Scale InvarianceScale Invariance

2 we introduced structures that are spatially scale invariant: fractals. These structures evolve and are related to one another at successive times by changes of spatial scale. Note that the relaxation of the system acts exactly like an ...

Author: Annick LESNE

Publisher: Springer Science & Business Media

ISBN: 9783642151231


Page: 400

View: 951

During a century, from the Van der Waals mean field description (1874) of gases to the introduction of renormalization group (RG techniques 1970), thermodynamics and statistical physics were just unable to account for the incredible universality which was observed in numerous critical phenomena. The great success of RG techniques is not only to solve perfectly this challenge of critical behaviour in thermal transitions but to introduce extremely useful tools in a wide field of daily situations where a system exhibits scale invariance. The introduction of scaling, scale invariance and universality concepts has been a significant turn in modern physics and more generally in natural sciences. Since then, a new "physics of scaling laws and critical exponents", rooted in scaling approaches, allows quantitative descriptions of numerous phenomena, ranging from phase transitions to earthquakes, polymer conformations, heartbeat rhythm, diffusion, interface growth and roughening, DNA sequence, dynamical systems, chaos and turbulence. The chapters are jointly written by an experimentalist and a theorist. This book aims at a pedagogical overview, offering to the students and researchers a thorough conceptual background and a simple account of a wide range of applications. It presents a complete tour of both the formal advances and experimental results associated with the notion of scaling, in physics, chemistry and biology.

Scale Invariance and BeyondScale Invariance and Beyond

behaves in the same way at the planetary or the laboratory scales and is thus a scaleinvariant phenomenon. On the other hand, at a critical point in a phase transition, fluctuations do not rely on a characteristic scale so that ...

Author: B. Dubrulle

Publisher: Springer Science & Business Media

ISBN: 9783662097991


Page: 287

View: 624

This book is an excellent introduction to the concept of scale invariance, which is a growing field of research with wide applications. It describes where and how symmetry under scale transformation (and its various forms of partial breakdown) can be used to analyze solutions of a problem without the need to explicitly solve it. The first part gives descriptions of tools and concepts; the second is devoted to recent attempts to go beyond the invariance or symmetry breaking, to discuss causes and consequences, and to extract useful information about the system. Examples are carefully worked out in fields as diverse as condensed matter physics, population dynamics, earthquake physics, turbulence, cosmology and finance.

Scale InvarianceScale Invariance

Thus, in a statistically stationary state, we expect Self-Similarity or scale invariance to be expressed for the structure functions according to the Lie group method as Sn = Sn (θ,φ)Lσen(δ−α)R. (8.42) In this expression α and δ have ...

Author: Richard N. Henriksen

Publisher: John Wiley & Sons

ISBN: 9783527687336


Page: 304

View: 684

Bringing the concepts of dimensional analysis, self-similarity, and fractal dimensions together in a logical and self-contained manner, this book reveals the close links between modern theoretical physics and applied mathematics. The author focuses on the classic applications of self-similar solutions within astrophysical systems, with some general theory of self-similar solutions, so as to provide a framework for researchers to apply the principles across all scientific disciplines. He discusses recent advances in theoretical techniques of scaling while presenting a uniform technique that encompasses these developments, as well as applications to almost any branch of quantitative science. The result is an invaluable reference for active scientists, featuring examples of dimensions and scaling in condensed matter physics, astrophysics, fluid mechanics, and general relativity, as well as in mathematics and engineering.

Scale Invariance Interfaces and Non Equilibrium DynamicsScale Invariance Interfaces and Non Equilibrium Dynamics

can be anticipated, and for actually calculating the exponents that govern the algebraic decays of scale invariant chaotic systems. For example, the idea that one can straightforwardly calculate from Langevin equations the asymptotic ...

Author: Alan McKane

Publisher: Springer Science & Business Media

ISBN: 9781489914217


Page: 344

View: 283

The NATO Advanced Study Institute on "Scale Invariance, Interfaces and Non Equilibrium Dynamics" was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK from 20-30 June 1994. The topics discussed at the Institute were all concerned with the origin and nature of complex structures found far from equilibrium. Examples ranged from reaction diffusion systems and hydrodynamics through to surface growth due to deposition. A common theme was that of scale invariance due to the self-similarity of the underly ing structures. The topics that were covered can be broadly classified as pattern for mation (theoretical, computational and experimental aspects), the non-equilibrium dynamics of the growth of interfaces and other manifolds, coarsening phenomena, generic scale invariance in driven systems and the concept of self-organized critical ity. The main feature of the Institute was the four one-hour-Iong lectures given each day by invited speakers. In addition to thirty-seven of these lectures, two contributed lectures were also given. The many questions that were asked after the lectures attested to the excitement and interest that the lecturers succeeded in generating amongst the students. In addition to the discussions initiated by lectures, an im portant component of the meeting were the poster sessions, where participants were able to present their own work, which took place on three of the afternoons. The list of titles given at the end of these proceedings gives some idea of the range and scope of these posters.

Introduction to Conformal Invariance and Its Applications to Critical PhenomenaIntroduction to Conformal Invariance and Its Applications to Critical Phenomena

Their role was taken into account in the subsequent developments leading to the scaling theories of critical phenomena and the ... Conformal invariance has been known for almost a century in connection with scale invariance.

Author: Philippe Christe

Publisher: Springer Science & Business Media

ISBN: 9783540475750


Page: 260

View: 221

The history of critical phenomena goes back to the year 1869 when Andrews discovered the critical point of carbon dioxide, located at about 31°C and 73 atmospheres pressure. In the neighborhood ofthis point the carbon dioxide was observed to become opalescent, that is, light is strongly scattered. This is nowadays interpreted as comingfrom the strong fluctuations of the system close to the critical point. Subsequently, a wide varietyofphysicalsystems were realized to display critical points as well. Ofparticular importance was the observation of a critical point in ferromagnetic iron by Curie. Further examples include multicomponent fluids and alloys, superfluids, superconductors, polymers and may even extend to the quark-gluon plasmaand the early universe as a whole. Early theoretical investigationstried to reduce the problem to a very small number of degrees of freedom, such as the van der Waals equation and mean field approximations and culminating in Landau's general theory of critical phenomena. In a dramatic development, Onsager's exact solutionofthe two-dimensional Ising model made clear the important role of the critical fluctuations. Their role was taken into account in the subsequent developments leading to the scaling theories of critical phenomena and the renormalization group. These developements have achieved a precise description of the close neighborhood of the critical point and results are often in good agreement with experiments. In contrast to the general understanding a century ago, the presence of fluctuations on all length scales at a critical point is today emphasized.

Local Invariant Feature DetectorsLocal Invariant Feature Detectors

It is therefore naturally invariant to rotation. It is also well adapted for detecting blob- like structures. The experimental evaluation in [138] shows this function is well suited for automatic scale selection. The scale invariance of ...

Author: Tinne Tuytelaars

Publisher: Now Publishers Inc

ISBN: 9781601981387


Page: 122

View: 859

Local Invariant Features Detectors is an overview of invariant interest point detectors, how they evolved over time, how they work, and what their respective strengths and weaknesses are.

From Current Algebra to Quantum ChromodynamicsFrom Current Algebra to Quantum Chromodynamics

Scale anomaly and renormalization group It is interesting to note that while scale invariance is unavoidably broken by renormalization effects in an anomalous way,3 the renormalization group equation that is underlain by the scale ...

Author: Tian Yu Cao

Publisher: Cambridge University Press

ISBN: 9781139491600



View: 437

The advent of quantum chromodynamics (QCD) in the early 1970s was one of the most important events in twentieth-century science. This book examines the conceptual steps that were crucial to the rise of QCD, placing them in historical context against the background of debates that were ongoing between the bootstrap approach and composite modeling, and between mathematical and realistic conceptions of quarks. It explains the origins of QCD in current algebra and its development through high-energy experiments, model-building, mathematical analysis and conceptual synthesis. Addressing a range of complex physical, philosophical and historiographical issues in detail, this book will interest graduate students and researchers in physics and in the history and philosophy of science.

Nonlinear Dynamics of the Lithosphere and Earthquake PredictionNonlinear Dynamics of the Lithosphere and Earthquake Prediction

t - 00 lim q « ( 1 , t ) = q * ( 1 ) , qk = lim q ( 1 , t ) = 9 ( 1 ) tThe scaling properties of the stationary solution ... Now we consider domains of stability , scale invariance , and catastrophe to describe the scaling properties of ...

Author: Vladimir Keilis-Borok

Publisher: Springer Science & Business Media

ISBN: 354043528X


Page: 358

View: 353

The vulnerability of our civilization to earthquakes is rapidly growing, rais ing earthquakes to the ranks of major threats faced by humankind. Earth quake prediction is necessary to reduce that threat by undertaking disaster preparedness measures. This is one of the critically urgent problems whose solution requires fundamental research. At the same time, prediction is a ma jor tool of basic science, a source of heuristic constraints and the final test of theories. This volume summarizes the state-of-the-art in earthquake prediction. Its following aspects are considered: - Existing prediction algorithms and the quality of predictions they pro vide. - Application of such predictions for damage reduction, given their current accuracy, so far limited. - Fundamental understanding of the lithosphere gained in earthquake prediction research. - Emerging possibilities for major improvements of earthquake prediction methods. - Potential implications for predicting other disasters, besides earthquakes. Methodologies. At the heart of the research described here is the inte gration of three methodologies: phenomenological analysis of observations; "universal" models of complex systems such as those considered in statistical physics and nonlinear dynamics; and Earth-specific models of tectonic fault networks. In addition, the theory of optimal control is used to link earthquake prediction with earthquake preparedness.

Cosmological Implications of Quantum AnomaliesCosmological Implications of Quantum Anomalies

B 648, 312–317 (2007). Y. Fujii, Scalar-tensor theory of gravitation and spontaneous breakdown of scale invariance. Phys. Rev. D 9, 874–876 (1974). F.

Author: Neil David Barrie

Publisher: Springer

ISBN: 9783319947150


Page: 140

View: 559

The successes of the standard models of particle physics and cosmology are many, but have proven incapable of explaining all the phenomena that we observe. This book investigates the potentially important role of quantum physics, particularly quantum anomalies, in various aspects of modern cosmology, such as inflation, the dynamical generation of the visible and dark matter in the universe, and gravitational waves. By doing so, the authors demonstrate that exploring the links between cosmology and particle physics is key to helping solve the mysteries of our Universe.

Statistical Physics of FieldsStatistical Physics of Fields

9.4 Generic scale invariance in equilibrium systems We live in a world full of complex spatial patterns and structures such as ... These phenomena lack natural length and time scales and exhibit scale invariance and self-similarity.

Author: Mehran Kardar

Publisher: Cambridge University Press

ISBN: 9781139855884



View: 692

While many scientists are familiar with fractals, fewer are familiar with scale-invariance and universality which underlie the ubiquity of their shapes. These properties may emerge from the collective behaviour of simple fundamental constituents, and are studied using statistical field theories. Initial chapters connect the particulate perspective developed in the companion volume, to the coarse grained statistical fields studied here. Based on lectures taught by Professor Kardar at MIT, this textbook demonstrates how such theories are formulated and studied. Perturbation theory, exact solutions, renormalization groups, and other tools are employed to demonstrate the emergence of scale invariance and universality, and the non-equilibrium dynamics of interfaces and directed paths in random media are discussed. Ideal for advanced graduate courses in statistical physics, it contains an integrated set of problems, with solutions to selected problems at the end of the book and a complete set available to lecturers at