Vladimir V. Uchaikin - Böcker

The first derivative of a particle coordinate means its velocity, the second means its acceleration, but what does a fractional order derivative mean? Where does it come from, how does it work, where does it lead to? The two-volume book written on high didactic level answers these questions. Fractional Derivatives for Physicists and Engineers— The first volume contains a clear introduction into such a modern branch of analysis as the fractional calculus. The second develops a wide panorama of applications of the fractional calculus to various physical problems. This book recovers new perspectives in front of the reader dealing with turbulence and semiconductors, plasma and thermodynamics, mechanics and quantum optics, nanophysics and astrophysics.  The book is addressed to students, engineers and physicists, specialists in theory of probability and statistics, in mathematical modeling and numerical simulations, to everybody who doesn't wish to stay apart from the new mathematical methods becoming more and more popular.Prof. Vladimir V. UCHAIKIN is a known Russian scientist and pedagogue, a Honored Worker of Russian High School, a member of the Russian Academy of Natural Sciences. He is the author of about three hundreds articles and more than a dozen books (mostly in Russian)  in Cosmic ray physics, Mathematical physics, Levy stable statistics, Monte Carlo methods with applications to anomalous processes in complex systems of various levels: from quantum dots to the Milky Way galaxy.

This book is first of its kind describing a new direction in modeling processes taking place in interplanetary and interstellar space (magnetic fields, plasma, cosmic rays, etc.). This method is based on a special mathematical analysis — fractional calculus. The reader will find in this book clear physical explanation of the fractional approach and will become familiar with basic rules in this calculus and main results obtained in frame of this approach. In spite of its profound subject, the book is not overloaded by mathematical details. It contains many illustrations, rich citation and remains accessible to a wide circle of physicists.This book is addressed to graduate and postgraduate students, young and mature researchers specializing in applications of fractional calculus, astrophysics, solar-terrestrial science and physics of cosmic rays.

The standard (Markovian) transport model based on the Boltzmann equation cannot describe some non-equilibrium processes called anomalous that take place in many disordered solids. Causes of anomality lie in non-uniformly scaled (fractal) spatial heterogeneities, in which particle trajectories take cluster form. Furthermore, particles can be located in some domains of small sizes (traps) for a long time. Estimations show that path length and waiting time distributions are often characterized by heavy tails of the power law type. This behavior allows the introduction of time and space derivatives of fractional orders. Distinction of path length distribution from exponential is interpreted as a consequence of media fractality, and analogous property of waiting time distribution as a presence of memory.In this book, a novel approach using equations with derivatives of fractional orders is applied to describe anomalous transport and relaxation in disordered semiconductors, dielectrics and quantum dot systems. A relationship between the self-similarity of transport, the Levy stable limiting distributions and the kinetic equations with fractional derivatives is established. It is shown that unlike the well-known Scher-Montroll and Arkhipov-Rudenko models, which are in a sense alternatives to the normal transport model, fractional differential equations provide a unified mathematical framework for describing normal and dispersive transport. The fractional differential formalism allows the equations of bipolar transport to be written down and transport in distributed dispersion systems to be described. The relationship between fractional transport equations and the generalized limit theorem reveals the probabilistic aspects of the phenomenon in which a dispersive to Gaussian transport transition occurs in a time-of-flight experiment as the applied voltage is decreased and/or the sample thickness increased. Recent experiments devoted to studies of transport in quantum dot arrays are discussed in the framework of dispersive transport models. The memory phenomena in systems under consideration are discussed in the analysis of fractional equations.It is shown that the approach based on the anomalous transport models and the fractional kinetic equations may be very useful in some problems that involve nano-sized systems. These are photon counting statistics of blinking single quantum dot fluorescence, relaxation of current in colloidal quantum dot arrays, and some others.

This book brings new perspectives in front of the reader dealing with turbulence and semiconductors, plasma and thermodynamics, mechanics and quantum optics, nanophysics and astrophysics. The first derivative of a particle coordinate means its velocity, the second means its acceleration, but what does a fractional order derivative mean? Where does it come from, how does it work, where does it lead to? The two-volume book written on high didactic level answers these questions. The first volume (ISBN: 978-3-642-33910-3) contains a clear introduction into such a modern branch of analysis as fractional calculus. This second volume develops a wide panorama of applications of the fractional calculus to various physical problems.This book is addressed to students, engineers and physicists, specialists in theory of probability and statistics, in mathematical modeling and numerical simulations, to everybody who doesn't wish to stay apart from the new mathematical methods becoming more and more popular.

This book demonstrates the practical application of an alternative approach to current problems in high-energy astrophysics. In high-energy astrophysical processes, single collisions are accompanied by the appearance of many secondary particles with different properties. To describe the infinitesimal evolution of such a system at a measurement instant, as is commonly done when deriving the kinetic equation for the system with conserved number of particles, one must know either its prehistory or the infinite family of many-particle distributions. An alternative to this approach is to use an adjoint (in the sense of Lagrange) mathematical formalism, where the independent active variable is the phase position of a primary particle generating the cascade, and the dependent variable has the form of a functional of the cascade as a whole, interpreted as the reading of some not necessarily linear (additive) detector. This approach is characterized by mathematical efficiency: no matter how many particles are formed in a cascade, the active argument of the desired functional always remains one particle. The second advantage is its generality: the formalization of the readings of the detector, which performs real measurements through the functionality of a random implementation of the cascade, allows it to be applied to a wide range of actually used devices and installations.Readers will be able to master the fundamentals of particle astrophysics in the context of the latest developments in the field. It will benefit graduate students and established researchers alike, equipping them with the knowledge and tools necessary to design and interpret their own experiments and, ultimately, to address a number of questions about the nature and origin of cosmic particles that have arisen in recent research.