This monograph contains original results in the field of mathematical and numerical modeling of mechanical behavior of granular materials and materials with different strengths. It proposes new models helping to define zones of the strain localization. The book shows how to analyze processes of the propagation of elastic and elastic-plastic waves in loosened materials, and constructs models of mixed type, describing the flow of granular materials in the presence of quasi-static deformation zones. In a last part, the book studies a numerical realization of the models on multiprocessor computer systems. The book is intended for scientific researchers, lecturers of universities, post-graduates and senior students, who specialize in the field of the deformable materials mechanics, mathematical modeling and adjacent fields of applied and calculus mathematics.

This book is essentially based on the lecture course on "Statistical Physics", which was taught by the author at the physical faculty of the Ural State University in Ekaterinburg since 1992. This course was intended for all physics students, not especially for those specializing in theoretical physics. In this sense the material presented here contains the necessary minimum of knowledge of statistical physics (also often called statistical mechanics), which is in author's opinion necessary for every person wishing to obtain a general education in the field of physics. This posed the rather difficult problem of the choice of material and compact enough presentation. At the same time it necessarily should contain all the basic principles of statistical physics, as well as its main applications to different physical problems, mainly from the field of the theory of condensed matter. Extended version of these lectures were published in Russian in 2003. For the present English edition, some of the material was rewritten and several new sections and paragraphs were added, bringing contents more up to date and adding more discussion on some more difficult cases.

This volume provides a compact presentation of modern statistical physics at an advanced level, from the foundations of statistical mechanics to the main modern applications of statistical physics. Special attention is given to new approaches, such as quantum field theory methods and non-equilibrium problems. This second, revised edition is expanded with biographical notes contextualizing the main results in statistical physics.

This book discusses the main concepts of the Standard Model of elementary particles in a compact and straightforward way. The theoretical results are derived using the physical phenomena as a starting point. This inductive approach allows a deep understanding of the methods used for solving problems in this field. This second, revised edition is expanded with biographical notes contextualizing the main results in quantum field theory.

This monograph contains original results in the field of mathematical and numerical modeling of mechanical behavior of granular materials and materials with different strengths. It proposes new models helping to define zones of the strain localization. The book shows how to analyze processes of the propagation of elastic and elastic-plastic waves in loosened materials, and constructs models of mixed type, describing the flow of granular materials in the presence of quasi-static deformation zones. In a last part, the book studies a numerical realization of the models on multiprocessor computer systems. The book is intended for scientific researchers, lecturers of universities, post-graduates and senior students, who specialize in the field of the deformable materials mechanics, mathematical modeling and adjacent fields of applied and calculus mathematics.

A condensing (or densifying) operator is a mapping under which the image of any set is in a certain sense more compact than the set itself. The degree of noncompactness of a set is measured by means of functions called measures of noncompactness. The contractive maps and the compact maps [i.e., in this Introduction, the maps that send any bounded set into a relatively compact one; in the main text the term "compact" will be reserved for the operators that, in addition to having this property, are continuous, i.e., in the authors' terminology, for the completely continuous operators] are condensing. For contractive maps one can take as measure of noncompactness the diameter of a set, while for compact maps can take the indicator function of a family of non-relatively com pact sets. The operators of the form F( x) = G( x, x), where G is contractive in the first argument and compact in the second, are also condensing with respect to some natural measures of noncompactness. The linear condensing operators are characterized by the fact that almost all of their spectrum is included in a disc of radius smaller than one. The examples given above show that condensing operators are a sufficiently typical phenomenon in various applications of functional analysis, for example, in the theory of differential and integral equations. As is turns out, the condensing operators have properties similar to the compact ones.

The introduction of quantum field theory methods has led to a kind of 'revolution' in condensed matter theory, resulting in the increased importance of Feynman diagrams or diagram technique. So, it has now become imperative for professionals in condensed matter theory to have a thorough knowledge of this method.The book is intended to teach students, postdocs and young theorists to use diagrammatic quantum field theory methods applied to different problems of modern condensed matter theory, using specific examples of such problems. This latest edition is extended by the inclusion of some new material on superconductivity and diagram combinatorics.

This book is a compilation of reprints of the major works by the prominent Soviet and Russian theoretical physicists, Leonid V Keldysh. He made important contributions to condensed matter theory, developing new approaches and methods, and discovering beautiful new physical effects later confirmed by experiments.Keldysh developed the consistent theory of phonon assisted tunneling in semiconductors and calculated the electric field induced shift of absorption edge in semiconductors, what is now called Franz-Keldysh effect. He also developed the original theory of deep levels in semiconductors. One of his most famous works was the 1964 theory of multiple photon ionization of atoms by intensive electromagnetic waves, thereby laying the foundation for the entire field of intense laser radiation interaction with atoms, ions, molecules, and solids. This theory introduced optical tunneling and beyond — threshold ionization, experimentally observed about 15 years later.His name is probably best known due to his 1964 work on general diagram technique for non-equilibrium processes. Introducing Green's functions with time — ordering along what is now known as Keldysh contour, he was able to construct the standard Feynman diagrams for these Green's functions at finite temperatures and for general non-equilibrium states. Since then, this approach has become a standard one, with multiple applications in solid state theory, the theory of quantum liquids, quantum field theory and even quantum cosmology.Since 1964, Keldysh's interests moved to the problem of many excitons in semiconductors. He introduced the new concept of excitonic insulator. Actually this was a new mechanism of metal-insulator transitions. In later works by Keldysh and his collaborators it was definitely shown, that there are no superfluidity properties in this model, as was initially suspected by some authors and he moved to the studies of non-equilibrium system of excitons, appearing under intensive laser pumping of semiconductors, where superfluidity of excitons was shown to be possible.In 1968, Keldysh realized that in most semiconductors (with multiple bands), the non-equilibrium system of many excitons actually transforms into electron — hole quantum liquid (with excitons destroyed), forming electron-hole drops. This idea immediately stimulated the experimental studies and electron-hole drops were soon discovered, leading to many further experimental and theoretical works on this new state of matter. Keldysh was awarded numerous awards, including the Lenin prize (1974), the Hewlett-Packard Prize (1975), Alexander von Humboldt Prize (1994), Rusnanoprize (2009), Eugene Feinberg Memorial Medal (2011), Pomeranchuk Prize (2014) and the Grand Lomonosov Gold Medal of the Russian Academy of Sciences (2015). He was elected the foreign member of the USA National Academy of Sciences (1995) and the Fellow of the American Physical Society (1996).