Trajectory branching in Liouville space as the source ofirreversibility
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From: "Juan R." González-Álvarez on 31 May 2010 10:56 TRAJECTORY BRANCHING IN LIOUVILLE SPACE AS THE SOURCE OF IRREVERSIBILITY An introduction to irreversibility was given in the first part: The quest for the ultimate theory of time, an introduction to irreversibility. In this event, the author want to communicate the recent advances done in our comprehension of irreversibility. This information is derived from a research program in progress and must be taken with caution. However, the research results obtained at the Center seem highly reasonable. It is realized by an increasing number of physicists and chemists that the solution of the problem of the "arrow of time" requires an extension of classical and quantum mechanics. The XXIth Solvay Conference on Physics has been devoted to Dynamical Systems and Irreversibility. There has been no unanimity in choosing the way in which the extension of classical or quantum mechanics has to be performed to include the dynamical description of irreversible processes. A selection of main fine-grained approaches is given in a recent monograph: Resonances, instability, and irreversibility [1]. This includes the work of the following authors —ordered by appearance on the content index—: T. Petrosky & I. Prigogine; E. C. G. Sudarshan, Charles B. Chiu & G. Bhamathi; Erkki J. Brändas; E. Eisenberg & L. P. Horwitz; I. Antoniou & Z. Suchanecki; V. V. Kocharovsky, Vl. V. Kocharovsky, & S. Tasaki; Pierre Gaspard; and C. A. Chazidimitriou-Dreismann. In their work Unstable systems in generalized quantum theory, Sudarshan et al. present us a generalization of quantum mechanics, beyond the standard Dirac formulation, aimed to study the decay of unstable particles, radioactive nuclei, and excited atoms, beyond phenomenological treatments as the Breit & Wigner model. Eisenberg & Horwitz review the recently developed quantum theory using the ideas and results of Lax & Phillips. Their new framework for the description of quantum unstable systems is based in a direct integral Hilbert space. Two characteristic features are the generalization of the dimensional time t = x^0 to the evolution parameter \tau —not to be confounded with proper time!— and of the generator of time translations H --> K, where H is the Hamiltonian. This generalized structure strongly resembles that on the theory of Stuëckelberg, Horwitz, & Piron. An analysis of the classical mechanics and electrodynamics of Stuëckelberg, Horwitz, & Piron was given in a previous report CSR:20093v1 —you may find additional information, in a post-publication note by Horwitz, in the matching event—. An analysis of the quantum versions is now being prepared; at the time of writing this, the draft is completed above the 80%. The general problem of irreversibility is discussed in the papers by T. Petrosky & I. Prigogine, and by I. Antoniou & Z. Suchanecki. They belong to the Brussels-Austin School. An analysis of their last work is given by Robert C. Bishop [2]. Internal analysis, by another member of the School, is found in the section Toward a Grand Theory of Irreversibility? in a last book by Radu Balescu [3]. The new work presented here solves many of the objections and corrects some common misunderstandings: for instance, many authors have misinterpreted the Brussels-Austin statements about the collapse of trajectories. Here and thereafter, the discussion is highly technical and focused to experts in this highly-sophisticated subject. Research reports and Perspectives for the general public will be prepared later. LIOUVILLE SPACE BIFURCATION POINTS The existence of bifurcation points —for which σ_E[t_b] · σ_S[t_b] + + g[t_b] = σ_E[t_b] · σ_S[t_b] is fulfilled— is inferred from the topological properties of the fundamental interactions for a N-body system: L'_ES[t_b] = 0. This is studied in the topological structural clusters program and may be understood as a generalization to condensed phase systems of the Stosszahlansatz of the kinetic theory of gases. However, no approximation is being done in the TSC formalism. Branching in finite volumes is an emergent property, because no bifurcation point exists in the Liouville space for one and two-body particle systems. For those simple systems, one recovers the cluster decomposition principle of quantum field theory as a special case of branching valid in the limit of infinite volumes [4]. Its emergency explains why particle physics could not solve the problem of irreversibility. Branching introduces irreversibility because trajectories are broken. This confirms a main result obtained by the Brussels-Austin School, except that the fundamental source for "the collapse of trajectories" are not Poincaré resonances, but bifurcation points in Liouville space. Irreversibility may be intuitively understood as follow —see the figure below—. Before and after a bifurcation point, the propagator is time-reversible and deterministic; the usual trajectory theory applies and the equations are just the Liouville equations 1, 1', and 2. However, at bifurcation points the evolution is not invertible. For instance, if at t = t_b the factorization is σ_E[t_b] σ_S[t_b] = {6·2 --> 12}, its inversion gives the collection of all the possible factorizations {12 --> 6·2}, {12 --> 3·4}, {12 --> 12·1}, {12 --> 2·6}, {12 --> (1/2)·24},..., with the associated probabilities, instead of just {12 --> 6·2} with probability one —indeed, this is the deep reason which one sums or integrates over all the possible final states compatible with a given initial state in the usual master equations—. As a consequence of the no-invertibility at t = tb, the global evolution is neither time-reversible nor deterministic. The whole equation of motion 3 can be obtained from analyzing the overall evolution, including the bifurcation points at t = t_b and t = t'_b. As a help to infer the exact form of the whole propagator, I have used a slight generalization of Balescu diagrams [3]. The first two terms are the free-motion propagator and the open Y-vertex giving a Vlasov evolution. Both terms are reversible and correspond to the mechanical term M in the canonical equation of motion —see the postulate (iii) in canonical theory as a unified Cosmovision—. The terms inside the curly brackets are irreversible and correspond to those terms of the canonical equation of motion denoted with curly brackets too. A fundamental difference with Keizer's original canonical equation is that the Markovian approximation is no longer used, which means that now \Omega = \Omega[t]. {{{{ FIGURE }}}} A FOUNDATION FOR THE BRUSSELS-AUSTIN THEORY Several years ago, in one of our conversations, I asked Prigogine a bunch of questions. One of them was: "Are Poincaré resonances the basis for the Second law?". His response (9 May 2003) was: "The questions that you ask are very difficult" and added "I wish you much success". He passed away nineteen days after. I only can imagine his reaction to those recent research results if he was living today, but I believe that he would be very pleased by them. From the analysis of the overall evolution sketched in the above section, one can obtain the equation for the average correlations in 2 as g = L_V/(D − L_0) σ_E σ_S + O(L_V^2) The Markovian regime is obtained setting D = 0. Multiplying and dividing by i and switching to the Brussels-Austin School (BAS) notation, L^BAS == iL, yields g = L_V^BAS/(i0 − L_0^BAS) σ_E σ_S + O((L_V^BAS)^2) This is essentially equivalent to the BAS result for the class of LPSs. They obtain it from a different way [1]. They start from the spectral decomposition for LPSs, which will include singular denominators associated to a continuous spectrum L_0^BAS --> 0. Prigogine and coworkers regularize them adding a small \pmi\epsilon with \epsilon a positive infinitesimal \epsilon --> 0+. The new denominator (L_0^BAS \pmi\epsilon) can be understood as arising from a complex eigenvalue extension of the usual Liouvillian. There is two possible analytic continuations: the positive and the negative branches. One of them is compatible with the Second law and the other is not. Prigogine and coworkers then choose the branch compatible with observations [1]. Note that this is not a derivation of the Second law from microscopic details but that the law is being used to eliminate the ambiguity associated to the Poincaré resonances of the BAS formalism. This is easy to understand; resonances are essentially an energetic concept and the Second law defines irreversibility in terms of entropy, not of energy. Alternatively, we can obtain the correct branch +iε directly from the condition D > 0 for 3, without any appeal to the Second law. Irreversibility for the finite systems around us is thus rooted in the existence of bifurcation points. In the limit of an infinite system, an alternative description in terms of a complex extended Liouvillian with Im(Z^BAS) =< 0 can be obtained. However, no real system is infinite and the new theory for finite systems is preferred. Other BAS main results can be re-derived here. For instance, the new theory gives generalized Lippmann & Schwinger equations. For a LPS one sets D = 0 and L_0^BAS = 0, and the generalized equations reduce to σ_E · σ_S + g --> σ_E · σ_S + L_V^BAS/(i0 − L_V^BAS) σ_E σ_S This implies that the Hilbert space norm vanishes <<σ_E · σ_S + g | σ_E · σ_S + g>> --> 0 in the limit of infinite system. The BAS takes this result as the confirmation that distributions of this kind do not belong to the Hilbert space [1]. They propose a Rigged Hilbert space as a more adequate algebraic structure. However, the Hilbert space norm is nonzero for the more realistic finite systems studied in the new theory. Another interesting result is the explanation of the factorization σ = σ_1 · σ_2 ··· σ_N for a LPS. This result follows directly from the TSC theory, without any appeal to special cosmological boundaries as in the BAS approach [1]. Furthermore, the cosmological models derived from canonical science are much more general and, as pointed in previous events, explain observations that none other approach does. There exist other advantages over the BAS formalism but are not directly associated to irreversibility and will be not discussed here. In the third part, Toward a grand theory of irreversibility, I will sketch how this new theory reproduces the known pragmatic and semi-phenomenological quantum approaches developed in the last decades by at least six communities: NMR chemists, quantum optics, condensed matter physicists, mathematical physicists, astrophysicists, and condensed phase chemical physicists [5]. And will sketch the advantages and generalizations over the current approaches. REFERENCES AND NOTES [1] Resonances, instability, and irreversibility 1997: Adv. Chem. Phys. 99. Several authors. [2] Brussels-Austin Nonequilibrium Statistical Mechanics in the Later Years: Large Poincaré Systems and Rigged Hilbert Space 2003: arXiv:physics/0304020v3. Bishop, Robert C. [3] Statistical Dynamics, Matter out of Equilibrium 1997: Imperial College Press, London. Balescu, Radu. [4] The Cluster Decomposition Principle In The Quantum Theory of Fields, Volume 1 Foundations 1996: Cambridge University Press, Cambridge; Reprinted with corrections. Weinberg, Steven. [5] Phase space approach to theories of quantum dissipation 1997: J. Chem. Phys. 107(13), 5236—5253. Kohen, D.; Marston, C. C.; Tannor, D. J. ############## NEWS AND BLOG: ############## http://www.canonicalscience.org/publications/canonicalsciencetoday/canonicalsciencetoday.html http://www.canonicalscience.org/publications/canonicalsciencetoday/20100531.html -- http://www.canonicalscience.org/ BLOG: http://www.canonicalscience.org/publications/canonicalsciencetoday/canonicalsciencetoday.html
From: "Juan R." González-Álvarez on 1 Jun 2010 12:40
Surfer wrote on Tue, 01 Jun 2010 11:22:57 +0930: > On Mon, 31 May 2010 14:56:55 +0000 (UTC), "Juan R." González-Álvarez > <nowhere(a)canonicalscience.com> wrote: > > >>TRAJECTORY BRANCHING IN LIOUVILLE SPACE AS THE SOURCE OF IRREVERSIBILITY >> > Original text here: > http://www.canonicalscience.org/publications/canonicalsciencetoday/20100531.html Yes, the above is the permanent link. However, comments only can be submitted here http://www.canonicalscience.org/publications/canonicalsciencetoday/canonicalsciencetoday.html Comments will be closed the next day 02. -- http://www.canonicalscience.org/ BLOG: http://www.canonicalscience.org/publications/canonicalsciencetoday/canonicalsciencetoday.html |