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III. An Organic, Conducive, Habitable MultiUniVerse

F. Systems Cosmology: Fractal SpaceTimeMatter

Sylos Labini, Francesco, et al. Persistent Fluctuations in the Distributions of Galaxies from the Two-degree Field Galaxy Redshift Survey. EPL. 85/29002, 2009. A team of Italian and Russian astronomers further quantify self-similar geometries across the celestial realms.

We apply the scale-length method to several three-dimensional samples of the Two-degree Field Galaxy Redshift Survey. This method allows us to map in a quantitative and powerful way large scale structures in the distribution of galaxies controlling systematic effects. By determining the probability density function of conditional fluctuations we show that large-scale structures are quite typical and correspond to large fluctuations in the galaxy density field. We do not find a convergence to homogeneity up to the samples sizes, i.e. 75 Mpc/h. We then measure, at scales r

Tatekawa, Takayuki and Kei-chi Maeda. Primordial Fractal Density Perturbations and Structure Formation in the Universe. The Astrophysical Journal. 547/531, 2001. A technical paper on how such recurrently ordered forms appear and ramify in the developing cosmos.

One of the most plausible explanations is the nonlinear dynamics of the perturbations will provide such a scale-free structure during the evolution of the universe. (531)

Theel, Friethjof, et al. The Fractal Geometry of Hartree-Fock. Chaos. 27/12, 2017. When this section went online in 2004, scientific perceptions of a natural self-similarity from atomic depths to cosmic breadth were spurious and rudimentary, with a smattering of evidence. A decade and a half later, as this entry by University of Hamburg physicists, and many citations herein now testify, iterative fractal forms are quantified and known to array across these reaches, and everywhere in between. Circa 2018, by a natural philosophy view, our worldwide humankinder seems to be well finding a new genesis universe graced by these intrinsic phenomenal qualities. OK

The Hartree-Fock method is an important approximation for the ground-state electronic wave function of atoms and molecules so that its usage is widespread in computational chemistry and physics. The Hartree-Fock method is an iterative procedure in which the electronic wave functions of the occupied orbitals are determined. The set of functions found in one step builds the basis for the next iteration step. In this work, we interpret the Hartree-Fock method as a dynamical system since dynamical systems are iterations where iteration steps represent the time development of the system, as encountered in the theory of fractals. The focus is put on the convergence behavior of the dynamical system as a function of a suitable control parameter. An investigation of the convergence behavior depending on the parameter λ is performed for helium, neon, and argon. We observe fractal structures in the complex λ-plane, which resemble the well-known Mandelbrot set, determine their fractal dimension, and find that with increasing nuclear charge, the fragmentation increases as well. (Abstract)

Vijar, Sagar, et al. A New Kind of Topological Quantum Order. arXiv:1505.02576. With a Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations subtitle, by way of clever mathematics, MIT physicists SV, Jeongwan Haah, and Liang Fu advance deep understandings about how cosmic nature’s is actually suffused by intrinsic structural geometries. See also Haah’s 2011 original paper Local Stabilizer Codes in Three Dimensions without String Logical Operators at 1101.1962 with much set off this quest.

We introduce exactly solvable models of interacting (Majorana) fermions in d≥3 spatial dimensions that realize a new kind of topological quantum order, building on a model presented in ref. [1]. These models have extensive topological ground-state degeneracy and a hierarchy of point-like, topological excitations that are only free to move within sub-manifolds of the lattice. In particular, one of our models has fundamental excitations that are completely stationary. To demonstrate these results, we introduce a powerful polynomial representation of commuting Majorana Hamiltonians. Remarkably, the physical properties of the topologically-ordered state are encoded in an algebraic variety, defined by the common zeros of a set of polynomials over a finite field. This provides a "geometric" framework for the emergence of topological order. (Abstract)

Von Korff, Modest and Thomas Sander. Molecular Complexity Calculated by Fractal Dimension. Nature Scientific Reports. 9/967, 2019. Scientific Computing Drug Discovery, Idorsia Pharmaceuticals, Switzerland researchers achieve another novel recognition that nature’s proclivity to adopt and display a self-similar, iterative essence can be traced even to molecular and atomic forms and sub-structures.

Molecular complexity is an important characteristic of organic molecules for drug discovery. How to calculate molecular complexity has been discussed in the scientific literature for decades. It was known from early on that the numbers of substructures that can be cut out of a molecular graph are of importance. However, it was never realized that the cut-out substructures show self-similarity to the parent structures. Such a series shows self-similarity similar to fractal objects. The fractal dimension of a molecule is a new matter constant that incorporates all features that are currently known to be important for describing molecular complexity.(Abstract)

During our work on repetitive molecular features we realized that every molecular substructure displays self-similarity to its parent structure. Self-similarity means that an object is similar to a part of itself. Let us derive the concept of self-similarity for organic molecules at the example of n-hexane, a linear alkane. A linear alkane consists of a chain of carbon atoms saturated with hydrogen atoms. Removing one of the two outmost carbon-atoms and the connecting bond creates a new chain with one bond less. To the chain carbon-atom from where the bond was removed, a hydrogen atom has to be added for completing the saturation of the chain. (2)

Vrobel, Susie, et al, eds. Simultaneity: Temporal Structures and Observer Perspectives. Hackensack, NJ: World Scientific, 2008. With co-editors Otto Rossler and Terry Marks-Tarlow, a volume emanating from an Institute for Fractal Research which seems to get wound up in its own vernacular claiming “the next revolution in physics” due to novel appreciations of human observership and the inherent “fractality of space and time.” An eclectic collection, with various contributions by the editors, Peter Allen on a hierarchy of evolutionary systems, Uri Fidelman on complementary brain hemispheres whereof the right side achieves its simultaneous synthesis, and on to a brush with econophysics. In any event, still another sign of coming universe change.

Walcher,, C. J., et al. Self-Similarity in the Chemical Evolution of Galaxies. arXiv:1607.00015. A ten person team from Germany and South America report signs of a universal geometric repetition across material, stellar and galactic realms that is “more than just a coincidence.”

Recent improvements in the age dating of stellar populations and single stars allow us to study the ages and abundance of stars and galaxies with unprecedented accuracy. We here compare the relation between age and \alpha-element abundances for stars in the solar neighborhood to that of local, early-type galaxies. This quantitative similarity seems surprising, given the different types of galaxies and scales involved. The data are consistent with a power law delay time distribution. We thus confirm that the delay time distribution inferred for the Milky Way from chemical evolution arguments also must apply to massive early-type galaxies.

Wang, Xin and Alex Szalay. On the Nonlinear Evolution of Cosmic Web. arXiv.1411.4117. Johns Hopkins University astrophysicists explain “cosmic morphologies of the large-structure” by way of Lagrangian dynamics, a technical finesse of statistical mechanics. We cite because by natural philosophy wonder, how fantastic is it that human folks can altogether suddenly traverse and quantify such infinite vistas. You might read Johns Hopkins (1795-1873) biography on Wikipedia for some context. Surely we peoples ought to grant ourselves a central significance to the course and fate of this genesis universe.

Wang, Yi and Robert Brandenberger. Scale-Invariant Fluctuations from Galilean Genesis. arXiv:1007.0027. Posted June 2012, McGill University physicists take up the work of Paolo Creminelli, et al, Abdus Salam International Centre for Theoretical Physics, posted October 2010 (search arXiv) to glean further insights upon a fractal cosmos. We include both Abstracts. “Inflation on Trial” by Alexandra Witze in Science News for July 25, 2012 reports on these 21st century revisions of the 1980s and 1990s instant inflation theories.

We study the spectrum of cosmological fluctuations in scenarios such as Galilean Genesis in which a spectator scalar field acquires a scale-invariant spectrum of perturbations during an early phase which asymptotes in the far past to Minkowski space-time. In the case of minimal coupling to gravity and standard scalar field Lagrangian, the induced curvature fluctuations depend quadratically on the spectator field and are hence non-scale-invariant and highly non-Gaussian. We show that if higher dimensional operators are considered, a linear coupling between background and spectator field fluctuations is induced which leads to scale-invariant and Gaussian curvature fluctuations. (Wang, Brandenberger)

We propose a novel cosmological scenario, in which standard inflation is replaced by an expanding phase with a drastic violation of the Null Energy Condition (NEC). The model is based on the recently introduced Galileon theories, which allow NEC violating solutions without instabilities. The unperturbed solution describes a Universe that is asymptotically Minkowski in the past, expands with increasing energy density until it exits the regime of validity of the effective field theory and reheats. This solution is a dynamical attractor and the Universe is driven to it, even if it is initially contracting. The study of perturbations of the Galileon field reveals some subtleties, related to the gross violation of the NEC and it shows that adiabatic perturbations are cosmologically irrelevant. The model, however, suggests a new way to produce a scale invariant spectrum of isocurvature perturbations, which can later be converted to adiabatic: the Galileon is forced by symmetry to couple to the other fields as a dilaton; the effective metric it yields on the NEC violating solution is that of de Sitter space, so that all light scalars will automatically acquire a nearly scale-invariant spectrum of perturbations. (Creminelli, et al)

Watkins, Nicholas, et al. 25 Years of Self-Organized Criticality. Space Science Reviews. Online Summer, 2015. The document is also at arXiv:1504.04991. It is a consummate paper to be published in a special retrospective issue along with Marcus Aschwanden, et al, 25 Years of SOC: Space and Laboratory Plasmas (search), James McAteer, et al, 25 Years of SOC: Numerical Detection, and other articles. As the Aschwanden posting explains, the once and future review stems from a series of International Space Science Institute workshops. In accord with a 2015 emergent synthesis of constant work and progress since the 1980s, across these widest natural domains is affirmed a robust “ubiquity, universality, generality” (26). See also a commentary on the work SOC Revisited by Mark Buchanan in Nature Physics for June 2015.

Introduced by the late Per Bak and his colleagues, self-organized criticality (SOC) has been one of the most stimulating concepts to come out of statistical mechanics and condensed matter theory in the last few decades, and has played a significant role in the development of complexity science. SOC, and more generally fractals and power laws, have attracted much comment, ranging from the very positive to the polemical. The other papers in this special issue showcase the considerable body of observations in solar, magnetospheric and fusion plasma inspired by the SOC idea, and expose the fertile role the new paradigm has played in approaches to modeling and understanding multiscale plasma instabilities. This very broad impact, and the necessary process of adapting a scientific hypothesis to the conditions of a given physical system, has meant that SOC as studied in these fields has sometimes differed significantly from the definition originally given by its creators. One aim of the present review is to address the dichotomy between the great reception SOC has received in some areas, and its shortcomings, as they became manifest in the controversies it triggered. Our article tries to clear up what we think are misunderstandings of SOC in fields more remote from its origins in statistical mechanics, condensed matter and dynamical systems by revisiting Bak, Tang and Wiesenfeld's original papers. (Abstract)

Wei, Zong-Wen, et al. Renormalization and Small-World Model of Fractal Quantum Repeater Networks. Nature Scientific Reports. 3/1222, 2013. Hangzhou Normal University, and University of Science and Technology of China, system physicists achieve a number of syntheses in this innovative paper. Traditional statistical mechanics is melded with nonlinear science to join renormalization theories with dynamic scale-free networks. By this approach, even quantum depths can be found to contain and express the same complex, self-similar system phenomena as everywhere else in nature and society. Such a novel, holistic vista can even more reveal and implicate, albeit in arcane terms than beg translation, an independent, creative, universal source.

Quantum networks provide access to exchange of quantum information. The primary task of quantum networks is to distribute entanglement between remote nodes. Although quantum repeater protocol enables long distance entanglement distribution, it has been restricted to one-dimensional linear network. Here we develop a general framework that allows application of quantum repeater protocol to arbitrary quantum repeater networks with fractal structure. Entanglement distribution across such networks is mapped to renormalization. Furthermore, we demonstrate that logarithmical times of recursive such renormalization transformations can trigger fractal to small-world transition, where a scalable quantum small-world network is achieved. Our result provides new insight into quantum repeater theory towards realistic construction of large-scale quantum networks. (Abstract)

To summarize, we have generalized one-dimensional quantum repeaters to high fractal dimension by introducing an approach called CR (coupled renormalization), which relates entanglement distribution over arbitrary fractal QRN (quantum repeater network) to recursive renormalization transformations. We assume that QRN is fractal and scale-free, which is the case for a large number of real-world networks. In spite of the large size of QRN, there exists a relatively small critical nesting level for CR, at which small-world is obtained. Small-world seems to be a necessary element for a scalable quantum network. Our study suggests that concepts and tools from statistical physics will play an important role in the joint study. It has conceived another significant direction which may open new avenue to address the outstanding issue of complexity. That is, quantum simulation of dynamic process on complex networks or design of quantum algorithms for sophisticated questions in network science. All of these attempts may dramatically alter the landscape of both fields. (5)

Weil, Melinda and Ralph Pudritz. Cosmological Evolution of Supergiant Star-Forming Clouds. The Astrophysical Journal. 556/164, 2001. Galaxies form into hierarchical clusters due to a “robust power-law mass spectrum.”

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