The book surveys the state-of-the-art methods that are currently available to model and simulate the presence of rigid particles in a fluid flow. For particles that are very small relative to the characteristic flow scales and move without interaction with other particles, effective equations of motion for particle tracking are formulated and applied (e.g. in gas-solid flows). For larger particles, for particles in liquid-solid flows and for particles that interact with each other or possibly modify the overall flow detailed model are presented. Special attention is given to the description of the approximate force coupling method (FCM) as a more general treatment for small particles, and derivations in the context of low Reynolds numbers for the particle motion as well as application at finite Reynolds numbers are provided. Other topics discussed in the book are the relation to higher resolution immersed boundary methods, possible extensions to non-spherical particles and examples of applications of such methods to dispersed multiphase flows.

This book lays out a vision for a coherent framework for understanding complex systems. By developing the genuine idea of Brownian agents, the author combines concepts from informatics, such as multiagent systems, with approaches of statistical many-particle physics. It demonstrates that Brownian agent models can be successfully applied in many different contexts, ranging from physicochemical pattern formation to swarming in biological systems.

This book introduces a class of alignment models based on the so-called Cucker-Smale system as well as its kinetic and hydrodynamic counterparts. Cutting edge research in the area of collective behavior is presented, including emerging techniques from fluid mechanics, fractional analysis, and kinetic theory. Analytical aspects are highlighted throughout, such as regularity theory and long time behavior of solutions. Featuring open problems, readers will be motivated to apply these breakthrough methods to future research. The chapters offer an overview of state of the art research with introductions to core concepts. Chapter One introduces the central focus of the book: The agent-based Cucker-Smale system. Further agent-based systems and alignment systems are covered in chapters Two and Three. Following this are chapters covering the kinetic and hydrodynamic variants of the Cucker-Smale system. The core well-posedness theory of both smooth and singular models is then presented. Chapter Eight discusses the fully developed one-dimensional theory. The final chapter presents some of the known partial results concerning the regularity of multidimensional Euler Alignment systems. Dynamics and Analysis of Alignment Models of Collective Behavior is ideal for graduate students and researchers studying PDEs, especially those interested in the active areas of collective behavior and alignment models.

Using examples from finance and modern warfare to the flocking of birds and the swarming of bacteria, the collected research in this volume demonstrates the common methodological approaches and tools for modeling and simulating collective behavior. The topics presented point toward new and challenging frontiers of applied mathematics, making the volume a useful reference text for applied mathematicians, physicists, biologists, and economists involved in the modeling of socio-economic systems.

Understanding the behavior of particles suspended in a fluid has many important applications across a range of fields, including engineering and geophysics. Comprising two main parts, this book begins with the well-developed theory of particles in viscous fluids, i.e. microhydrodynamics, particularly for single- and pair-body dynamics. Part II considers many-body dynamics, covering shear flows and sedimentation, bulk flow properties and collective phenomena. An interlude between the two parts provides the basic statistical techniques needed to employ the results of the first (microscopic) in the second (macroscopic). The authors introduce theoretical, mathematical concepts through concrete examples, making the material accessible to non-mathematicians. They also include some of the many open questions in the field to encourage further study. Consequently, this is an ideal introduction for students and researchers from other disciplines who are approaching suspension dynamics for the first time.

Multiscale models in social applications combine mean-field and kinetic equations with either microscopic or macroscopic level descriptions. In this book the reader will find not only a wide spectrum of multiscale analysis results (like convergence proofs), but also practically important information such as derivations of mean-field equations, methods to handle hard contacts numerically, to model group behavior, to quantitative estimate microscopic/macroscopic segregation of competing species, to quantitative understand the limits of validity of mass-action kinetics for simple reactions.

This book is a new edition of Roederer’s classic Dynamics of Geomagnetically Trapped Radiation, updated and considerably expanded. The main objective is to describe the dynamic properties of magnetically trapped particles in planetary radiation belts and plasmas and explain the physical processes involved from the theoretical point of view. The approach is to examine in detail the orbital and adiabatic motion of individual particles in typical configurations of magnetic and electric fields in the magnetosphere and, from there, derive basic features of the particles’ collective “macroscopic” behavior in general planetary environments. Emphasis is not on the “what” but on the “why” of particle phenomena in near-earth space, providing a solid and clear understanding of the principal basic physical mechanisms and dynamic processes involved. The book will also serve as an introduction to general space plasma physics, with abundant basic examples to illustrate and explain the physical origin of different types of plasma current systems and their self-organizing character via the magnetic field. The ultimate aim is to help both graduate students and interested scientists to successfully face the theoretical and experimental challenges lying ahead in space physics in view of recent and upcoming satellite missions and an expected wealth of data on radiation belts and plasmas.

A pedagogical review of the mathematical modelling in fluid dynamics necessary to understand the motility of most microorganisms on Earth.

This is an introductory book dealing with collective phenomena in many-body systems. A gas of bosons or fermions can show oscillations of various types of density. These are described by different combinations of field variables. Especially delicate is the competition of these variables. In superfluid 3He, for example, the atoms can be attracted to each other by molecular forces, whereas they are repelled from each other at short distance due to a hardcore repulsion. The attraction gives rise to Cooper pairs, and the repulsion is overcome by paramagnon oscillations. The combination is what finally led to the discovery of superfluidity in 3He. In general, the competition between various channels can most efficiently be studied by means of a classical version of the Hubbard-Stratonovich transformation.A gas of electrons is controlled by the interplay of plasma oscillations and pair formation. In a system of rod- or disc-like molecules, liquid crystals are observed with directional orientations that behave in unusual five-fold or seven-fold symmetry patterns. The existence of such a symmetry was postulated in 1975 by the author and K Maki. An aluminium material of this type was later manufactured by Dan Shechtman which won him the 2014 Nobel prize. The last chapter presents some solvable models, one of which was the first to illustrate the existence of broken supersymmetry in nuclei.

The volume covers most of the topics addressed and discussed during the Workshop INdAM "Recent advances in kinetic equations and applications", which took place in Rome (Italy), from November 11th to November 15th, 2019. The volume contains results on kinetic equations for reactive and nonreactive mixtures and on collisional and noncollisional Vlasov equations for plasmas. Some contributions are devoted to the study of phase transition phenomena, kinetic problems with nontrivial boundary conditions and hierarchies of models. The book, addressed to researchers interested in the mathematical and numerical study of kinetic equations, provides an overview of recent advances in the field and future research directions.