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This book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*-algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as mathematical interest that are covered in detail include symmetry (and its "spontaneous" breaking), the measurement problem, the Kochen-Specker, Free Will, and Bell Theorems, the Kadison-Singer conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory.

Quantum physics --- Mathematical physics --- Matrix theory --- Algebra

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Resonance is a common phenomenon, which is observed both in nature and in numerous devices and structures. It occurs in literally all types of vibrations. To mention just a few examples, acoustic, mechanical, or electromagnetic resonance can be distinguished. In the present book, 12 chapters dealing with different aspects of resonance phenomena have been presented.

Physical Sciences, Engineering and Technology --- Mathematics --- Applied Mathematics --- Mathematical Physics

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Molecular Dynamics is a two-volume compendium of the ever-growing applications of molecular dynamics simulations to solve a wider range of scientific and engineering challenges. The contents illustrate the rapid progress on molecular dynamics simulations in many fields of science and technology, such as nanotechnology, energy research, and biology, due to the advances of new dynamics theories and the extraordinary power of today's computers. This first book begins with a general description of underlying theories of molecular dynamics simulations and provides extensive coverage of molecular dynamics simulations in nanotechnology and energy. Coverage of this book includes: Recent advances of molecular dynamics theory Formation and evolution of nanoparticles of up to 106 atoms Diffusion and dissociation of gas and liquid molecules on silicon, metal, or metal organic frameworks Conductivity of ionic species in solid oxides Ion solvation in liquid mixtures Nuclear structures

Physical Sciences, Engineering and Technology --- Mathematics --- Applied Mathematics --- Mathematical Physics

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Historically, the notion of entropy emerged in conceptually very distinct contexts. This book deals with the connection between entropy, probability, and fractional dynamics as they appeared, for example, in solar neutrino astrophysics since the 1970's (Mathai and Rathie 1975, Mathai and Pederzoli 1977, Mathai and Saxena 1978, Mathai, Saxena, and Haubold 2010).The original solar neutrino problem, experimentally and theoretically, was resolved through the discovery of neutrino oscillations and was recently enriched by neutrino entanglement entropy. To reconsider possible new physics of solar neutrinos, diffusion entropy analysis, utilizing Boltzmann entropy, and standard deviation analysis was undertaken with Super-Kamiokande solar neutrino data. This analysis revealed a non-Gaussian signal with harmonic content. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the Super-Kamiokande data deviate considerably from the value of ½, which indicates that the statistics of the underlying phenomenon is anomalous. Here experiment may provide guidance about the generalization of theory of Boltzmann statistical mechanics. Arguments in the so-called Boltzmann-Planck-Einstein discussion related to Planck's discovery of the black-body radiation law are recapitulated mathematically and statistically and emphasize from this discussion is pursued that a meaningful implementation of the complex ‘entropy-probability-dynamics’ may offer two ways for explaining the results of diffusion entropy analysis and standard deviation analysis. One way is to consider an anomalous diffusion process that needs to use the fractional space-time diffusion equation (Gorenflo and Mainardi) and the other way is to consider a generalized Boltzmann entropy by assuming a power law probability density function. Here new mathematical framework, invented by sheer thought, may provide guidance for the generalization of Boltzmann statistical mechanics. In this book Boltzmann entropy, generalized by Tsallis and Mathai, is considered. The second one contains a varying parameter that is used to construct an entropic pathway covering generalized type-1 beta, type-2 beta, and gamma families of densities. Similarly, pathways for respective distributions and differential equations can be developed. Mathai's entropy is optimized under various conditions reproducing the well-known Boltzmann distribution, Raleigh distribution, and other distributions used in physics. Properties of the entropy measure for the generalized entropy are examined. In this process the role of special functions of mathematical physics, particularly the H-function, is highlighted.

special functions --- fractional calculus --- entropic functional --- mathematical physics --- applied analysis --- statistical distributions --- geometrical probabilities --- multivariate analysis

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During the past four decades or so, various operators of fractional calculus, such as those named after Riemann–Liouville, Weyl, Hadamard, Grunwald–Letnikov, Riesz, Erdelyi–Kober, Liouville–Caputo, and so on, have been found to be remarkably popular and important due mainly to their demonstrated applications in numerous diverse and widespread fields of the mathematical, physical, chemical, engineering, and statistical sciences. Many of these fractional calculus operators provide several potentially useful tools for solving differential, integral, differintegral, and integro-differential equations, together with the fractional-calculus analogues and extensions of each of these equations, and various other problems involving special functions of mathematical physics, as well as their extensions and generalizations in one and more variables. In this Special Issue, we invite and welcome review, expository, and original research articles dealing with the recent advances in the theory of fractional calculus and its multidisciplinary applications.

operators of fractional calculus --- chaos and fractional dynamics --- fractional differential --- fractional differintegral equations --- fractional integro-differential equations --- fractional integrals --- fractional derivatives associated with special functions of mathematical physics --- applied mathematics --- identities and inequalities involving fractional integrals --- fractional derivatives