Through the development of an exact path integral for use in transferring information from observations to a model of the observed system, the author provides a general framework for the discussion of model building and evaluation across disciplines. Through many illustrative examples drawn from models in neuroscience, geosciences, and nonlinear electrical circuits, the concepts are exemplified in detail. Practical numerical methods for approximate evaluations of the path integral are explored, and their use in designing experiments and determining a model's consistency with observations is explored.

This book provides a theoretical, step-by-step comprehensive explanation of superconductivity for undergraduate and graduate students who have completed elementary courses on thermodynamics and quantum mechanics. To this end, it adopts the unique approach of starting with the statistical mechanics of quantum ideal gases and successively adding and clarifying elements and techniques indispensible for understanding it. They include the spin-statistics theorem, second quantization, density matrices, the Bloch-De Dominicis theorem, the variational principle in statistical mechanics, attractive interaction and bound states. Ample examples of their usage are also provided in terms of topics from advanced statistical mechanics such as two-particle correlations of quantum ideal gases, derivation of the Hartree-Fock equations, and Landau's Fermi-liquid theory, among others. With these preliminaries, the fundamental mean-field equations of superconductivity are derived with maximum mathematical clarity based on a coherent state in terms of the Cooper-pair creation operator, a quasiparticle field for describing the excitation and the variational principle in statistical mechanics. They have the advantage that the phase coherence due to the Cooper-pair condensation can be clearly seen making the superfluidity comprehensible naturally. Subsequently, they are applied to homogeneous cases to describe the BCS theory for classic s-wave superconductors and its extension to the p-wave superfluidity of 3He. Later, the mean-field equations are simplified to the Eilenberger and Ginzburg-Landau equations so as to describe inhomogeneous superconductivity such as Abrikosov's flux-line lattice concisely and transparently. Chapters provide the latest studies on the quasiclassical theory of superconductivity and a discovery of p-wave superfluidity in liquid 3He. The book serves as a standard reference for advanced courses of statistical mechanics with exercises along with detailed answers.

Data assimilation is a hugely important mathematical technique, relevant in fields as diverse as geophysics, data science, and neuroscience. This modern book provides an authoritative treatment of the field as it relates to several scientific disciplines, with a particular emphasis on recent developments from machine learning and its role in the optimisation of data assimilation. Underlying theory from statistical physics, such as path integrals and Monte Carlo methods, are developed in the text as a basis for data assimilation, and the author then explores examples from current multidisciplinary research such as the modelling of shallow water systems, ocean dynamics, and neuronal dynamics in the avian brain. The theory of data assimilation and machine learning is introduced in an accessible and unified manner, and the book is suitable for undergraduate and graduate students from science and engineering without specialized experience of statistical physics.

Thermodynamics is used increasingly in ecology to understand the system properties of ecosystems because it is a basic science that describes energy transformation from a holistic view. In the last decade, many contributions to ecosystem theory based on thermodynamics have been published, therefore an important step toward integrating these theories and encouraging a more wide spread use of them is to present them in one volume.
An ecosystem consists of interdependent living organisms that are also interdependent with their environment, all of which are involved in a constant transfer of energy and mass within a general state of equilibrium or dis-equilibrium. Thermodynamics can quantify exactly how "organized" or "disorganized" a system is - an extremely useful to know when trying to understand how a dynamic ecosystem is behaving.
A part of the Environmental and Ecological (Math) Modeling series, Thermodynamics and Ecology is a book-length study - the first of its kind - of the current thinking on how an ecosystem can be explained and predicted in terms of its thermodynamical behavior. After the introductory chapters on the fundamentals of thermodynamics, the book explains how thermodynamic theory can be specifically applied to the "measurement" of an ecosystem, including the assessment of its state of entropy and enthalpy. Additionally, it will show economists how to put these theories to use when trying to quantify the movement of goods and services through another type of complex living system - a human society.

This proceedings volume gathers together original articles and survey works that originate from presentations given at the conference Transient Transcendence in Transylvania, held in Brasov, Romania, from May 13th to 17th, 2019. The conference gathered international experts from various fields of mathematics and computer science, with diverse interests and viewpoints on transcendence. The covered topics are related to algebraic and transcendental aspects of special functions and special numbers arising in algebra, combinatorics, geometry and number theory. Besides contributions on key topics from invited speakers, this volume also brings selected papers from attendees.

This edited volume collects six surveys that present state-of-the-art results on modeling, qualitative analysis, and simulation of active matter, focusing on specific applications in the natural sciences. Following the previously published Active Particles volumes, these chapters are written by leading experts in the field and reflect the diversity of subject matter in theory and applications within an interdisciplinary framework. Topics covered include: Variability and heterogeneity in natural swarms Multiscale aspects of the dynamics of human crowds Mathematical modeling of cell collective motion triggered by self-generated gradients Clustering dynamics on graphs Random Batch Methods for classical and quantum interacting particle systems The consensus-based global optimization algorithm and its recent variants Mathematicians and other members of the scientific community interested in active matter and its many applications will find this volume to be a timely, authoritative, and valuable resource.

This classic book marks the beginning of an era of vigorous mathematical progress in equilibrium statistical mechanics. Its treatment of the infinite system limit has not been superseded, and the discussion of thermodynamic functions and states remains basic for more recent work. The conceptual foundation provided by the Rigorous Results remains invaluable for the study of the spectacular developments of statistical mechanics in the second half of the 20th century.

Emergence and complexity refer to the appearance of higher-level properties and behaviours of a system that obviously comes from the collective dynamics of that system's components. These properties are not directly deducible from the lower-level motion of that system. Emergent properties are properties of the "whole'' that are not possessed by any of the individual parts making up that whole. Such phenomena exist in various domains and can be described, using complexity concepts and thematic knowledges. This book highlights complexity modelling through dynamical or behavioral systems. The pluridisciplinary purposes, developed along the chapters, are able to design links between a wide-range of fundamental and applicative Sciences. Developing such links - instead of focusing on specific and narrow researches - is characteristic of the Science of Complexity that we try to promote by this contribution.

This monograph introduces an exact model for a critical spin chain with arbitrary spin S, which includes the Haldane--Shastry model as the special case S=1/2. While spinons in the Haldane-Shastry model obey abelian half-fermi statistics, the spinons in the general model introduced here obey non-abelian statistics. This manifests itself through topological choices for the fractional momentum spacings. The general model is derived by mapping exact models of quantized Hall states onto spin chains. The book begins with pedagogical review of all the relevant models including the non-abelian statistics in the Pfaffian Hall state, and is understandable to every student with a graduate course in quantum mechanics.

This text provides a uniform and consistent approach to diversified problems encountered in the study of dynamical processes in condensed phase molecular systems. Given the broad interdisciplinary aspect of this subject, the book focuses on three themes: coverage of needed background material, in-depth introduction of methodologies, and analysis of several key applications. The uniform approach and common language used in all discussions help to develop general understanding and insight on condensed phases chemical dynamics. The applications discussed are among the most fundamental processes that underlie physical, chemical and biological phenomena in complex systems.
The first part of the book starts with a general review of basic mathematical and physical methods (Chapter 1) and a few introductory chapters on quantum dynamics (Chapter 2), interaction of radiation and matter (Chapter 3) and basic properties of solids (chapter 4) and liquids (Chapter 5). In the second part the text embarks on a broad coverage of the main methodological approaches. The central role of classical and quantum time correlation functions is emphasized in Chapter 6. The presentation of dynamical phenomena in complex systems as stochastic processes is discussed in Chapters 7 and 8. The basic theory of quantum relaxation phenomena is developed in Chapter 9, and carried on in Chapter 10 which introduces the density operator, its quantum evolution in Liouville space, and the concept of reduced equation of motions. The methodological part concludes with a discussion of linear response theory in Chapter 11, and of the spin-boson model in chapter 12. The third part of the book applies the methodologies introducedearlier to several fundamental processes that underlie much of the dynamical behaviour of condensed phase molecular systems. Vibrational relaxation and vibrational energy transfer (Chapter 13), Barrier crossing and diffusion controlled reactions (Chapter 14), solvation dynamics (Chapter 15), electron transfer in bulk solvents (Chapter 16) and at electrodes/electrolyte and metal/molecule/metal junctions (Chapter 17), and several processes pertaining to molecular spectroscopy in condensed phases (Chapter 18) are the main subjects discussed in this part.

This book discusses all three formalisms used in the study of finite temperature field theory, namely the imaginary time formalism, the closed time formalism and thermofield dynamics. Applications of the formalisms are worked out in detail. Gauge field theories and symmetry restoration at finite temperature are among the practical examples discussed in depth. The question of gauge dependence of the effective potential and the Nielsen identities are explained. The nonrestoration of some symmetries at high temperature (such as supersymmetry) and theories on nonsimply connected space-times are also described thoroughly. Other topics include (1+1)- and (2+1)-dimensional field theories at finite temperature and phase transitions, derivative expansion, linear response theory and the question of infrared divergences at finite temperature. In addition, examples of nonequilibrium phenomena are discussed with the disoriented chiral condensates as an illustration.This book is a very useful tool for graduate students, teachers and researchers in theoretical physics.

Bringing together the key ideas from nonequilibrium statistical mechanics and powerful methodology from quantum field theory, this book captures the essence of nonequilibrium quantum field theory. Beginning with the foundational aspects of the theory, the book presents important concepts and useful techniques, discusses issues of basic interest, and shows how thermal field, linear response, kinetic theories and hydrodynamics emerge. It also illustrates how these concepts and methodology are applied to current research topics including nonequilibrium phase transitions, thermalization in relativistic heavy ion collisions, the nonequilibrium dynamics of Bose-Einstein condensation, and the generation of structures from quantum fluctuations in the early Universe. Divided into five parts, with each part addressing a particular stage in the conceptual and technical development of the subject, this self-contained book is a valuable reference for graduate students and researchers in particle physics, gravitation, cosmology, atomic-optical and condensed matter physics.

Potts Models and Related Problems in Statistical Mechanics

This thesis make significant contributions to both the numerical and analytical aspects of particle physics, reducing the noise associated with matrix calculations in quantum chromodynamics (QCD) and modeling multi-quark mesonic matters that could be used to investigate particles previously unseen in nature. Several methods are developed that can reduce the statistical uncertainty in the extraction of hard-to-detect lattice QCD signals from disconnected diagrams. The most promising technique beats competing methods by 1700 percent, leading to a potential decrease in the computation time of quark loop quantities by an order of magnitude. This not only increases efficiency but also works for QCD matrices with almost-zero eigenvalues, a region where most QCD algorithms break down. This thesis also develops analytical solutions used to investigate exotic particles, specifically the Thomas-Fermi quark model, giving insight into possible new states formed from mesonic matter. The main benefit of this model is that it can work for a large number of quarks which is currently almost impossible with lattice QCD. Patterns of single-quark energies are observed which give the first a priori indication that stable octa-quark and hexadeca-quark versions of the charmed and bottom Z-meson exist.

The material for these volumes has been selected from the past twenty years' examination questions for graduate students at University of California at Berkeley, Columbia University, the University of Chicago, MIT, State University of New York at Buffalo, Princeton University and University of Wisconsin.

The material for these volumes has been selected from the past twenty years' examination questions for graduate students at University of California at Berkeley, Columbia University, the University of Chicago, MIT, State University of New York at Buffalo, Princeton University and University of Wisconsin.

This book is intended as a supplementary text to the standard course books on theoretical physics and astrophysics, addressing applications and selected problems in theoretical physics and astrophysics, most of which are to a greater or lesser extent associated with electrodynamics.

This book presents a new approach to learning the dynamics of particles and rigid bodies at an intermediate to advanced level. There are three distinguishing features of this approach. First, the primary emphasis is to obtain the equations of motion of dynamical systems and to solve them numerically. As a consequence, most of the analytical exercises and homework found in traditional dynamics texts written at this level are replaced by MATLAB (R)-based simulations. Second, extensive use is made of matrices. Matrices are essential to define the important role that constraints have on the behavior of dynamical systems. Matrices are also key elements in many of the software tools that engineers use to solve more complex and practical dynamics problems, such as in the multi-body codes used for analyzing mechanical, aerospace, and biomechanics systems. The third and feature is the use of a combination of Newton-Euler and Lagrangian (analytical mechanics) treatments for solving dynamics problems. Rather than discussing these two treatments separately, Engineering Dynamics 2.0 uses a geometrical approach that ties these two treatments together, leading to a more transparent description of difficult concepts such as "virtual" displacements. Some important highlights of the book include: Extensive discussion of the role of constraints in formulating and solving dynamics problems. Implementation of a highly unified approach to dynamics in a simple context suitable for a second-level course. Descriptions of non-linear phenomena such as parametric resonances and chaotic behavior. A treatment of both dynamic and static stability. Overviews of the numerical methods (ordinary differential equation solvers, Newton-Raphson method) needed to solve dynamics problems. An introduction to the dynamics of deformable bodies and the use of finite difference and finite element methods. Engineering Dynamics 2.0 provides a unique, modern treatment of dynamics problems that is directly useful in advanced engineering applications. It is a valuable resource for undergraduate and graduate students and for practicing engineers.

Physics underlies all complexity, including our own existence: how is this possible? How can our own lives emerge from interactions of electrons, protons, and neutrons? This book considers the interaction of physical and non-physical causation in complex systems such as living beings, and in particular in the human brain, relating this to the emergence of higher levels of complexity with real causal powers. In particular it explores the idea of top-down causation, which is the key effect allowing the emergence of true complexity and also enables the causal efficacy of non-physical entities, including the value of money, social conventions, and ethical choices.

Probability theory has been extraordinarily successful at describing a variety of phenomena, from the behaviour of gases to the transmission of messages, and is, besides, a powerful tool with applications throughout mathematics. At its heart are a number of concepts familiar in one guise or another to many: Gauss' bell-shaped curve, the law of averages, and so on, concepts that crop up in so many settings they are in some sense universal. This universality is predicted by probability theory to a remarkable degree. This book explains that theory and investigates its ramifications. Assuming a good working knowledge of basic analysis, real and complex, the author maps out a route from basic probability, via random walks, Brownian motion, the law of large numbers and the central limit theorem, to aspects of ergodic theorems, equilibrium and nonequilibrium statistical mechanics, communication over a noisy channel, and random matrices. Numerous examples and exercises enrich the text.

This book is the productof my research at the EndowedChair of Corporate Finance and Capital Markets at the European Business School. Many people devoted their inspiration, time, and knowledge in order to support me with my interdisciplinary research. Thank you all for your support. Especially, I thank my academic supervisor Prof. Ulrich Hommel, Ph.D., who provided me with the freedom to pursue my research interests and who always s- ported me generously. In addition, I am very grateful to my co-supervisor Prof. Dr. Susanne Strahringer (TU Dresden), who graciously accepted the task of refer- ing my book. Also, I thank Prof. Dr. Matthias Krause of the Chair for Theoretical Computer Science in Mannheim for the technical and computer scienti?c support. During my research visits I received generous support from Prof. Yaneer Bar- Yam (NECSI and Harvard) and Prof. John D. Sterman (MIT), who introduced me to the ?eld of System Dynamics and Complex Systems, as well as from Prof. Victor Mossotti (U.S. Geological Survey), Prof. Hiroki Sayama (NECSI and Bingh- ton), Prof. Markus de Aguiar (Universidade Estadual de Campinas), Prof. Michel Baranger (MIT), Prof. Jay W. Forrester (MIT), and Prof. Brian D. Josephson (Cambridge). Thank you very much for numerous inspiring discussions.

Humans engage in a seemingly endless variety of different behaviors, of which some are found across species, while others are conceived of as typically human. Most generally, behavior comes about through the interplay of various constraints - informational, mechanical, neural, metabolic, and so on - operating at multiple scales in space and time. Over the years, consensus has grown in the research community that, rather than investigating behavior only from bottom up, it may be also well understood in terms of concepts and laws on the phenomenological level. Such top down approach is rooted in theories of synergetics and self-organization using tools from nonlinear dynamics. The present compendium brings together scientists from all over the world that have contributed to the development of their respective fields departing from this background. It provides an introduction to deterministic as well as stochastic dynamical systems and contains applications to motor control and coordination, visual perception and illusion, as well as auditory perception in the context of speech and music.

This textbook provides an exposition of equilibrium thermodynamics and its applications to several areas of physics with particular attention to phase transitions and critical phenomena. The applications include several areas of condensed matter physics and include also a chapter on thermochemistry. Phase transitions and critical phenomena are treated according to the modern development of the field, based on the ideas of universality and on the Widom scaling theory. For each topic, a mean-field or Landau theory is presented to describe qualitatively the phase transitions. These theories include the van der Waals theory of the liquid-vapor transition, the Hildebrand-Heitler theory of regular mixtures, the Griffiths-Landau theory for multicritical points in multicomponent systems, the Bragg-Williams theory of order-disorder in alloys, the Weiss theory of ferromagnetism, the Neel theory of antiferromagnetism, the Devonshire theory for ferroelectrics and Landau-de Gennes theory of liquid crystals. This new edition presents expanded sections on phase transitions, liquid crystals and magnetic systems, for all problems detailed solutions are provided. It is intended for students in physics and chemistry and provides a unique combination of thorough theoretical explanation and presentation of applications in both areas. Chapter summaries, highlighted essentials and problems with solutions enable a self sustained approach and deepen the knowledge. It is intended for students in physics and chemistry and provides a unique combination of thorough theoretical explanation and presentation of applications in both areas. Chapter summaries, highlighted essentials and problems with solutions enable a self sustained approach and deepen the knowledge.

The European Conference on Complex Systems, held under the patronage of the Complex Systems Society, is an annual event that has become the leading European conference devoted to complexity science. ECCS'12, its ninth edition, took place in Brussels, during the first week of September 2012. It gathered about 650 scholars representing a wide range of topics relating to complex systems research, with emphasis on interdisciplinary approaches. More specifically, the following tracks were covered:

1. Foundations of Complex Systems

2. Complexity, Information and Computation

3. Prediction, Policy and Planning, Environment

4. Biological Complexity

5. Interacting Populations, Collective Behavior

6. Social Systems, Economics and Finance

This book contains a selection of the contributions presented at the conference and its satellite meetings. Its contents reflect the extent, diversity and richness of research areas in the field, both fundamental and applied.

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