence of the theory. For instance, does not discuss the general equation of motion or how one derive Pauli, or generalized Puli equations, does not present the complex spectral decomposition, neither introduce details of application to systems of physical interests: gases, Friedrich model, unstable particles, etc. Whereas, Bishop analizes critical points of the Brussels formalims (a good point) and cites adittional references and other reviews (another good point), He apparently fails to understand the basis of the formalism and motivations. also Bishop fails to discern basic points like intrinsic and extrinsic irreversibility (the kaon example is *extrinsic*); difference between a coarse-grained and a fine-grained approach (contrary to Bishop comments, the coarse-grained approach is in conflict with the second law, precisely this was the historical reason for the search of fine- grained theories avoiding the conflict and letting us the formulation of a consistent non-equilibrium statistical mechanics.), etc. > > The best way is to follow directly > > the authors' literature [2]. > > This is a very long list. Surely not all of them are equally relevant. > Which references there gives the key to this theory? Abstracts usually differentiate reviews from applications to specific problems. In some of the listed works, even one does not need to read abstract. Take for instance the title of paper "Semigroup Representation of the Vlasov Evolution" It is clear this is application of the semigroup formalism of the theory to classical electrodynamics system: the Vlasov equation is a Boltzman-like equation very popular for modelling plasmas. I already cited two comprehensive reviews: one for quantum dynamics and the other for classical dynamics. You would begin from here and then move to other references in basis to your own preferences and research agenda. > > Spectral decomposition for an usual quantum system (e.g. atom): > [quoted text clipped - 17 lines] > > All this looks to me to be the same, apart from a change in notation. Nothing more far from reality. For instance, |PHI> is a vector on a Hilbert space, whereas |RHO_B>> is a vector in a more general class of space, called the Liouville space. Physically, the vector |RHO_B>> describes both pure and mixed states, whereas its Hilbert counterpart |PHI> only works for pure quantum states. On a simple look to the standard spectral decomposition, one can notice the Hilbert basis symmetry. However, in the novel theory, the basis (now for the Liouville space) contains the vector <<TILDE F_alpha^(v)|. See references cited for definition of the TILDE operation. >From a physical view, the asymmetry between 'bras' and 'kets' on the Liouville space is related to decay times of unstable systems for instance. The description of unstable quantum systems in a Liouville space is more general than using Gamow vectors for instance. > Arnold Neumaier [1] Adv. Chem. Phys. 1997, 99, 1. Petrosky, Tomio; Prigogine, Ilya. |
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