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This course-based open access textbook delves into percolation theory, examining the physical properties of random media—materials characterized by varying sizes of holes and pores. The focus is on both the mathematical foundations and the computational and statistical methods used in this field. Designed as a practical introduction, the book places particular emphasis on providing a comprehensive set of computational tools necessary for studying percolation theory. Readers will learn how to generate, analyze, and comprehend data and models, with detailed theoretical discussions complemented by accessible computer codes. The book's structure ensures a complete exploration of worked examples, encompassing theory, modeling, implementation, analysis, and the resulting connections between theory and analysis. Beginning with a simplified model system—a model porous medium—whose mathematical theory is well-established, the book subsequently applies the same framework to realistic random systems. Key topics covered include one- and infinite-dimensional percolation, clusters, scaling theory, diffusion in disordered media, and dynamic processes. Aimed at graduate students and researchers, this textbook serves as a foundational resource for understanding essential concepts in modern statistical physics, such as disorder, scaling, and fractal geometry.

This open access book presents a comprehensive exploration of diffusion metamaterials that control energy and mass diffusion. Currently, if from the perspective of governing equations, diffusion metamaterials and wave metamaterials (pioneered by J. B. Pendry in the 1990s) are recognised as the two most prominent branches in the field of metamaterials. These two branches differ in their emphasis on the diffusion equation (as the governing equation) and time-dependent characteristic lengths in diffusion metamaterials, as opposed to the wave equation (as the governing equation) and time-independent characteristic lengths in wave metamaterials. Organized into three distinct parts – 'Thermal Diffusion Metamaterials', 'Particle Diffusion Metamaterials', and 'Plasma Diffusion Metamaterials' – this book offers a rigorous exploration spanning physics, engineering, and materials science, aimed at advancing our understanding of diffusion processes controlled by diffusion metamaterials. Incorporating foundational theory, computational simulations, and laboratory experiments, the book equips researchers and scholars across these disciplines with comprehensive methods, insights, and results pivotal to the advancement of diffusion control. Beyond facilitating interdisciplinary discourse, the book serves as a catalyst for innovative breakthroughs at the crossroads of physics, thermodynamics, and materials science. Essentially, readers will acquire profound insights that empower them to spearhead advancements in diffusion science (diffusionics) and the engineering of metamaterials.

This open access book focuses on how scientific methodologies can help industrial managers, entrepreneurs and policymakers handle the 17 Sustainable Development Goals in an efficient and realistic way. It also offers an operative scheme for scientists to overcome their discipline barriers. Is interdisciplinarity an intrinsic research value or is it merely instrumental for handling the increasing flux of open problems that sustainability poses to science?Can these problems of sustainability be solved with what the authors already know? Is it just a matter of having the right people at the table and giving them sufficient resources, or is it something more? Is meeting the needs of the present without compromising those of future generations a scientific definition of sustainable development? Questions similar to those posed in the sixties regarding complexity must be asked about sustainability today. In addition, the new data science includes powerful tools for making novel quantitative predictions about future sustainability indicators, an open problem that the book discusses. This book is primarily addressed to Ph.D. students, postdocs and senior researchers in the Life and Hard Science (LHS) and Social Sciences and Humanities (SSH) disciplines, as well as professionals of the primary, secondary and tertiary industrial sectors.

Students of particle physics often find it difficult to locate resources to learn calculational techniques. Intermediate steps are not usually given in the research literature. To a certain extent, this is also the case even in some of the textbooks. In this book of worked problems we have made an effort to provide enough details so that a student starting in the field will understand the solution in each case. Our hope is that with this step-by-step guidance, students (after first attempting the solution themselves) can develop their skill, and confidence in their ability, to work out particle theory problems.

This book explores Lieb's Simplified approach to the ground state of systems of interacting bosons. While extensive research has delved into the behavior of interacting bosons, persistent challenges, such as proving Bose-Einstein condensation, remain. Introduced by Lieb in 1963, the Simplified approach has been the object of renewed attention in recent years, revealing surprising and promising results. Notably, this approach provides ground state energy predictions that agree with many-body systems asymptotically at both low and high densities. It further predicts a condensate fraction and correlation function that agree with Bogolyubov theory at low densities, and numerical predictions match quantum Monte Carlo simulations across all densities. This suggests that Lieb's Simplified approach could serve as a potent tool for reimagining the study of interacting bosons. The book defines Lieb's Simplified approach, discusses its predictions, and presents known analytical and numerical results. It is designed for advanced students and young researchers working in the fields of mathematical physics, quantum many-body physics and Bose-Einstein condensates.

This book offers a fundamental description of physics using classical probabilities for events occurring at all times and locations throughout the Universe. The laws of quantum mechanics emerge naturally when focusing on a specific moment in time. Each step is explained in detail through simple systems, such as bits, Ising spins, or fermionic occupation numbers, allowing readers to grasp the core concepts of the modern functional integral approach to fundamental physics without requiring prior knowledge. Notably, probabilistic cellular automata are presented as intriguing quantum systems or as representations of fermionic quantum field theories. Embedding quantum mechanics within classical statistics opens up new avenues for computing, particularly through the use of correlation functions. A small neuromorphic computer, for instance, can learn basic quantum operations without the typical requirement for extreme isolation, as is the case with conventional quantum computers. On a philosophical level, the book proposes a fresh perspective on science. The content is aimed at readers with a foundational understanding of physics, suitable for advanced students.

A unified introduction to information theory for scientistsApplications of information theory span a broad range of disciplines today. This book presents a unified treatment of the subject for students and practitioners in the sciences. It teaches the tools universally used by physicists working on quantum computers and black holes, engineers designing self-driving cars, traders perfecting market strategies, chemists playing with molecules, biologists studying cells and living beings, linguists analyzing languages, and neuroscientists figuring out how the brain works. No matter what area of science you specialize in, The Physical Nature of Information unlocks the power of information theory to test the limits imposed by uncertainty.Provides a panoramic approach to information theoryDraws on examples from physics, engineering, biology, economics, and linguisticsApplications range from thermodynamics and statistical mechanics to dynamical chaos, information and communication theories, and quantum informationIncludes materials for lectures and tutorials along with exercises with detailed solutionsCan be used to design a one-semester introductory courseIdeal for self-study by graduate students and advanced undergraduatesInvaluable for scholars seeking new research opportunities

This book provides a modern pedagogical exposition of the mechanical approach to statistical mechanics initiated by Boltzmann with his early works (1866–1871). Despite the later contribution by Helmholtz, Boltzmann himself (1884–1887), Gibbs, P. Hertz, and Einstein, the mechanical approach remained almost unknown to the modern reader, in favour of the celebrated combinatorial approach, developed by Boltzmann himself during his probabilistic turn (1876–1884). The brief constitutes an ideal continuation of a graduate course of classical mechanics and requires knowledge of basic calculus in many dimensions (including differential forms), thermodynamics, and probability theory, besides Hamiltonian mechanics. The cornerstone of the whole presentation is the ergodic hypothesis. Special attention is devoted to Massieu potentials (the Legendre transforms of the entropy) which are most natural in statistical mechanics and also allow for a more direct treatment of the topic of ensemble equivalence. In this second edition, a chapter is added that addresses the long-debated question of how the second law of thermodynamics can be reconciled with mechanics, by using modern methods of non-equilibrium statistical mechanics.

This book delves into the foundational principles governing the treatment of molecular networks and'chemical space'—the comprehensive domain encompassing all physically achievable molecules—from the perspectives of vector space, graph theory, and data science. It explores similarity kernels, network measures, spectral graph theory, and random matrix theory, weaving intriguing connections between these diverse subjects. Notably, it emphasizes the visualization of molecular networks. The exploration continues by delving into contemporary generative deep learning models, increasingly pivotal in the pursuit of new materials possessing specific properties, showcasing some of the most compelling advancements in this field. Concluding with a discussion on the meanings of discovery, creativity, and the role of artificial intelligence (AI) therein.Its primary audience comprises senior undergraduate and graduate students specializing in physics, chemistry, and materials science. Additionally, it caters to those interested in the potential transformation of material discovery through computational, network, AI, and machine learning (ML) methodologies.

This open access book offers a concise overview of how data from large scale experiments are analyzed and how technological tools are used in practice, as in the search for new elementary particles. It focuses on interconnects between physics and detector technology in experimental particle physics, and includes descriptions of mathematical approaches. Readers find all the important steps in analysis, including reconstruction of the momentum and energy of particles from detector information, particle identification, and also the general concept of simulating particle production from collisions and detector responses. As the scale of scientific experiments becomes larger and data-intensive science emerges, the techniques used in the data analysis become ever more complicated, making it difficult for beginners to grasp the overall picture. The book provides an explanation of the idea and concepts behind the methods, helping readers understand journal articles on high energy physics. This book is engaging as it does not overemphasize mathematical formalism and it gives a lively example of how such methods have been applied to the Higgs particle discovery in the Large Hadron Collider (LHC) experiments, which led to Englert and Higgs being awarded the Nobel Prize in Physics for 2013. Graduate students and young researchers can easily obtain the required knowledge on how to start data analyses from these notes, without having to spend time in consulting many experts or digesting huge amounts of literature.

This textbook offers an introduction to statistical mechanics, special relativity, and quantum physics, developed from lecture notes for the'Quantum Physics'course at the University of Padua. Beginning with a brief review of classical statistical mechanics in the first chapter, the book explores special and general relativity in the second chapter. The third chapter delves into the historical analysis of light quantization, while the fourth chapter discusses Niels Bohr's quantization of energy levels and electromagnetic transitions. The Schrödinger equation is investigated in the fifth chapter. Chapter Six covers applications of quantum mechanics, including the quantum particle in a box, quantum particle in harmonic potential, quantum tunneling, stationary perturbation theory, and time-dependent perturbation theory. Chapter Seven outlines the basic axioms of quantum mechanics. Chapter Eight focuses on quantum atomic physics, emphasizing electron spin and utilizing the Dirac equation for theoretical justification. The ninth chapter explains quantum mechanics principles for identical particles at zero temperature, while Chapter Ten extends the discussion to quantum particles at finite temperature. Chapter Eleven provides insights into quantum information and entanglement, and the twelfth chapter explains the path integral approach to quantum mechanics.

This text presents a collection of mathematical exercises with the aim of guiding readers to study topics in statistical physics, equilibrium thermodynamics, information theory, and their various connections. It explores essential tools from linear algebra, elementary functional analysis, and probability theory in detail and demonstrates their applications in topics such as entropy, machine learning, error-correcting codes, and quantum channels. The theory of communication and signal theory are also in the background, and many exercises have been chosen from the theory of wavelets and machine learning. Exercises are selected from a number of different domains, both theoretical and more applied. Notes and other remarks provide motivation for the exercises, and hints and full solutions are given for many. For senior undergraduate and beginning graduate students majoring in mathematics, physics, or engineering, this text will serve as a valuable guide as theymove on to more advanced work.

This edited volume collects nine surveys that present the state-of-the-art in modeling, qualitative analysis, and simulation of active particles, focusing on specific applications in the natural sciences. As in the preceding Active Particles volumes, it blends diverse applications that demonstrate the interdisciplinary nature of the subject and the various mathematical tools available. Contributions were selected with the aim of covering a variety of viewpoints, from modeling the interactions in collective dynamics of animals and in population dynamics; through neural-networks, semi-supervised learning, and Monte Carlo methods in optimization; to kinetic and continuum theories with applications to aggregations and birth-and-death processes. 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 textbook is based on lecture notes that the author delivered at Qiuzhen College (Tsinghua University), a Chinese institution known for its exceptionally talented mathematics students. The book's intended audience shapes its character. It introduces Statistical Mechanics from the ground up, offering a fully self-contained presentation that aims for mathematical precision. It distinguishes rigorous results from controlled approximations and provides physical insights into phenomena. Despite its concise nature (suited for a one-semester basic course), this book covers several topics typically not found in introductory texts. These include Shannon's information-theoretic interpretation of entropy, the gauge approach to order-disorder duality in the Ising model, the Yang-Lee theory, and the quantum dissipation-fluctuation theorem. Additionally, it explores frustrated and quenched systems, including an introduction to the celebrated Parisi solution of the Sherrington-Kirkpatrick model of spin glasses. The path integral formalism is extensively discussed from various perspectives to suit different applications. Chapter 2 approaches path integrals through the Feynman-Kac formula and second quantization. In Chapter 5, they are examined within the context of effective field theories like Landau-Ginzburg theory, while Chapter 6 delves into their connection with Brownian motion, Langevin stochastic differential equations, and Fokker-Planck diffusion PDEs. The book also explores the relationship between stochastic processes and supersymmetry. Various techniques for computing path integrals, especially functional determinants, are introduced throughout the relevant chapters, offering the most suitable computational tools for each application.

A look at Statistical Physics by renowned Russia-Soviet physicist Yuri Klimontovich of the Moscow State University, MSU Faculty of Physics.

This book serves as an introduction to the growing field of quantum many-body transport in interacting nanojunctions. It delves into a theoretical approach based on a general density-matrix formulation for open quantum systems. In the book, relevant transport observables, like the current or its higher order cumulants, are obtained by evaluating quantum statistical averages. This approach requires the knowledge of the reduced density matrix of the interacting nanosystems. The formulation for addressing transport problems, based on the evolution of the reduced density operator in Liouville space, is highly versatile. It enables the treatment of charge and spin transport across various realistic nanostructures. Topics encompass standard Coulomb blockade, cotunneling phenomena in quantum dots, vibrational and Franck-Condon effects in molecular junctions, as well as many-body interference observed in double quantum dots or carbon nanotubes. Derived from lectures tailored for graduate and advanced students at the University of Regensburg in Germany, this book is enriched with exercises and step-by-step derivations.

This book presents the experimental development of an information-powered engine inspired by the famous thought experiment, Maxwell's demon, to understand its potential to produce energy for practical purposes. The development of an engine based on Maxwell's demon was for a long time inconceivable, but technological advances have led to novel investigations into theoretical and practical applications. The built information engine consists of a micron-sized glass bead trapped in a tightly focused laser beam. It rectifies the bead's Brownian motion by controlling the laser's position and generates a unidirectional motion against gravity without doing any work, thus converting thermal heat into stored gravitational potential energy. A theoretical model based on a spring-mass system describes the engine's dynamics and was then used to find optimum parameters to improve the engine's performance. Experimentally implementing these optimization strategies led to engine output powers comparable to those measured in biological motors. This book also highlights performance improvements made in the presence of measurement noise and presents important guiding principles to design information engines to operate in non-equilibrium environments. By focusing on practical applications, the book overall aims to broaden the scope of information-engine investigations.

This book presents an introduction to the main concepts of statistical physics, followed by applications to specific problems and more advanced concepts, selected for their pedagogical or practical interest. Particular attention has been devoted to the presentation of the fundamental aspects, including the foundations of statistical physics, as well as to the discussion of important physical examples. Comparison of theoretical results with the relevant experimental data (with illustrative curves) is present through the entire textbook. This aspect is facilitated by the broad range of phenomena pertaining to statistical physics, providing example issues from domains as varied as the physics of classical and quantum liquids, condensed matter, liquid crystals, magnetic systems, astrophysics, atomic and molecular physics, superconductivity and many more. This textbook is intended for graduate students (MSc and PhD) and for those teaching introductory or advanced courses on statistical physics.Key Features: A rigorous and educational approach of statistical physics illustrated with concrete examples. A clear presentation of fundamental aspects of statistical physics. Many exercises with detailed solutions. Nicolas Sator is Associate Professor at Sorbonne University, Paris, France. He is a member of the Laboratory of Theoretical Physics of Condensed Matter (LPTMC) and his research focuses on the physics of liquids.Nicolas Pavloff is Professor at Paris-Saclay University, France. He is a member of Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS) and his domain of research is quantum fluid theory.Lénaïc Couëdel is Professor at the University of Sasktchewan, Saskatoon, Canada and researcher at CNRS, France. His research area is plasma physics with a focus on complex plasma crystals.

This up-to-date overview of Bayesian nonparametric statistics provides both an introduction to the field and coverage of recent research topics, including deep neural networks, high-dimensional models and multiple testing, Bernstein-von Mises theorems and variational Bayes approximations, many of which have previously only been accessible through research articles. Although Bayesian posterior distributions are widely applied in astrophysics, inverse problems, genomics, machine learning and elsewhere, their theory is still only partially understood, especially in complex settings such as nonparametric or semiparametric models. Here, the available theory on the frequentist analysis of posterior distributions is outlined in terms of convergence rates, limiting shape results and uncertainty quantification. Based on lecture notes for a course given at the St-Flour summer school in 2023, the book is aimed at researchers and graduate students in statistics and probability.

This book provides a captivating journey through the realms of classical and quantum systems as it unravels the profound influence that noise may have on their static and dynamic properties. The first part of the book offers succinct yet enlightening discussions on foundational topics related to noise. The second part focuses on a variety of applications, where a diverse spectrum of noise effects in physical systems comes to life, meticulously presented and thoughtfully analyzed. Whether you are a curious student or a dedicated researcher, this book is your key to gaining invaluable insights into noise effects in physical systems. “The book has the merit of presenting several topics scattered in the literature and could become a very useful reference.” Giovanni Jona-Lasinio, Sapienza – Università di Roma, Italy

This book analyses the physics of complex systems to elaborate the problems encountered in teaching and research. Inspired by the of Kurt Gödel (including his incompleteness theorems) it considers the concept of time, the idea of models and the concept of complexity before trying to assess the state of physics in general. Using both general and practical examples, the idea of information is discussed, emphasizing its physical interpretation, debates ideas in depth using examples and evidence to provide detailed considerations on the topics. Based on the authors'own research on these topics, this book puts forward the idea that the application of information measures can provide new results in the study of complex systems. Helpful for those already familiar with the concepts who wish to deepen their critical understanding, Physics of Complex Systems will be extremely valuable both for people that are already involved in complex systems and also readers beginning their journey into the subject. This work will encourage readers to follow and continue these ideas, enabling them to investigate the various topics further.

This thesis combines methods from statistical physics and nonlinear dynamics to advance research on the pattern formation in active fluids in several directions. In particular, it focuses on mesoscale turbulence, a state observed in microswimmer suspensions, which is characterized by the emergence of dynamic vortex patterns. The first major contribution concerns the bottom-up derivation of a frequently used continuum model of mesoscale turbulence from a set of particle-resolved stochastic equations. Utilizing the model, mesoscale turbulence is shown to induce nontrivial transport properties including a regime of optimal diffusion. The thesis then explores possible strategies of control. One of these relies on an external field that leads to stripe-like structures and can even suppress patterns entirely. The other involves geometric confinement realized by strategically placed obstacles that can reorganize the flow into a variety of ordered vortex structures. The turbulence transition inside an obstacle lattice is shown to have an intriguing analogy to an equilibrium transition in the Ising universality class. As a whole, this thesis provides important contributions to the understanding and control of turbulence in active fluids, as well as outlining exciting future directions, including applications. It includes a substantial introduction to the topic, which is suitable for newcomers to the field.

This textbook serves to teach readers about the origins of life, the probabilistic process of self-assembly underpinning all living systems, from a biophysics perspective. The author cohesively summarizes the various organizing principles that led to the development of an ordered physical basis on which the evolution of life operates. This book answers critical questions, such as why life depends on the properties of inanimate objects and how the laws of physics, chemistry, and biology convolved to spontaneously produce the periodic table and, of course, life itself. Readers are provided with an introduction to probability distributions as well as detailed descriptions of important concepts in thermodynamics, statistical mechanics, and quantum mechanics. As the book progresses, an understanding for the inevitability of life is developed through topics such as stellar nucleosynthesis and prebiotic evolution. Each chapter also includes problems for readers to gain a better understanding of the material. This textbook is accessible to students and researchers of all levels and serves as a comprehensive guide on the physics behind abiogenesis.

This book presents experimental studies of nonequilibrium phase transitions induced by ac and dc forces in collectively interacting systems—a superconducting vortex system with random pinning. It first shows that a phase transition from reversible to irreversible flow occurs by increasing vortex density as well as amplitude of ac shear, which is indicative of the universality of the reversible-irreversible transition. Two distinct flow regimes are also found in the reversible phase. Next, the book presents new methods for dc driven experiments—transverse mode-locking and transverse current-voltage measurements—and provides convincing evidence of the second-order dynamical transition from disordered plastic to anisotropically ordered smectic flow. Lastly it reports on the first experimental demonstration of the Kibble-Zurek mechanism for the nonequilibrium phase transition.The experimental results indicate that both the reversible-irreversible transition and the dynamical ordering transition belong to the directed percolation universality class which is one of the fundamental classes of nonequilibrium phase transitions. Hence, the findings will be generalized to other nonequilibrium systems and stimulate research on nonequilibrium physics.

This book explores topics in multivariate statistical analysis, relevant in the real and complex domains. It utilizes simplified and unified notations to render the complex subject matter both accessible and enjoyable, drawing from clear exposition and numerous illustrative examples. The book features an in-depth treatment of theory with a fair balance of applied coverage, and a classroom lecture style so that the learning process feels organic. It also contains original results, with the goal of driving research conversations forward. This will be particularly useful for researchers working in machine learning, biomedical signal processing, and other fields that increasingly rely on complex random variables to model complex-valued data. It can also be used in advanced courses on multivariate analysis. Numerous exercises are included throughout.

This volume celebrates the over fifty-year career in non-equilibrium statistical physics of Professor Paolo Grigolini of the Center for Nonlinear Science at the University of North Texas. It begins by positioning Grigolini in a five-dimensional science-personality space with the following axes: Sleeper, Keeper, Leaper, Creeper and Reaper. This introduction to the person is followed by a sequence of papers in the various areas of science where his work has had impact, including subtle questions concerned with the connection between classical and quantum systems; a two-level atom coupled to a radiation field; classical probability calculus; anomalous diffusion that is Brownian yet non-Gaussian; a new method for detecting scaling in time series; and the effect of strong Anderson localization on ultrasound transmission, among other topics.

This book could not have been timelier. It describes a multidisciplinary experimental work reported in the literature from 2015 to 2022, supported by a theoretical proposal from 2006, exploiting random lasers and random fiber lasers as a photonic platform to perform statistical physics, as Lévy-like statistics and extreme events, as well as complex systems, including turbulence, replica symmetry breaking (RSB) and Floquet states. Most of the theoretical grounds for these subjects date back to the 1970s. Of particular relevance for the timing for this book is the fact that two of the Nobel Prize winners of 2021 have their work connected through the experimental and theoretical work exploiting random lasers. In fact, the very first demonstration of RSB, a theory proposed by Giorgio Parisi, one of the 2021 Nobel winners, was first experimentally demonstrated in 2015 using random lasers. The scope of the book relies on the description of the already vast literature starting in 2006, but with an experimental explosion since 2015. The book describes the basis of random lasers and random fibers, theoretical background and connection between magnetism and photonics related to RSB, and theoretical backgrounds for experiments in Lévy statistics, turbulence, and Floquet states. The contributors are from three of the groups with most contributions in the field.

Hermite's theorem makes it known that there are three levels of mathematical frames in which a simple addition formula is valid. They are rational, q-analogue, and elliptic-analogue. Based on the addition formula and associated mathematical structures, productive studies have been carried out in the process of q-extension of the rational (classical) formulas in enumerative combinatorics, theory of special functions, representation theory, study of integrable systems, and so on. Originating from the paper by Date, Jimbo, Kuniba, Miwa, and Okado on the exactly solvable statistical mechanics models using the theta function identities (1987), the formulas obtained at the q-level are now extended to the elliptic level in many research fields in mathematics and theoretical physics. In the present monograph, the recent progress of the elliptic extensions in the study of statistical and stochastic models in equilibrium and nonequilibrium statistical mechanics and probability theory is shown. At the elliptic level, many special functions are used, including Jacobi's theta functions, Weierstrass elliptic functions, Jacobi's elliptic functions, and others. This monograph is not intended to be a handbook of mathematical formulas of these elliptic functions, however. Thus, use is made only of the theta function of a complex-valued argument and a real-valued nome, which is a simplified version of the four kinds of Jacobi's theta functions. Then, the seven systems of orthogonal theta functions, written using a polynomial of the argument multiplied by a single theta function, or pairs of such functions, can be defined. They were introduced by Rosengren and Schlosser (2006), in association with the seven irreducible reduced affine root systems. Using Rosengren and Schlosser's theta functions, non-colliding Brownian bridges on a one-dimensional torus and an interval are discussed, along with determinantal point processes on a two-dimensional torus. Their scaling limitsare argued, and the infinite particle systems are derived. Such limit transitions will be regarded as the mathematical realizations of the thermodynamic or hydrodynamic limits that are central subjects of statistical mechanics.

This book presents a broad and well-structured overview of various non-Fourier heat conduction models. The classical Fourier heat conduction model is valid for most macroscopic problems. However, it fails when the wave nature of the heat propagation becomes dominant and memory or non-local spatial effects become significant; e.g., during ultrafast heating, heat transfer at the nanoscale, in granular and porous materials, at extremely high values of the heat flux, or in heat transfer in biological tissues. The book looks at numerous non-Fourier heat conduction models that incorporate time non-locality for materials with memory, such as hereditary materials, including fractional hereditary materials, and/or spatial non-locality, i.e. materials with a non-homogeneous inner structure. Beginning with an introduction to classical transport theory, including phase-lag, phonon, and thermomass models, the book then looks at various aspects of relativistic and quantum transport, including approaches based on the Landauer formalism as well as the Green-Kubo theory of linear response. Featuring an appendix that provides an introduction to methods in fractional calculus, this book is a valuable resource for any researcher interested in theoretical and numerical aspects of complex, non-trivial heat conduction problems.

This book is about the Random Field Ising Model (RFIM) – a paradigmatic spin model featuring a frozen disordering field. The focus is on the second-order phase transition between the paramagnetic and ferromagnetic phases, and the associated critical exponents. The book starts by summarizing the current knowledge about the RFIM from experiments, numerical simulations and rigorous mathematical results. It then reviews the classic theoretical works from the 1970's which suggested a property of dimensional reduction – that the RFIM critical exponents should be the same as for the ordinary, non-disordered, Ising model of lower dimensionality, and related this an emergent Parisi-Sourlas supersymmetry. As is now known, these remarkable properties only hold when the spatial dimensionality of the model is larger than a critical dimension. The book presents a method to estimate the critical dimension, using standard tools such as the replica trick and perturbative renormalization group, whose result is in agreement with the numerical simulations. Some more elementary steps in the derivations are left as exercises for the readers. This book is of interest to researchers, PhD students and advanced master students specializing in statistical field theory.

Stochastic mechanics is a theory that holds great promise in resolving the mathematical and interpretational issues encountered in the canonical and path integral formulations of quantum theories. It provides an equivalent formulation of quantum theories, but substantiates it with a mathematically rigorous stochastic interpretation by means of a stochastic quantization prescription.The book builds on recent developments in this theory, and shows that quantum mechanics can be unified with the theory of Brownian motion in a single mathematical framework. Moreover, it discusses the extension of the theory to curved spacetime using second order geometry, and the induced Itô deformations of the spacetime symmetries.The book is self-contained and provides an extensive review of stochastic mechanics of the single spinless particle. The book builds up the theory on a step by step basis. It starts, in chapter 2, with a review of the classical particle subjected to scalar and vector potentials. In chapter 3, the theory is extended to the study of a Brownian motion in any potential, by the introduction of a Gaussian noise. In chapter 4, the Gaussian noise is complexified. The result is a complex diffusion theory that contains both Brownian motion and quantum mechanics as a special limit. In chapters 5, the theory is extended to relativistic diffusion theories. In chapter 6, the theory is further generalized to the context of pseudo-Riemannian geometry. Finally, in chapter 7, some interpretational aspects of the stochastic theory are discussed in more detail. The appendices concisely review relevant notions from probability theory, stochastic processes, stochastic calculus, stochastic differential geometry and stochastic variational calculus.The book is aimed at graduate students and researchers in theoretical physics and applied mathematics with an interest in the foundations of quantum theory andBrownian motion. The book can be used as reference material for courses on and further research in stochastic mechanics, stochastic quantization, diffusion theories on curved spacetimes and quantum gravity.

This textbook presents the fundamental concepts and theories in thermal physics and elementary statistical mechanics in a very simple, systematic and comprehensive way. This book is written in a way that it presents the topics in a holistic manner with end-of-chapter exercises and examples where concepts are supported by numerous solved examples and multiple-choice questions to aid self-learning. The textbook also contains illustrated diagrams for better understanding of the concepts. The book will benefit students who are taking introductory courses in thermal physics, thermodynamics and statistical mechanics.

This volume of the Encyclopedia of Complexity and Systems Science, Second Edition, focuses on current challenges in the field from materials and mechanics to applications of statistical and nonlinear physics in the life sciences. Challenges today are mostly in the realm of non-equilibrium systems, although certain equilibrium systems also present serious hurdles. Where possible, pairwise articles focus on a single topic, one from a theoretical perspective and the other from an experimental one, providing valuable insights. In other cases, theorists and experimentalists have collaborated on a single article. Coverage includes both quantum and classical systems, and emphasizes 1) mature fields that are not covered in the current specialist literature, (2) topics that fall through the cracks in disciplinary journals/books, or (3) developing areas where the knowledge base is large and robust and upon which future developments will depend. The result is an invaluable resource for condensed matter physicists, material scientists, engineers and life scientists.

This book investigates statistical observables for anomalous and nonergodic dynamics, focusing on the dynamical behaviors of particles modelled by non-Brownian stochastic processes in the complex real-world environment.Statistical observables are widely used for anomalous and nonergodic stochastic systems, thus serving as a key to uncover their dynamics. This study explores the cutting edge of anomalous and nonergodic diffusion from the perspectives of mathematics, computer science, statistical and biological physics, and chemistry. With this interdisciplinary approach, multiple physical applications and mathematical issues are discussed, including stochastic and deterministic modelling, analyses of (stochastic) partial differential equations (PDEs), scientific computations and stochastic analyses, etc. Through regularity analysis, numerical scheme design and numerical experiments, the book also derives the governing equations for the probability density function of statistical observables, linking stochastic processes with PDEs.The book will appeal to both researchers of electrical engineering expert in the niche area of statistical observables and stochastic systems and scientists in a broad range of fields interested in anomalous diffusion, especially applied mathematicians and statistical physicists.

This first of a kind textbook provides computational tools in Fortran 90 that are fundamental to quantum information, quantum computing, linear algebra and one dimensional spin half condensed matter systems. Over 160 subroutines are included, and the numerical recipes are aided by detailed flowcharts. Suitable for beginner and advanced readers alike, students and researchers will find this textbook to be a helpful guide and a compendium.Key Features: Includes 160 subroutines all of which can be used either as a standalone program or integrated with any other main program without any issues. Every parameter in the input, output and execution has been provided while keeping both beginner and advanced users in mind. The output of every program is explained thoroughly with detailed examples. A detailed dependency chart is provided for every recipe.

This book addresses research challenges in the rapidly developing area of nonequilibrium thermodynamics and fluctuation kinetics. This cross-disciplinary field comprises various topics, ranging from fundamental problems of nonequilibrium statistical mechanics and thermodynamics to multiple applications in plasma, fluid mechanics, nonlinear science, systems of dissipative particles, and high-Q resonators. The purpose of this book is to bring together world-leading experts in the above fields to initiate a cross-fertilization among these active research areas. The book is dedicated to and honours the memory of Professor Slava Belyi who passed away unexpectedly on May 20, 2020. He was pioneering the theory of nonequilibrium fluctuations, in particular the application of the Callen-Welton fluctuation-dissipation theorem to nonequilibrium systems and its generalization. This and related problems also feature in the book.

To understand phenomena in nature, it is important to focus not only on properties of stationary states, but also their changes in time, that is, the dynamics between bistable states. This book reviews the mechanics of first-order phase transitions and discusses relaxation and collapses of metastable states from various viewpoints, including Kramers'method for the lifetime of metastability, Langer's analysis on the singularity, effects of thermal fluctuation studied by Néel and Brown, and eigenvalue structures of the transfer-matrix for the phase transitions. The book also goes into the mechanics of metastability in quantum systems from the viewpoints of the eigenvalue problem of the Hamiltonian and the Liouvillian for a dynamical process and discusses relations between quantum tunneling processes and metastability therein. Lastly, the coercivity of magnets consisting of an ensemble of grains is reviewed. The book is beneficial for those new in the field as a primer on first-order phase transition from modern perspectives. The comprehensive content offers overviews of related topics and allows readers to quickly catch up with developments in the field.

This book highlights the importance of Electron Statistics (ES), which occupies a singular position in the arena of solid state sciences, in heavily doped (HD) nanostructures by applying Heisenberg's Uncertainty Principle directly without using the complicated Density-of-States function approach as given in the literature. The materials considered are HD quantum confined nonlinear optical, III-V, II-VI, IV-VI, GaP, Ge, PtSb2, stressed materials, GaSb, Te, II-V, Bi2Te3, lead germanium telluride, zinc and cadmium diphosphides, and quantum confined III-V, IV-VI, II-VI and HgTe/CdTe super-lattices with graded interfaces and effective mass super-lattices. The presence of intense light waves in optoelectronics and strong electric field in nano-devices change the band structure of materials in fundamental ways, which have also been incorporated in the study of ES in HD quantized structures of optoelectronic compounds that control the studies of the HD quantum effect devices under strong fields. The influence of magnetic quantization, magneto size quantization, quantum wells, wires and dots, crossed electric and quantizing fields, intense electric field, and light waves on the ES in HD quantized structures and superlattices are discussed. The content of this book finds six different applications in the arena of nano-science and nanotechnology and the various ES dependent electronic quantities, namely the effective mass, the screening length, the Einstein relation and the elastic constants have been investigated. This book is useful for researchers, engineers and professionals in the fields of Applied Sciences, solid state and materials science, nano-science and technology, condensed matter physics, and allied fields, including courses in semiconductor nanostructures. ​

The study of most scientific fields now relies on an ever-increasing amount of data, due to instrumental and experimental progress in monitoring and manipulating complex systems made of many microscopic constituents. How can we make sense of such data, and use them to enhance our understanding of biological, physical, and chemical systems? Aimed at graduate students in physics, applied mathematics, and computational biology, the primary objective of this textbook is to introduce the concepts and methods necessary to answer this question at the intersection of probability theory, statistics, optimisation, statistical physics, inference, and machine learning. The second objective of this book is to provide practical applications for these methods, which will allow students to assimilate the underlying ideas and techniques. While readers of this textbook will need basic knowledge in programming (Python or an equivalent language), the main emphasis is not on mathematical rigour, but on the development of intuition and the deep connections with statistical physics.

This book covers recent developments in the understanding, quantification, and exploitation of entanglement in spin chain models from both condensed matter and quantum information perspectives. Spin chain models are at the foundation of condensed matter physics and quantum information technologies and elucidate many fundamental phenomena such as information scrambling, quantum phase transitions, and many-body localization. Moreover, many quantum materials and emerging quantum devices are well described by spin chains. Comprising accessible, self-contained chapters written by leading researchers, this book is essential reading for graduate students and researchers in quantum materials and quantum information. The coverage is comprehensive, from the fundamental entanglement aspects of quantum criticality, non-equilibrium dynamics, classical and quantum simulation of spin chains through to their experimental realizations, and beyond into machine learning applications.

This Primer focusses on the statistical physics of classical and quantum systems. The course text explores the three cornerstones of statistical physics which include the Boltzmann, Fermi-Dirac, and Bose-Einstein distribution laws. It also provides a highly useful and in-depth investigation of the thermal properties of paradigmatically important systems such as classical ideal gas, electron gas and phonon gas. The structure of this text is tailored to facilitate the planning of a one-semester course: this volume provides bachelor students with a main teaching tool. More specifically, each part identifies an independent teaching module, while each chapter corresponds to approximately two weeks of lecturing.Key FeaturesFills a gap for a self-contained undergraduate textbook in statistical physics of condensed matter systemsTailored for a one-semester courseFocuses on a selected set of topics, whilst also providing substantial, in-depth coverage of the subjectEmphasises phenomenology rather than mathematics/formalismUses various pedagogical features, including chapter outline, proof-of-concept pictures

This book offers an interdisciplinary theoretical approach based on non-equilibrium statistical thermodynamics and control theory for mathematically modeling shock-induced out-of-equilibrium processes in condensed matter. The book comprises two parts. The first half of the book establishes the theoretical approach, reviewing fundamentals of non-equilibrium statistical thermodynamics and control theory of adaptive systems. The latter half applies the presented approach to a problem on shock-induced plane wave propagation in condensed matter. The result successfully reproduces the observed feature of waveform propagation in experiments, which conventional continuous mechanics cannot access. Further, the consequent stress–strain relationships derived with relaxation and inertia effect in elastic–plastic transition determines material properties in transient regimes.

This book highlights a comprehensive introduction to the fundamental statistical mechanics underneath the inner workings of neural networks. The book discusses in details important concepts and techniques including the cavity method, the mean-field theory, replica techniques, the Nishimori condition, variational methods, the dynamical mean-field theory, unsupervised learning, associative memory models, perceptron models, the chaos theory of recurrent neural networks, and eigen-spectrums of neural networks, walking new learners through the theories and must-have skillsets to understand and use neural networks. The book focuses on quantitative frameworks of neural network models where the underlying mechanisms can be precisely isolated by physics of mathematical beauty and theoretical predictions. It is a good reference for students, researchers, and practitioners in the area of neural networks.

It is generally believed that collisions of particles reduce the self-diffusion coefficient. In this book, Erik Kalz shows that in classical systems under the effect of Lorentz force, which are characterized by diffusion tensors with antisymmetric elements, collisions surprisingly can enhance self-diffusion. In these systems, due to an inherent curving effect, the motion of particles is facilitated, instead of hindered by collisions. Consistent with this the author finds that the collective diffusion remains unaffected. Using a geometric model, he theoretically predicts a magnetic field governed crossover from a reduced to an enhanced self-diffusion. The physical interpretation is quantitatively supported by the force autocorrelation function, which turns negative with increasing the magnetic field. Using Brownian-dynamics simulations, he validates the predictions.